
# Paolo: I propose to have shorter & more interpretable strings than those generated by DBLP

@inproceedings{zeng2021parameter,
  title={Is Parameter Learning via Weighted Model Integration Tractable?},
  author={Zeng, Zhe and Morettin, Paolo and Yan, Fanqi and Vergari, Antonio and Passerini, Andrea and Van den Broeck, Guy and others},
  booktitle={4th Workshop on Tractable Probabilistic Modeling},
  year={2021},
  organization={UAI}
}


@inproceedings{wmisurvey,
  author       = {Paolo Morettin and
                  Pedro Zuidberg Dos Martires and
                  Samuel Kolb and
                  Andrea Passerini},
  editor       = {Zhi{-}Hua Zhou},
  title        = {Hybrid Probabilistic Inference with Logical and Algebraic Constraints:
                  a Survey},
  booktitle    = {Proceedings of the Thirtieth International Joint Conference on Artificial
                  Intelligence, {IJCAI} 2021, Virtual Event / Montreal, Canada, 19-27
                  August 2021},
  pages        = {4533--4542},
  publisher    = {ijcai.org},
  year         = {2021},
  url          = {https://doi.org/10.24963/ijcai.2021/617},
  doi          = {10.24963/IJCAI.2021/617},
  timestamp    = {Tue, 15 Oct 2024 16:43:28 +0200},
  biburl       = {https://dblp.org/rec/conf/ijcai/MorettinMKP21.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{cdgm,
  author       = {Mihaela C. Stoian and
                  Salijona Dyrmishi and
                  Maxime Cordy and
                  Thomas Lukasiewicz and
                  Eleonora Giunchiglia},
  title        = {How Realistic Is Your Synthetic Data? Constraining Deep Generative
                  Models for Tabular Data},
  booktitle    = {The Twelfth International Conference on Learning Representations,
                  {ICLR} 2024, Vienna, Austria, May 7-11, 2024},
  publisher    = {OpenReview.net},
  year         = {2024},
  url          = {https://openreview.net/forum?id=tBROYsEz9G},
  timestamp    = {Wed, 07 Aug 2024 17:11:53 +0200},
  biburl       = {https://dblp.org/rec/conf/iclr/StoianDCLG24.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{pishield,
  author       = {Mihaela C. Stoian and
                  Alex Tatomir and
                  Thomas Lukasiewicz and
                  Eleonora Giunchiglia},
  title        = {PiShield: {A} PyTorch Package for Learning with Requirements},
  booktitle    = {Proceedings of the Thirty-Third International Joint Conference on
                  Artificial Intelligence, {IJCAI} 2024, Jeju, South Korea, August 3-9,
                  2024},
  pages        = {8805--8809},
  publisher    = {ijcai.org},
  year         = {2024},
  url          = {https://www.ijcai.org/proceedings/2024/1037},
  timestamp    = {Fri, 18 Oct 2024 20:55:30 +0200},
  biburl       = {https://dblp.org/rec/conf/ijcai/StoianTLG24.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{incal,
  author       = {Samuel Kolb and
                  Stefano Teso and
                  Andrea Passerini and
                  Luc De Raedt},
  editor       = {J{\'{e}}r{\^{o}}me Lang},
  title        = {Learning {SMT(LRA)} Constraints using {SMT} Solvers},
  booktitle    = {Proceedings of the Twenty-Seventh International Joint Conference on
                  Artificial Intelligence, {IJCAI} 2018, July 13-19, 2018, Stockholm,
                  Sweden},
  pages        = {2333--2340},
  publisher    = {ijcai.org},
  year         = {2018},
  url          = {https://doi.org/10.24963/ijcai.2018/323},
  doi          = {10.24963/IJCAI.2018/323},
  timestamp    = {Sun, 06 Oct 2024 21:07:57 +0200},
  biburl       = {https://dblp.org/rec/conf/ijcai/KolbTPR18.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}




@inproceedings{lariat,
  author       = {Paolo Morettin and
                  Samuel Kolb and
                  Stefano Teso and
                  Andrea Passerini},
  title        = {Learning Weighted Model Integration Distributions},
  booktitle    = {The Thirty-Fourth {AAAI} Conference on Artificial Intelligence, {AAAI}
                  2020, The Thirty-Second Innovative Applications of Artificial Intelligence
                  Conference, {IAAI} 2020, The Tenth {AAAI} Symposium on Educational
                  Advances in Artificial Intelligence, {EAAI} 2020, New York, NY, USA,
                  February 7-12, 2020},
  pages        = {5224--5231},
  publisher    = {{AAAI} Press},
  year         = {2020},
  url          = {https://doi.org/10.1609/aaai.v34i04.5967},
  doi          = {10.1609/AAAI.V34I04.5967},
  timestamp    = {Mon, 04 Sep 2023 16:50:26 +0200},
  biburl       = {https://dblp.org/rec/conf/aaai/MorettinKTP20.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{multiplexnet,
  author       = {Nick Hoernle and
                  Rafael{-}Michael Karampatsis and
                  Vaishak Belle and
                  Kobi Gal},
  title        = {MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural
                  Networks},
  booktitle    = {Thirty-Sixth {AAAI} Conference on Artificial Intelligence, {AAAI}
                  2022, Thirty-Fourth Conference on Innovative Applications of Artificial
                  Intelligence, {IAAI} 2022, The Twelveth Symposium on Educational Advances
                  in Artificial Intelligence, {EAAI} 2022 Virtual Event, February 22
                  - March 1, 2022},
  pages        = {5700--5709},
  publisher    = {{AAAI} Press},
  year         = {2022},
  url          = {https://doi.org/10.1609/aaai.v36i5.20512},
  doi          = {10.1609/AAAI.V36I5.20512},
  timestamp    = {Mon, 04 Sep 2023 16:50:24 +0200},
  biburl       = {https://dblp.org/rec/conf/aaai/HoernleKBG22.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@article{nesysurvey,
  author       = {Giuseppe Marra and
                  Sebastijan Dumancic and
                  Robin Manhaeve and
                  Luc De Raedt},
  title        = {From statistical relational to neurosymbolic artificial intelligence:
                  {A} survey},
  journal      = {Artif. Intell.},
  volume       = {328},
  pages        = {104062},
  year         = {2024},
  url          = {https://doi.org/10.1016/j.artint.2023.104062},
  doi          = {10.1016/J.ARTINT.2023.104062},
  timestamp    = {Tue, 07 May 2024 20:26:20 +0200},
  biburl       = {https://dblp.org/rec/journals/ai/MarraDMR24.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{deepsets,
  author       = {Manzil Zaheer and
                  Satwik Kottur and
                  Siamak Ravanbakhsh and
                  Barnab{\'{a}}s P{\'{o}}czos and
                  Ruslan Salakhutdinov and
                  Alexander J. Smola},
  editor       = {Isabelle Guyon and
                  Ulrike von Luxburg and
                  Samy Bengio and
                  Hanna M. Wallach and
                  Rob Fergus and
                  S. V. N. Vishwanathan and
                  Roman Garnett},
  title        = {Deep Sets},
  booktitle    = {Advances in Neural Information Processing Systems 30: Annual Conference
                  on Neural Information Processing Systems 2017, December 4-9, 2017,
                  Long Beach, CA, {USA}},
  pages        = {3391--3401},
  year         = {2017},
  url          = {https://proceedings.neurips.cc/paper/2017/hash/f22e4747da1aa27e363d86d40ff442fe-Abstract.html},
  timestamp    = {Thu, 21 Jan 2021 15:15:21 +0100},
  biburl       = {https://dblp.org/rec/conf/nips/ZaheerKRPSS17.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{deeplattice,
  author       = {Seungil You and
                  David Ding and
                  Kevin Robert Canini and
                  Jan Pfeifer and
                  Maya R. Gupta},
  editor       = {Isabelle Guyon and
                  Ulrike von Luxburg and
                  Samy Bengio and
                  Hanna M. Wallach and
                  Rob Fergus and
                  S. V. N. Vishwanathan and
                  Roman Garnett},
  title        = {Deep Lattice Networks and Partial Monotonic Functions},
  booktitle    = {Advances in Neural Information Processing Systems 30: Annual Conference
                  on Neural Information Processing Systems 2017, December 4-9, 2017,
                  Long Beach, CA, {USA}},
  pages        = {2981--2989},
  year         = {2017},
  url          = {https://proceedings.neurips.cc/paper/2017/hash/464d828b85b0bed98e80ade0a5c43b0f-Abstract.html},
  timestamp    = {Thu, 21 Jan 2021 15:15:21 +0100},
  biburl       = {https://dblp.org/rec/conf/nips/YouDCPG17.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}




@inproceedings{deepsade,
  author       = {Kshitij Goyal and
                  Sebastijan Dumancic and
                  Hendrik Blockeel},
  editor       = {Michael J. Wooldridge and
                  Jennifer G. Dy and
                  Sriraam Natarajan},
  title        = {DeepSaDe: Learning Neural Networks That Guarantee Domain Constraint
                  Satisfaction},
  booktitle    = {Thirty-Eighth {AAAI} Conference on Artificial Intelligence, {AAAI}
                  2024, Thirty-Sixth Conference on Innovative Applications of Artificial
                  Intelligence, {IAAI} 2024, Fourteenth Symposium on Educational Advances
                  in Artificial Intelligence, {EAAI} 2014, February 20-27, 2024, Vancouver,
                  Canada},
  pages        = {12199--12207},
  publisher    = {{AAAI} Press},
  year         = {2024},
  url          = {https://doi.org/10.1609/aaai.v38i11.29109},
  doi          = {10.1609/AAAI.V38I11.29109},
  timestamp    = {Tue, 02 Apr 2024 16:32:09 +0200},
  biburl       = {https://dblp.org/rec/conf/aaai/GoyalDB24.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@article{deepproblog,
  author       = {Robin Manhaeve and
                  Sebastijan Dumancic and
                  Angelika Kimmig and
                  Thomas Demeester and
                  Luc De Raedt},
  title        = {Neural probabilistic logic programming in DeepProbLog},
  journal      = {Artif. Intell.},
  volume       = {298},
  pages        = {103504},
  year         = {2021},
  url          = {https://doi.org/10.1016/j.artint.2021.103504},
  doi          = {10.1016/J.ARTINT.2021.103504},
  timestamp    = {Sun, 02 Oct 2022 15:26:59 +0200},
  biburl       = {https://dblp.org/rec/journals/ai/ManhaeveDKDR21.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{semanticloss1,
  author       = {Jingyi Xu and
                  Zilu Zhang and
                  Tal Friedman and
                  Yitao Liang and
                  Guy Van den Broeck},
  editor       = {Jennifer G. Dy and
                  Andreas Krause},
  title        = {A Semantic Loss Function for Deep Learning with Symbolic Knowledge},
  booktitle    = {Proceedings of the 35th International Conference on Machine Learning,
                  {ICML} 2018, Stockholmsm{\"{a}}ssan, Stockholm, Sweden, July
                  10-15, 2018},
  series       = {Proceedings of Machine Learning Research},
  volume       = {80},
  pages        = {5498--5507},
  publisher    = {{PMLR}},
  year         = {2018},
  url          = {http://proceedings.mlr.press/v80/xu18h.html},
  timestamp    = {Wed, 03 Apr 2019 18:17:30 +0200},
  biburl       = {https://dblp.org/rec/conf/icml/XuZFLB18.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{semanticloss2,
  author       = {Kareem Ahmed and
                  Kai{-}Wei Chang and
                  Guy Van den Broeck},
  editor       = {Alice Oh and
                  Tristan Naumann and
                  Amir Globerson and
                  Kate Saenko and
                  Moritz Hardt and
                  Sergey Levine},
  title        = {A Pseudo-Semantic Loss for Autoregressive Models with Logical Constraints},
  booktitle    = {Advances in Neural Information Processing Systems 36: Annual Conference
                  on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans,
                  LA, USA, December 10 - 16, 2023},
  year         = {2023},
  timestamp    = {Fri, 01 Mar 2024 16:26:20 +0100},
  biburl       = {https://dblp.org/rec/conf/nips/AhmedCB23.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{DBLP:conf/aaai/StewartE17,
  author       = {Russell Stewart and
                  Stefano Ermon},
  editor       = {Satinder Singh and
                  Shaul Markovitch},
  title        = {Label-Free Supervision of Neural Networks with Physics and Domain
                  Knowledge},
  booktitle    = {Proceedings of the Thirty-First {AAAI} Conference on Artificial Intelligence,
                  February 4-9, 2017, San Francisco, California, {USA}},
  pages        = {2576--2582},
  publisher    = {{AAAI} Press},
  year         = {2017},
  url          = {https://doi.org/10.1609/aaai.v31i1.10934},
  doi          = {10.1609/AAAI.V31I1.10934},
  timestamp    = {Mon, 04 Sep 2023 16:50:27 +0200},
  biburl       = {https://dblp.org/rec/conf/aaai/StewartE17.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{dl2,
  author       = {Marc Fischer and
                  Mislav Balunovic and
                  Dana Drachsler{-}Cohen and
                  Timon Gehr and
                  Ce Zhang and
                  Martin T. Vechev},
  editor       = {Kamalika Chaudhuri and
                  Ruslan Salakhutdinov},
  title        = {{DL2:} Training and Querying Neural Networks with Logic},
  booktitle    = {Proceedings of the 36th International Conference on Machine Learning,
                  {ICML} 2019, 9-15 June 2019, Long Beach, California, {USA}},
  series       = {Proceedings of Machine Learning Research},
  volume       = {97},
  pages        = {1931--1941},
  publisher    = {{PMLR}},
  year         = {2019},
  url          = {http://proceedings.mlr.press/v97/fischer19a.html},
  timestamp    = {Thu, 02 Sep 2021 14:46:54 +0200},
  biburl       = {https://dblp.org/rec/conf/icml/FischerBDGZV19.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}






@article{sbr,
  author       = {Michelangelo Diligenti and
                  Marco Gori and
                  Claudio Sacc{\`{a}}},
  title        = {Semantic-based regularization for learning and inference},
  journal      = {Artif. Intell.},
  volume       = {244},
  pages        = {143--165},
  year         = {2017},
  url          = {https://doi.org/10.1016/j.artint.2015.08.011},
  doi          = {10.1016/J.ARTINT.2015.08.011},
  timestamp    = {Sat, 27 May 2017 14:24:42 +0200},
  biburl       = {https://dblp.org/rec/journals/ai/DiligentiGS17.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}






@article{DBLP:journals/corr/abs-2402-04823,
  author       = {Mihaela Catalina Stoian and
                  Salijona Dyrmishi and
                  Maxime Cordy and
                  Thomas Lukasiewicz and
                  Eleonora Giunchiglia},
  title        = {How Realistic Is Your Synthetic Data? Constraining Deep Generative
                  Models for Tabular Data},
  journal      = {CoRR},
  volume       = {abs/2402.04823},
  year         = {2024},
  url          = {https://doi.org/10.48550/arXiv.2402.04823},
  doi          = {10.48550/ARXIV.2402.04823},
  eprinttype    = {arXiv},
  eprint       = {2402.04823},
  timestamp    = {Wed, 14 Feb 2024 16:31:41 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2402-04823.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@incollection{barrettsst21,
  title = {Satisfiability {{Modulo Theories}}},
  booktitle = {Handbook of {{Satisfiability}}},
  edition = {2nd},
  author = {Barrett, Clark W. and Sebastiani, Roberto and Seshia, Sanjit A. and Tinelli, Cesare},
  @editor = {Biere, Armin and Heule, Marijn and van Maaren, Hans and Walsh, Toby},
  year = {2021},
  @series = {Frontiers in {{Artificial Intelligence}} and {{Applications}}},
  volume = {336},
  chapter = {33},
  pages = {1267--1329},
  publisher = {{IOS Press}},
  @doi = {10.3233/FAIA201017}
}

@incollection{barrett2018satisfiability,
  title={Satisfiability modulo theories},
  author={Barrett, Clark and Tinelli, Cesare},
  booktitle={Handbook of Model Checking},
  pages={305--343},
  year={2018},
  publisher={Springer}
}

@inproceedings{barrett2010smt,
  title={The {SMT-LIB} initiative and the rise of {SMT}},
  author={Barrett, Clark and de Moura, Leonardo and Ranise, Silvio and Stump, Aaron and Tinelli, Cesare},
  booktitle={Proceedings of the 6th international conference on Hardware and software: verification and testing},
  pages={3--3},
  year={2010},
  organization={Springer-Verlag}
}

@inproceedings{dantam2016incremental,
  title={Incremental Task and Motion Planning: A Constraint-Based Approach.},
  author={Dantam, Neil T and Kingston, Zachary K and Chaudhuri, Swarat and Kavraki, Lydia E},
  booktitle={Robotics: Science and systems},
  volume={12},
  pages={00052},
  year={2016},
  organization={Ann Arbor, MI, USA}
}

@article{chistikov2017approximate,
  title={Approximate counting in SMT and value estimation for probabilistic programs},
  author={Chistikov, Dmitry and Dimitrova, Rayna and Majumdar, Rupak},
  journal={Acta Informatica},
  volume={54},
  number={8},
  pages={729--764},
  year={2017},
  publisher={Springer}
}

@article{albarghouthifairness,
  title={Fairness as a Program Property},
  author={Albarghouthi, Aws and D’Antoni, Loris and Drews, Samuel and Nori, Aditya}
}

@inproceedings{dehnert2017storm,
  title={A storm is coming: A modern probabilistic model checker},
  author={Dehnert, Christian and Junges, Sebastian and Katoen, Joost-Pieter and Volk, Matthias},
  booktitle={International Conference on Computer Aided Verification},
  pages={592--600},
  year={2017},
  organization={Springer}
}

@inproceedings{barrett2011cvc4,
  title={Cvc4},
  author={Barrett, Clark and Conway, Christopher L and Deters, Morgan and Hadarean, Liana and Jovanovi{\'c}, Dejan and King, Tim and Reynolds, Andrew and Tinelli, Cesare},
  booktitle={International Conference on Computer Aided Verification},
  pages={171--177},
  year={2011},
  organization={Springer}
}


@inproceedings{DBLP:conf/aaai/HoernleKBG22,
  author       = {Nick Hoernle and
                  Rafael{-}Michael Karampatsis and
                  Vaishak Belle and
                  Kobi Gal},
  title        = {MultiplexNet: Towards Fully Satisfied Logical Constraints in Neural
                  Networks},
  booktitle    = {Thirty-Sixth {AAAI} Conference on Artificial Intelligence, {AAAI}
                  2022, Thirty-Fourth Conference on Innovative Applications of Artificial
                  Intelligence, {IAAI} 2022, The Twelveth Symposium on Educational Advances
                  in Artificial Intelligence, {EAAI} 2022 Virtual Event, February 22
                  - March 1, 2022},
  pages        = {5700--5709},
  publisher    = {{AAAI} Press},
  year         = {2022},
  url          = {https://doi.org/10.1609/aaai.v36i5.20512},
  doi          = {10.1609/AAAI.V36I5.20512},
  timestamp    = {Mon, 04 Sep 2023 16:50:24 +0200},
  biburl       = {https://dblp.org/rec/conf/aaai/HoernleKBG22.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/corr/abs-2104-12225,
  author       = {Priya L. Donti and
                  David Rolnick and
                  J. Zico Kolter},
  title        = {{DC3:} {A} learning method for optimization with hard constraints},
  journal      = {CoRR},
  volume       = {abs/2104.12225},
  year         = {2021},
  url          = {https://arxiv.org/abs/2104.12225},
  eprinttype    = {arXiv},
  eprint       = {2104.12225},
  timestamp    = {Mon, 03 May 2021 17:38:30 +0200},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2104-12225.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/corr/abs-1902-01785,
  author       = {Thomas Frerix and
                  Matthias Nie{\ss}ner and
                  Daniel Cremers},
  title        = {Linear Inequality Constraints for Neural Network Activations},
  journal      = {CoRR},
  volume       = {abs/1902.01785},
  year         = {2019},
  url          = {http://arxiv.org/abs/1902.01785},
  eprinttype    = {arXiv},
  eprint       = {1902.01785},
  timestamp    = {Tue, 21 May 2019 18:03:40 +0200},
  biburl       = {https://dblp.org/rec/journals/corr/abs-1902-01785.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/corr/abs-2307-08336,
  author       = {Jesus Tordesillas and
                  Jonathan P. How and
                  Marco Hutter},
  title        = {{RAYEN:} Imposition of Hard Convex Constraints on Neural Networks},
  journal      = {CoRR},
  volume       = {abs/2307.08336},
  year         = {2023},
  url          = {https://doi.org/10.48550/arXiv.2307.08336},
  doi          = {10.48550/ARXIV.2307.08336},
  eprinttype    = {arXiv},
  eprint       = {2307.08336},
  timestamp    = {Tue, 25 Jul 2023 16:04:08 +0200},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2307-08336.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{DBLP:conf/nips/Ahmed0TGCKSB021,
  author       = {Kareem Ahmed and
                  Tao Li and
                  Thy Ton and
                  Quan Guo and
                  Kai{-}Wei Chang and
                  Parisa Kordjamshidi and
                  Vivek Srikumar and
                  Guy Van den Broeck and
                  Sameer Singh},
  editor       = {Douwe Kiela and
                  Marco Ciccone and
                  Barbara Caputo},
  title        = {Pylon: {A} PyTorch Framework for Learning with Constraints},
  booktitle    = {NeurIPS 2021 Competitions and Demonstrations Track, 6-14 December
                  2021, Online},
  series       = {Proceedings of Machine Learning Research},
  volume       = {176},
  pages        = {319--324},
  publisher    = {{PMLR}},
  year         = {2021},
  url          = {https://proceedings.mlr.press/v176/ahmed22a.html},
  timestamp    = {Thu, 11 Aug 2022 16:34:36 +0200},
  biburl       = {https://dblp.org/rec/conf/nips/Ahmed0TGCKSB021.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{DBLP:conf/nips/LiHLY0CM023,
  author       = {Zenan Li and
                  Yunpeng Huang and
                  Zhaoyu Li and
                  Yuan Yao and
                  Jingwei Xu and
                  Taolue Chen and
                  Xiaoxing Ma and
                  Jian Lu},
  editor       = {Alice Oh and
                  Tristan Naumann and
                  Amir Globerson and
                  Kate Saenko and
                  Moritz Hardt and
                  Sergey Levine},
  title        = {Neuro-symbolic Learning Yielding Logical Constraints},
  booktitle    = {Advances in Neural Information Processing Systems 36: Annual Conference
                  on Neural Information Processing Systems 2023, NeurIPS 2023, New Orleans,
                  LA, USA, December 10 - 16, 2023},
  year         = {2023},
  url          = {http://papers.nips.cc/paper\_files/paper/2023/hash/4459c3c143db74ee52afebdf56836375-Abstract-Conference.html},
  timestamp    = {Tue, 19 Mar 2024 18:16:28 +0100},
  biburl       = {https://dblp.org/rec/conf/nips/LiHLY0CM023.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{DBLP:conf/pkdd/GoyalDB22,
  author       = {Kshitij Goyal and
                  Sebastijan Dumancic and
                  Hendrik Blockeel},
  editor       = {Massih{-}Reza Amini and
                  St{\'{e}}phane Canu and
                  Asja Fischer and
                  Tias Guns and
                  Petra Kralj Novak and
                  Grigorios Tsoumakas},
  title        = {SaDe: Learning Models that Provably Satisfy Domain Constraints},
  booktitle    = {Machine Learning and Knowledge Discovery in Databases - European Conference,
                  {ECML} {PKDD} 2022, Grenoble, France, September 19-23, 2022, Proceedings,
                  Part {V}},
  series       = {Lecture Notes in Computer Science},
  volume       = {13717},
  pages        = {410--425},
  publisher    = {Springer},
  year         = {2022},
  url          = {https://doi.org/10.1007/978-3-031-26419-1\_25},
  doi          = {10.1007/978-3-031-26419-1\_25},
  timestamp    = {Sat, 30 Sep 2023 09:54:46 +0200},
  biburl       = {https://dblp.org/rec/conf/pkdd/GoyalDB22.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@inproceedings{DBLP:conf/aaai/GoyalDB24,
  author       = {Kshitij Goyal and
                  Sebastijan Dumancic and
                  Hendrik Blockeel},
  editor       = {Michael J. Wooldridge and
                  Jennifer G. Dy and
                  Sriraam Natarajan},
  title        = {DeepSaDe: Learning Neural Networks That Guarantee Domain Constraint
                  Satisfaction},
  booktitle    = {Thirty-Eighth {AAAI} Conference on Artificial Intelligence, {AAAI}
                  2024, Thirty-Sixth Conference on Innovative Applications of Artificial
                  Intelligence, {IAAI} 2024, Fourteenth Symposium on Educational Advances
                  in Artificial Intelligence, {EAAI} 2014, February 20-27, 2024, Vancouver,
                  Canada},
  pages        = {12199--12207},
  publisher    = {{AAAI} Press},
  year         = {2024},
  url          = {https://doi.org/10.1609/aaai.v38i11.29109},
  doi          = {10.1609/AAAI.V38I11.29109},
  timestamp    = {Tue, 02 Apr 2024 16:32:09 +0200},
  biburl       = {https://dblp.org/rec/conf/aaai/GoyalDB24.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{morettin2024unified,
  title={A Unified Framework for Probabilistic Verification of AI Systems via Weighted Model Integration},
  author={Morettin, Paolo and Passerini, Andrea and Sebastiani, Roberto},
  journal={arXiv preprint arXiv:2402.04892},
  year={2024}
}

@article{chin2020efficient,
  title={An efficient method to integrate polynomials over polytopes and curved solids},
  author={Chin, Eric B and Sukumar, N},
  journal={Computer Aided Geometric Design},
  volume={82},
  pages={101914},
  year={2020},
  publisher={Elsevier}
}


@article{zeng2023collapsed,
  title={Collapsed inference for bayesian deep learning},
  author={Zeng, Zhe and Van den Broeck, Guy},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  pages={78394--78411},
  year={2023}
}


@inproceedings{deepseaproblog,
  author       = {De Smet, Lennert and
                  Pedro Zuidberg Dos Martires and
                  Robin Manhaeve and
                  Giuseppe Marra and
                  Angelika Kimmig and
                  Luc De Raedt},
  editor       = {Robin J. Evans and
                  Ilya Shpitser},
  title        = {Neural probabilistic logic programming in discrete-continuous domains},
  booktitle    = {Uncertainty in Artificial Intelligence, {UAI} 2023, July 31 - 4 August
                  2023, Pittsburgh, PA, {USA}},
  series       = {Proceedings of Machine Learning Research},
  volume       = {216},
  pages        = {529--538},
  publisher    = {{PMLR}},
  year         = {2023},
  url          = {https://proceedings.mlr.press/v216/de-smet23a.html},
  timestamp    = {Mon, 28 Aug 2023 17:23:08 +0200},
  biburl       = {https://dblp.org/rec/conf/uai/SmetMMMKR23.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/natmi/HessDPSB22,
  author       = {Philipp Hess and
                  Markus Dr{\"{u}}ke and
                  Stefan Petri and
                  Felix M. Strnad and
                  Niklas Boers},
  title        = {Physically constrained generative adversarial networks for improving
                  precipitation fields from Earth system models},
  journal      = {Nat. Mac. Intell.},
  volume       = {4},
  number       = {10},
  pages        = {828--839},
  year         = {2022},
  url          = {https://doi.org/10.1038/s42256-022-00540-1},
  doi          = {10.1038/S42256-022-00540-1},
  timestamp    = {Wed, 17 May 2023 21:56:52 +0200},
  biburl       = {https://dblp.org/rec/journals/natmi/HessDPSB22.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inbook{doi:https://doi.org/10.1002/9781119646181.ch20,
author = {Zanna, Laure and Bolton, Thomas},
publisher = {John Wiley \& Sons, Ltd},
isbn = {9781119646181},
title = {Deep Learning of Unresolved Turbulent Ocean Processes in Climate Models},
booktitle = {Deep Learning for the Earth Sciences},
chapter = {20},
pages = {298-306},
doi = {https://doi.org/10.1002/9781119646181.ch20},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/9781119646181.ch20},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781119646181.ch20},
year = {2021},
keywords = {climate models, data-driven parameterizations, deep learning, eddy field, machine learning, physics-driven parameterizations, unresolved ocean processes},
abstract = {Summary Climate models are an approximate representation of the laws of physics describing the evolution of the ocean and atmosphere dynamics. Due to limited computational resources, many ocean processes, which are crucial for the transport of heat and carbon, occur at scales smaller than the grid resolution of climate models. Therefore, we rely on approximations, called parameterizations, to represent these unresolved processes in climate models. Parameterizations, traditionally derived from semi-empirical or idealized theories, are often imperfect and can lead to biases in climate models. Machine learning algorithms, and deep learning (DL) algorithms in particular, could provide an avenue to improve the representation of unresolved processes in ocean models by efficiently extracting information from high-resolution ocean simulations and/or observational data, potentially enhancing the skill of parameterizations in climate models.}
}

@article{DBLP:journals/corr/abs-2207-11786,
  author       = {Paula Harder and
                  Duncan Watson{-}Parris and
                  Philip Stier and
                  Dominik Strassel and
                  Nicolas R. Gauger and
                  Janis Keuper},
  title        = {Physics-Informed Learning of Aerosol Microphysics},
  journal      = {CoRR},
  volume       = {abs/2207.11786},
  year         = {2022},
  url          = {https://doi.org/10.48550/arXiv.2207.11786},
  doi          = {10.48550/ARXIV.2207.11786},
  eprinttype    = {arXiv},
  eprint       = {2207.11786},
  timestamp    = {Sun, 12 Nov 2023 02:17:21 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2207-11786.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{Harder2022HardConstrainedDL,
  title={Hard-Constrained Deep Learning for Climate Downscaling},
  author={P. Harder and Venkatesh Ramesh and Alex Hern{\'a}ndez-Garc{\'i}a and Qidong Yang and Prasanna Sattigeri and Daniel Szwarcman and Campbell D. Watson and David Rolnick},
  year={2022},
  url={https://api.semanticscholar.org/CorpusID:256664860}
}

@article{Beucler2019EnforcingAC,
  title={Enforcing Analytic Constraints in Neural Networks Emulating Physical Systems.},
  author={Tom Beucler and Michael S. Pritchard and Stephan Rasp and Jordan Ott and Pierre Baldi and Pierre Gentine},
  journal={Physical review letters},
  year={2019},
  volume={126 9},
  pages={
          098302
        },
  url={https://api.semanticscholar.org/CorpusID:202236953}
}

@article{DBLP:journals/sj/PanCZL23,
  author       = {Xiang Pan and
                  Minghua Chen and
                  Tianyu Zhao and
                  Steven H. Low},
  title        = {DeepOPF: {A} Feasibility-Optimized Deep Neural Network Approach for
                  {AC} Optimal Power Flow Problems},
  journal      = {{IEEE} Syst. J.},
  volume       = {17},
  number       = {1},
  pages        = {673--683},
  year         = {2023},
  url          = {https://doi.org/10.1109/JSYST.2022.3201041},
  doi          = {10.1109/JSYST.2022.3201041},
  timestamp    = {Sat, 11 Mar 2023 00:13:35 +0100},
  biburl       = {https://dblp.org/rec/journals/sj/PanCZL23.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{DBLP:conf/iclr/ZhaoPCL23,
  author       = {Tianyu Zhao and
                  Xiang Pan and
                  Minghua Chen and
                  Steven H. Low},
  title        = {Ensuring {DNN} Solution Feasibility for Optimization Problems with
                  Linear Constraints},
  booktitle    = {The Eleventh International Conference on Learning Representations,
                  {ICLR} 2023, Kigali, Rwanda, May 1-5, 2023},
  publisher    = {OpenReview.net},
  year         = {2023},
  url          = {https://openreview.net/pdf?id=QVcDQJdFTG},
  timestamp    = {Wed, 16 Aug 2023 15:18:20 +0200},
  biburl       = {https://dblp.org/rec/conf/iclr/ZhaoPCL23.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{Kondo2024CGDCD,
  title={CGD: Constraint-Guided Diffusion Policies for UAV Trajectory Planning},
  author={Kota Kondo and Andrea Tagliabue and Xiaoyi Cai and Claudius T. Tewari and Olivia Garcia and Marcos Espitia-Alvarez and Jonathan P. How},
  year={2024},
  url={https://api.semanticscholar.org/CorpusID:269587884}
}

@article{DBLP:journals/corr/abs-1910-01213,
  author       = {Ahmed S. Zamzam and
                  Kyri Baker},
  title        = {Learning Optimal Solutions for Extremely Fast {AC} Optimal Power Flow},
  journal      = {CoRR},
  volume       = {abs/1910.01213},
  year         = {2019},
  url          = {http://arxiv.org/abs/1910.01213},
  eprinttype    = {arXiv},
  eprint       = {1910.01213},
  timestamp    = {Fri, 15 Jan 2021 13:33:15 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/abs-1910-01213.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}


@inproceedings{algebraicgibbs,
  author       = {Hadi Mohasel Afshar and
                  Scott Sanner and
                  Christfried Webers},
  editor       = {Dale Schuurmans and
                  Michael P. Wellman},
  title        = {Closed-Form Gibbs Sampling for Graphical Models with Algebraic Constraints},
  booktitle    = {Proceedings of the Thirtieth {AAAI} Conference on Artificial Intelligence,
                  February 12-17, 2016, Phoenix, Arizona, {USA}},
  pages        = {3287--3293},
  publisher    = {{AAAI} Press},
  year         = {2016},
  url          = {https://doi.org/10.1609/aaai.v30i1.10419},
  doi          = {10.1609/AAAI.V30I1.10419},
  timestamp    = {Mon, 04 Sep 2023 16:50:22 +0200},
  biburl       = {https://dblp.org/rec/conf/aaai/AfsharSW16.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@misc{cowtan2019climate,
  title={The Climate Data Guide: Global surface temperatures: BEST: Berkeley Earth Surface Temperatures},
  author={Cowtan, K and National Center for Atmospheric Research Staff},
  year={2019}
}



@inproceedings{wmireduction,
  author       = {Zhe Zeng and
                  Guy {Van den Broeck}},
  editor       = {Amir Globerson and
                  Ricardo Silva},
  title        = {Efficient Search-Based Weighted Model Integration},
  booktitle    = {Proceedings of the Thirty-Fifth Conference on Uncertainty in Artificial
                  Intelligence, {UAI} 2019, Tel Aviv, Israel, July 22-25, 2019},
  series       = {Proceedings of Machine Learning Research},
  volume       = {115},
  pages        = {175--185},
  publisher    = {{AUAI} Press},
  year         = {2019},
  url          = {http://proceedings.mlr.press/v115/zeng20a.html},
  timestamp    = {Tue, 15 Dec 2020 17:40:18 +0100},
  biburl       = {https://dblp.org/rec/conf/uai/ZengB19.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{loconte2024subtractive,
title={Subtractive Mixture Models via Squaring: Representation and Learning},
author={Loconte, Lorenzo and Aleksanteri, M. Sladek and Mengel, Stefan and Trapp, Martin and Solin, Arno and Gillis, Nicolas and Vergari, Antonio},
booktitle={The Twelfth International Conference on Learning Representations},
year={2024},
url={https://openreview.net/forum?id=xIHi5nxu9P}}

@inproceedings{loconte2025faster,
title={On Faster Marginalization with Squared Circuits via Orthonormalization},
author={Lorenzo Loconte and Antonio Vergari},
year={2025},
eprint={2412.07883},
url={https://openreview.net/forum?id=WTRBDY9m3n},
booktitle={CoLoRAI - The AAAI25 Workshop on Connecting Low-rank Representations in AI}
}

@article{raissi2019physics,
  title={Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
  author={Raissi, Maziar and Perdikaris, Paris and Karniadakis, George E},
  journal={Journal of Computational physics},
  volume={378},
  pages={686--707},
  year={2019},
  publisher={Elsevier}
}


@article{kroese2012monte,
  title={Monte carlo methods},
  author={Kroese, Dirk P and Rubinstein, Reuven Y},
  journal={Wiley Interdisciplinary Reviews: Computational Statistics},
  volume={4},
  number={1},
  pages={48--58},
  year={2012},
  publisher={Wiley Online Library}
}

@article{abboud2022approximate,
  title={Approximate weighted model integration on DNF structures},
  author={Abboud, Ralph and Ceylan, {\.I}smail {\.I}lkan and Dimitrov, Radoslav},
  journal={Artificial Intelligence},
  volume={311},
  pages={103753},
  year={2022},
  publisher={Elsevier}
}



@article{li2020fourier,
  title={Fourier neural operator for parametric partial differential equations},
  author={Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
  journal={arXiv preprint arXiv:2010.08895},
  year={2020}
}

@inproceedings{beltran2024galerkin,
  title={Galerkin meets Laplace: Fast uncertainty estimation in neural PDEs},
  author={Beltran, Christian Jimenez and Vergari, Antonio and Teckentrup, Aretha L and Zygalakis, Konstantinos C},
  booktitle={ICLR 2024 Workshop on AI4DifferentialEquations In Science},
  year={2024}
}


@article{sympy,
     title = {SymPy: symbolic computing in Python},
     author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony},
     year = 2017,
     month = jan,
     keywords = {Python, Computer algebra system, Symbolics},
     abstract = {
                SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
             },
     volume = 3,
     pages = {e103},
     journal = {PeerJ Computer Science},
     issn = {2376-5992},
     url = {https://doi.org/10.7717/peerj-cs.103},
     doi = {10.7717/peerj-cs.103}
    }


@inproceedings{chengmultilinear,
  title={Multilinear Operator Networks},
  author={Cheng, Yixin and Chrysos, Grigorios and Georgopoulos, Markos and Cevher, Volkan},
  booktitle={The Twelfth International Conference on Learning Representations},
  year={2024}
}


@article{loconte2025relationshiptensorfactorizationscircuits, title={What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?}, author={Lorenzo Loconte and Antonio Mari and Gennaro Gala and Robert Peharz and Cassio de Campos and Erik Quaeghebeur and Gennaro Vessio and Antonio Vergari}, 
year={2025},
journal={Transactions of Machine Learning Research},
url={https://openreview.net/forum?id=Y7dRmpGiHj}, 
}


@inproceedings{loconte2024sos,
title={Sum of Squares Circuits},
author={Lorenzo Loconte and Stefan Mengel and Antonio Vergari},
year={2025},
booktitle={The 39th Annual AAAI Conference on Artificial Intelligence}}

@inproceedings{DBLP:conf/ijcai/KolbMSBK18,
  author       = {Samuel Kolb and
                  Martin Mladenov and
                  Scott Sanner and
                  Vaishak Belle and
                  Kristian Kersting},
  editor       = {J{\'{e}}r{\^{o}}me Lang},
  title        = {Efficient Symbolic Integration for Probabilistic Inference},
  booktitle    = {Proceedings of the Twenty-Seventh International Joint Conference on
                  Artificial Intelligence, {IJCAI} 2018, July 13-19, 2018, Stockholm,
                  Sweden},
  pages        = {5031--5037},
  publisher    = {ijcai.org},
  year         = {2018},
  url          = {https://doi.org/10.24963/ijcai.2018/698},
  doi          = {10.24963/IJCAI.2018/698},
  timestamp    = {Tue, 20 Aug 2019 16:19:08 +0200},
  biburl       = {https://dblp.org/rec/conf/ijcai/KolbMSBK18.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{Defazio2024TheRL,
  title={The Road Less Scheduled},
  author={Aaron Defazio and Xingyu Yang and Harsh Mehta and Konstantin Mishchenko and Ahmed Khaled and Ashok Cutkosky},
  journal={ArXiv},
  year={2024},
  volume={abs/2405.15682},
  url={https://api.semanticscholar.org/CorpusID:270045085}
}

@inproceedings{DBLP:journals/corr/KingmaB14,
  author       = {Diederik P. Kingma and
                  Jimmy Ba},
  editor       = {Yoshua Bengio and
                  Yann LeCun},
  title        = {Adam: {A} Method for Stochastic Optimization},
  booktitle    = {3rd International Conference on Learning Representations, {ICLR} 2015,
                  San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings},
  year         = {2015},
  url          = {http://arxiv.org/abs/1412.6980},
  timestamp    = {Thu, 25 Jul 2019 14:25:37 +0200},
  biburl       = {https://dblp.org/rec/journals/corr/KingmaB14.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{wmipa,
  author       = {Paolo Morettin and
                  Andrea Passerini and
                  Roberto Sebastiani},
  editor       = {Carles Sierra},
  title        = {Efficient Weighted Model Integration via SMT-Based Predicate Abstraction},
  booktitle    = {Proceedings of the Twenty-Sixth International Joint Conference on
                  Artificial Intelligence, {IJCAI} 2017, Melbourne, Australia, August
                  19-25, 2017},
  pages        = {720--728},
  publisher    = {ijcai.org},
  year         = {2017},
  url          = {https://doi.org/10.24963/ijcai.2017/100},
  doi          = {10.24963/IJCAI.2017/100},
  timestamp    = {Fri, 09 Apr 2021 18:53:48 +0200},
  biburl       = {https://dblp.org/rec/conf/ijcai/MorettinPS17.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@misc{carlini2011solutionwaringsproblemmonomials,
      title={The Solution to Waring's Problem for Monomials}, 
      author={Enrico Carlini and Maria Virginia Catalisano and Anthony V. Geramita},
      year={2011},
      eprint={1110.0745},
      archivePrefix={arXiv},
      primaryClass={math.AC},
      url={https://arxiv.org/abs/1110.0745}, 
}

@article{Lasserre2019SimpleFF,
  title={Simple formula for integration of polynomials on a simplex},
  author={Jean Bernard Lasserre},
  journal={BIT Numerical Mathematics},
  year={2019},
  volume={61},
  pages={523 - 533},
  url={https://api.semanticscholar.org/CorpusID:201070379}
}

@article{Loera2011SoftwareFE,
  title={Software for exact integration of polynomials over polyhedra},
  author={Jes{\'u}s A. De Loera and Brandon E. Dutra and Matthias K{\"o}ppe and Stanislav Moreinis and Gregory Pinto and Jianqiu Wu},
  journal={ACM Commun. Comput. Algebra},
  year={2011},
  volume={45},
  pages={169-172},
  url={https://api.semanticscholar.org/CorpusID:6247343}
}

@article{latte,
  title={A user’s guide for LattE integrale v1. 7.2},
  author={Baldoni, Velleda and Berline, Nicole and De Loera, Jes{\'u}s A and Dutra, Brandon and K{\"o}ppe, Matthias and Moreinis, Stanislav and Pinto, Gregory and Vergne, Michele and Wu, Jianqiu},
  journal={Optimization},
  volume={22},
  number={2},
  year={2014}
}


@misc{Baldoni_Berline_Vergne, url={https://www.cmls.polytechnique.fr/perso/berline/Integral_over_simplex_users-guide.pdf}, journal={Integral of a monomial over a simplex.}, author={Baldoni, Velleda and Berline, Nicole and Vergne, Michele}} 

@article{chin2015numerical,
  title={Numerical integration of homogeneous functions on convex and nonconvex polygons and polyhedra},
  author={Chin, Eric B and Lasserre, Jean B and Sukumar, N},
  journal={Computational Mechanics},
  volume={56},
  pages={967--981},
  year={2015},
  publisher={Springer}
}

@article{lasserre1998integration,
  title={Integration on a convex polytope},
  author={Lasserre, Jean},
  journal={Proceedings of the American Mathematical Society},
  volume={126},
  number={8},
  pages={2433--2441},
  year={1998}
}

@article{LIN2020125140,
title = {A new recursive formula for integration of polynomial over simplex},
journal = {Applied Mathematics and Computation},
volume = {376},
pages = {125140},
year = {2020},
issn = {0096-3003},
doi = {https://doi.org/10.1016/j.amc.2020.125140},
url = {https://www.sciencedirect.com/science/article/pii/S0096300320301090},
author = {ShaoZhong Lin and ZhiQiang Xie},
keywords = {Analytical integration, Simplex integration, Recursive formula, Polynomials, Polyhedrons, Special operations of matrices},
abstract = {The simplex integration is a convenient method for the integration over complex domains. It automatically represents the ordinary integral over an arbitrary polyhedron as the algebraic sum of integrals over the oriented simplexes. An existing recursive formula for the integration of monomials over simplex, which was deduced based on special operations of matrices and was presented by the first author of this paper, has significant advantages: not only the computation amount is small, but also the integrals of all the lower order monomials are obtained while computing the integral of the highest order monomial. The extension to a polynomial can be obtained by the linearity of integrals. In this paper, the derivation of the existing recursive formula is detailed. A new recursive formula is emphatically proposed to simplify the existing recursive formula and further increase the speed of computation. The code for computer implementation is also presented. Examples show that the accuracy and efficiency of the recursive formula are higher than numerical integration.}
}

@article{Lasserre2001MultiDimVersion,
 ISSN = {00029890, 19300972},
 URL = {http://www.jstor.org/stable/2695528},
 author = {Jean B. Lasserre and Konstantin E. Avrachenkov},
 journal = {The American Mathematical Monthly},
 number = {2},
 pages = {151--154},
 publisher = {Mathematical Association of America},
 title = {The Multi-Dimensional Version of integral over a to b x**p dx},
 urldate = {2024-09-10},
 volume = {108},
 year = {2001}
}







@article{doi:10.1137/0715019,
author = {Grundmann, Axel and M\"{o}ller, H. M.},
title = {Invariant Integration Formulas for the n-Simplex by Combinatorial Methods},
journal = {SIAM Journal on Numerical Analysis},
volume = {15},
number = {2},
pages = {282-290},
year = {1978},
doi = {10.1137/0715019},

URL = { 
    
        https://doi.org/10.1137/0715019
    
    

},
eprint = { 
    
        https://doi.org/10.1137/0715019
    
    

}
,
    abstract = { For the n-simplex \$T\_n\$, integration formulas of arbitrary odd degree are derived and the monomial representations of the orthogonal polynomials corresponding to \$T\_n\$ are given. }
}

@article{Baldoni2008HowTI,
  title={How to integrate a polynomial over a simplex},
  author={Velleda Baldoni and Nicole Berline and Jes{\'u}s A. De Loera and Matthias K{\"o}ppe and Mich{\`e}le Vergne},
  journal={Math. Comput.},
  year={2008},
  volume={80},
  pages={297-325},
  url={https://api.semanticscholar.org/CorpusID:569190}
}

@inproceedings{wmi,
  author       = {Vaishak Belle and
                  Andrea Passerini and
                  Guy Van den Broeck},
  editor       = {Qiang Yang and
                  Michael J. Wooldridge},
  title        = {Probabilistic Inference in Hybrid Domains by Weighted Model Integration},
  booktitle    = {Proceedings of the Twenty-Fourth International Joint Conference on
                  Artificial Intelligence, {IJCAI} 2015, Buenos Aires, Argentina, July
                  25-31, 2015},
  pages        = {2770--2776},
  publisher    = {{AAAI} Press},
  year         = {2015},
  url          = {http://ijcai.org/Abstract/15/392},
  timestamp    = {Tue, 20 Aug 2019 16:17:58 +0200},
  biburl       = {https://dblp.org/rec/conf/ijcai/BellePB15.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DELOERA2013232,
title = {Software for exact integration of polynomials over polyhedra},
journal = {Computational Geometry},
volume = {46},
number = {3},
pages = {232-252},
year = {2013},
issn = {0925-7721},
doi = {https://doi.org/10.1016/j.comgeo.2012.09.001},
url = {https://www.sciencedirect.com/science/article/pii/S0925772112001113},
author = {J.A. {De Loera} and B. Dutra and M. Köppe and S. Moreinis and G. Pinto and J. Wu},
keywords = {Exact integration, Volume computation, Integration of polynomials, Valuations, Polyhedral computation},
abstract = {We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed-ups and extensions of the algorithms presented in previous work by some of the authors. We provide a new software implementation and benchmark computations. The computation of integrals of polynomials over polyhedral regions has many applications; here we demonstrate our algorithmic tools solving a challenge from combinatorial voting theory.}
}



@inproceedings{wmisapa,
  author       = {Giuseppe Spallitta and
                  Gabriele Masina and
                  Paolo Morettin and
                  Andrea Passerini and
                  Roberto Sebastiani},
  editor       = {James Cussens and
                  Kun Zhang},
  title        = {SMT-based weighted model integration with structure awareness},
  booktitle    = {Uncertainty in Artificial Intelligence, Proceedings of the Thirty-Eighth
                  Conference on Uncertainty in Artificial Intelligence, {UAI} 2022,
                  1-5 August 2022, Eindhoven, The Netherlands},
  series       = {Proceedings of Machine Learning Research},
  volume       = {180},
  pages        = {1876--1885},
  publisher    = {{PMLR}},
  year         = {2022},
  url          = {https://proceedings.mlr.press/v180/spallitta22a.html},
  timestamp    = {Sat, 15 Oct 2022 12:08:13 +0200},
  biburl       = {https://dblp.org/rec/conf/uai/SpallittaMMPS22.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@article{sae4wmi,
  author       = {Giuseppe Spallitta and
                  Gabriele Masina and
                  Paolo Morettin and
                  Andrea Passerini and
                  Roberto Sebastiani},
  title        = {Enhancing SMT-based Weighted Model Integration by structure awareness},
  journal      = {Artif. Intell.},
  volume       = {328},
  pages        = {104067},
  year         = {2024},
  url          = {https://doi.org/10.1016/j.artint.2024.104067},
  doi          = {10.1016/J.ARTINT.2024.104067},
  timestamp    = {Tue, 07 May 2024 20:26:20 +0200},
  biburl       = {https://dblp.org/rec/journals/ai/SpallittaMMPS24.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@misc{kaibel2002algorithmicproblemspolytopetheory,
      title={Some Algorithmic Problems in Polytope Theory}, 
      author={Volker Kaibel and Marc E. Pfetsch},
      year={2002},
      eprint={math/0202204},
      archivePrefix={arXiv},
      primaryClass={math.CO},
      url={https://arxiv.org/abs/math/0202204}, 
}

@article{doi:10.1137/0217060,
author = {Dyer, M. E. and Frieze, A. M.},
title = {On the Complexity of Computing the Volume of a Polyhedron},
journal = {SIAM Journal on Computing},
volume = {17},
number = {5},
pages = {967-974},
year = {1988},
doi = {10.1137/0217060},

URL = { 
        https://doi.org/10.1137/0217060
},
eprint = { 
        https://doi.org/10.1137/0217060
}
,
    abstract = { We show that computing the volume of a polyhedron given either as a list of facets or as a list of vertices is as hard as computing the permanent of a matrix. }
}

@article{10.1145/235815.235821,
author = {Barber, C. Bradford and Dobkin, David P. and Huhdanpaa, Hannu},
title = {The quickhull algorithm for convex hulls},
year = {1996},
issue_date = {Dec. 1996},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {22},
number = {4},
issn = {0098-3500},
url = {https://doi.org/10.1145/235815.235821},
doi = {10.1145/235815.235821},
abstract = {The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it used less memory. computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serous errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.},
journal = {ACM Trans. Math. Softw.},
month = dec,
pages = {469–483},
numpages = {15},
keywords = {Delaunay triangulation, Voronoi diagram, convex hull, halfspace intersection}
}

@article{DBLP:journals/corr/abs-1912-01703,
  author       = {Adam Paszke and
                  Sam Gross and
                  Francisco Massa and
                  Adam Lerer and
                  James Bradbury and
                  Gregory Chanan and
                  Trevor Killeen and
                  Zeming Lin and
                  Natalia Gimelshein and
                  Luca Antiga and
                  Alban Desmaison and
                  Andreas K{\"{o}}pf and
                  Edward Z. Yang and
                  Zach DeVito and
                  Martin Raison and
                  Alykhan Tejani and
                  Sasank Chilamkurthy and
                  Benoit Steiner and
                  Lu Fang and
                  Junjie Bai and
                  Soumith Chintala},
  title        = {PyTorch: An Imperative Style, High-Performance Deep Learning Library},
  journal      = {CoRR},
  volume       = {abs/1912.01703},
  year         = {2019},
  url          = {http://arxiv.org/abs/1912.01703},
  eprinttype    = {arXiv},
  eprint       = {1912.01703},
  timestamp    = {Tue, 02 Nov 2021 15:18:32 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/abs-1912-01703.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/rjour/ChalkisF21,
  author       = {Apostolos Chalkis and
                  Vissarion Fisikopoulos},
  title        = {volesti: Volume Approximation and Sampling for Convex Polytopes in
                  {R}},
  journal      = {R J.},
  volume       = {13},
  number       = {2},
  pages        = {561},
  year         = {2021},
  url          = {https://doi.org/10.32614/rj-2021-077},
  doi          = {10.32614/RJ-2021-077},
  timestamp    = {Sat, 12 Mar 2022 15:06:28 +0100},
  biburl       = {https://dblp.org/rec/journals/rjour/ChalkisF21.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{10.7717/peerj-cs.103,
     title = {SymPy: symbolic computing in Python},
     author = {Meurer, Aaron and Smith, Christopher P. and Paprocki, Mateusz and \v{C}ert\'{i}k, Ond\v{r}ej and Kirpichev, Sergey B. and Rocklin, Matthew and Kumar, AMiT and Ivanov, Sergiu and Moore, Jason K. and Singh, Sartaj and Rathnayake, Thilina and Vig, Sean and Granger, Brian E. and Muller, Richard P. and Bonazzi, Francesco and Gupta, Harsh and Vats, Shivam and Johansson, Fredrik and Pedregosa, Fabian and Curry, Matthew J. and Terrel, Andy R. and Rou\v{c}ka, \v{S}t\v{e}p\'{a}n and Saboo, Ashutosh and Fernando, Isuru and Kulal, Sumith and Cimrman, Robert and Scopatz, Anthony},
     year = 2017,
     month = jan,
     keywords = {Python, Computer algebra system, Symbolics},
     abstract = {
                SymPy is an open source computer algebra system written in pure Python. It is built with a focus on extensibility and ease of use, through both interactive and programmatic applications. These characteristics have led SymPy to become a popular symbolic library for the scientific Python ecosystem. This paper presents the architecture of SymPy, a description of its features, and a discussion of select submodules. The supplementary material provide additional examples and further outline details of the architecture and features of SymPy.
             },
     volume = 3,
     pages = {e103},
     journal = {PeerJ Computer Science},
     issn = {2376-5992},
     url = {https://doi.org/10.7717/peerj-cs.103},
     doi = {10.7717/peerj-cs.103}
    }

@inproceedings{DBLP:journals/jmlr/GlorotBB11,
  author       = {Xavier Glorot and
                  Antoine Bordes and
                  Yoshua Bengio},
  editor       = {Geoffrey J. Gordon and
                  David B. Dunson and
                  Miroslav Dud{\'{\i}}k},
  title        = {Deep Sparse Rectifier Neural Networks},
  booktitle    = {Proceedings of the Fourteenth International Conference on Artificial
                  Intelligence and Statistics, {AISTATS} 2011, Fort Lauderdale, USA,
                  April 11-13, 2011},
  series       = {{JMLR} Proceedings},
  volume       = {15},
  pages        = {315--323},
  publisher    = {JMLR.org},
  year         = {2011},
  url          = {http://proceedings.mlr.press/v15/glorot11a/glorot11a.pdf},
  timestamp    = {Thu, 14 Oct 2021 09:27:42 +0200},
  biburl       = {https://dblp.org/rec/journals/jmlr/GlorotBB11.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/corr/abs-2405-15682,
  author       = {Aaron Defazio and
                  Xingyu Yang and
                  Harsh Mehta and
                  Konstantin Mishchenko and
                  Ahmed Khaled and
                  Ashok Cutkosky},
  title        = {The Road Less Scheduled},
  journal      = {CoRR},
  volume       = {abs/2405.15682},
  year         = {2024},
  url          = {https://doi.org/10.48550/arXiv.2405.15682},
  doi          = {10.48550/ARXIV.2405.15682},
  eprinttype    = {arXiv},
  eprint       = {2405.15682},
  timestamp    = {Wed, 19 Jun 2024 08:52:57 +0200},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2405-15682.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{DBLP:conf/eccv/RobicquetSAS16,
  author       = {Alexandre Robicquet and
                  Amir Sadeghian and
                  Alexandre Alahi and
                  Silvio Savarese},
  editor       = {Bastian Leibe and
                  Jiri Matas and
                  Nicu Sebe and
                  Max Welling},
  title        = {Learning Social Etiquette: Human Trajectory Understanding In Crowded
                  Scenes},
  booktitle    = {Computer Vision - {ECCV} 2016 - 14th European Conference, Amsterdam,
                  The Netherlands, October 11-14, 2016, Proceedings, Part {VIII}},
  series       = {Lecture Notes in Computer Science},
  volume       = {9912},
  pages        = {549--565},
  publisher    = {Springer},
  year         = {2016},
  url          = {https://doi.org/10.1007/978-3-319-46484-8\_33},
  doi          = {10.1007/978-3-319-46484-8\_33},
  timestamp    = {Thu, 25 Apr 2024 15:20:40 +0200},
  biburl       = {https://dblp.org/rec/conf/eccv/RobicquetSAS16.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/pami/WuLZC23,
  author       = {Yu{-}Huan Wu and
                  Yun Liu and
                  Xin Zhan and
                  Ming{-}Ming Cheng},
  title        = {{P2T:} Pyramid Pooling Transformer for Scene Understanding},
  journal      = {{IEEE} Trans. Pattern Anal. Mach. Intell.},
  volume       = {45},
  number       = {11},
  pages        = {12760--12771},
  year         = {2023},
  url          = {https://doi.org/10.1109/TPAMI.2022.3202765},
  doi          = {10.1109/TPAMI.2022.3202765},
  timestamp    = {Thu, 26 Oct 2023 16:46:27 +0200},
  biburl       = {https://dblp.org/rec/journals/pami/WuLZC23.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/corr/abs-2005-01463,
  author       = {Chiyu Max Jiang and
                  Soheil Esmaeilzadeh and
                  Kamyar Azizzadenesheli and
                  Karthik Kashinath and
                  Mustafa Mustafa and
                  Hamdi A. Tchelepi and
                  Philip Marcus and
                  Prabhat and
                  Anima Anandkumar},
  title        = {MeshfreeFlowNet: {A} Physics-Constrained Deep Continuous Space-Time
                  Super-Resolution Framework},
  journal      = {CoRR},
  volume       = {abs/2005.01463},
  year         = {2020},
  url          = {https://arxiv.org/abs/2005.01463},
  eprinttype    = {arXiv},
  eprint       = {2005.01463},
  timestamp    = {Tue, 30 Jan 2024 08:41:53 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2005-01463.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{DBLP:journals/corr/abs-1711-10561,
  author       = {Maziar Raissi and
                  Paris Perdikaris and
                  George E. Karniadakis},
  title        = {Physics Informed Deep Learning (Part {I):} Data-driven Solutions of
                  Nonlinear Partial Differential Equations},
  journal      = {CoRR},
  volume       = {abs/1711.10561},
  year         = {2017},
  url          = {http://arxiv.org/abs/1711.10561},
  eprinttype    = {arXiv},
  eprint       = {1711.10561},
  timestamp    = {Mon, 13 Aug 2018 16:47:03 +0200},
  biburl       = {https://dblp.org/rec/journals/corr/abs-1711-10561.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{DBLP:conf/nips/KrishnapriyanGZ21,
  author       = {Aditi S. Krishnapriyan and
                  Amir Gholami and
                  Shandian Zhe and
                  Robert M. Kirby and
                  Michael W. Mahoney},
  editor       = {Marc'Aurelio Ranzato and
                  Alina Beygelzimer and
                  Yann N. Dauphin and
                  Percy Liang and
                  Jennifer Wortman Vaughan},
  title        = {Characterizing possible failure modes in physics-informed neural networks},
  booktitle    = {Advances in Neural Information Processing Systems 34: Annual Conference
                  on Neural Information Processing Systems 2021, NeurIPS 2021, December
                  6-14, 2021, virtual},
  pages        = {26548--26560},
  year         = {2021},
  url          = {https://proceedings.neurips.cc/paper/2021/hash/df438e5206f31600e6ae4af72f2725f1-Abstract.html},
  timestamp    = {Tue, 03 May 2022 16:20:49 +0200},
  biburl       = {https://dblp.org/rec/conf/nips/KrishnapriyanGZ21.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{DBLP:conf/icann/GneccoGMS13,
  author       = {Giorgio Gnecco and
                  Marco Gori and
                  Stefano Melacci and
                  Marcello Sanguineti},
  editor       = {Valeri M. Mladenov and
                  Petia D. Koprinkova{-}Hristova and
                  G{\"{u}}nther Palm and
                  Alessandro E. P. Villa and
                  Bruno Appollini and
                  Nikola K. Kasabov},
  title        = {Learning with Hard Constraints},
  booktitle    = {Artificial Neural Networks and Machine Learning - {ICANN} 2013 - 23rd
                  International Conference on Artificial Neural Networks, Sofia, Bulgaria,
                  September 10-13, 2013. Proceedings},
  series       = {Lecture Notes in Computer Science},
  volume       = {8131},
  pages        = {146--153},
  publisher    = {Springer},
  year         = {2013},
  url          = {https://doi.org/10.1007/978-3-642-40728-4\_19},
  doi          = {10.1007/978-3-642-40728-4\_19},
  timestamp    = {Mon, 03 Jan 2022 22:22:30 +0100},
  biburl       = {https://dblp.org/rec/conf/icann/GneccoGMS13.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@book{doi:10.1137/1.9781611977622,
author = {Beck, Amir},
title = {Introduction to Nonlinear Optimization: Theory, Algorithms, and Applications with Python and MATLAB, Second Edition},
publisher = {Society for Industrial and Applied Mathematics},
year = {2023},
doi = {10.1137/1.9781611977622},
address = {Philadelphia, PA},
edition   = {},
URL = {https://epubs.siam.org/doi/abs/10.1137/1.9781611977622},
eprint = {https://epubs.siam.org/doi/pdf/10.1137/1.9781611977622}
}

@inproceedings{DBLP:conf/icml/Liang0L23,
  author       = {Enming Liang and
                  Minghua Chen and
                  Steven H. Low},
  editor       = {Andreas Krause and
                  Emma Brunskill and
                  Kyunghyun Cho and
                  Barbara Engelhardt and
                  Sivan Sabato and
                  Jonathan Scarlett},
  title        = {Low Complexity Homeomorphic Projection to Ensure Neural-Network Solution
                  Feasibility for Optimization over (Non-)Convex Set},
  booktitle    = {International Conference on Machine Learning, {ICML} 2023, 23-29 July
                  2023, Honolulu, Hawaii, {USA}},
  series       = {Proceedings of Machine Learning Research},
  volume       = {202},
  pages        = {20623--20649},
  publisher    = {{PMLR}},
  year         = {2023},
  url          = {https://proceedings.mlr.press/v202/liang23a.html},
  timestamp    = {Mon, 28 Aug 2023 17:23:08 +0200},
  biburl       = {https://dblp.org/rec/conf/icml/Liang0L23.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{sampo,
  author       = {Pedro Zuidberg {Dos Martires} and
                  Anton Dries and
                  Luc De Raedt},
  title        = {Exact and Approximate Weighted Model Integration with Probability
                  Density Functions Using Knowledge Compilation},
  booktitle    = {The Thirty-Third {AAAI} Conference on Artificial Intelligence, {AAAI}
                  2019, The Thirty-First Innovative Applications of Artificial Intelligence
                  Conference, {IAAI} 2019, The Ninth {AAAI} Symposium on Educational
                  Advances in Artificial Intelligence, {EAAI} 2019, Honolulu, Hawaii,
                  USA, January 27 - February 1, 2019},
  pages        = {7825--7833},
  publisher    = {{AAAI} Press},
  year         = {2019},
  url          = {https://doi.org/10.1609/aaai.v33i01.33017825},
  doi          = {10.1609/AAAI.V33I01.33017825},
  timestamp    = {Mon, 04 Sep 2023 12:29:24 +0200},
  biburl       = {https://dblp.org/rec/conf/aaai/MartiresDR19.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{hinton2002training,
  title={Training products of experts by minimizing contrastive divergence},
  author={Hinton, Geoffrey E},
  journal={Neural computation},
  volume={14},
  number={8},
  pages={1771--1800},
  year={2002},
  publisher={MIT Press}
}


@book{Boyd_Vandenberghe_2004_simplex, place={Cambridge}, title={Convex Optimization},
    publisher={Cambridge University Press}, author={Boyd, Stephen and Vandenberghe, Lieven},
    year={2004},
    chapter={2.2.4},
    note={Simplex}
}

@inproceedings{DBLP:journals/jmlr/GlorotB10,
  author       = {Xavier Glorot and
                  Yoshua Bengio},
  editor       = {Yee Whye Teh and
                  D. Mike Titterington},
  title        = {Understanding the difficulty of training deep feedforward neural networks},
  booktitle    = {Proceedings of the Thirteenth International Conference on Artificial
                  Intelligence and Statistics, {AISTATS} 2010, Chia Laguna Resort, Sardinia,
                  Italy, May 13-15, 2010},
  series       = {{JMLR} Proceedings},
  volume       = {9},
  pages        = {249--256},
  publisher    = {JMLR.org},
  year         = {2010},
  url          = {http://proceedings.mlr.press/v9/glorot10a.html},
  timestamp    = {Thu, 14 Oct 2021 09:27:43 +0200},
  biburl       = {https://dblp.org/rec/journals/jmlr/GlorotB10.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{grivas2024taming,
  title={Taming the sigmoid bottleneck: Provably argmaxable sparse multi-label classification},
  author={Grivas, Andreas and Vergari, Antonio and Lopez, Adam},
  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
  volume={38},
  number={11},
  pages={12208--12216},
  year={2024}
}


@article{DBLP:journals/corr/HeZR015,
  author       = {Kaiming He and
                  Xiangyu Zhang and
                  Shaoqing Ren and
                  Jian Sun},
  title        = {Delving Deep into Rectifiers: Surpassing Human-Level Performance on
                  ImageNet Classification},
  journal      = {CoRR},
  volume       = {abs/1502.01852},
  year         = {2015},
  url          = {http://arxiv.org/abs/1502.01852},
  eprinttype    = {arXiv},
  eprint       = {1502.01852},
  timestamp    = {Wed, 25 Jan 2023 11:01:16 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/HeZR015.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{spl,
  title={Semantic probabilistic layers for neuro-symbolic learning},
  author={Ahmed, Kareem and Teso, Stefano and Chang, Kai-Wei and Van den Broeck, Guy and Vergari, Antonio},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={29944--29959},
  year={2022}
}


@article{bishop1994mixture,
  title={Mixture density networks},
  author={Bishop, Christopher M},
  year={1994},
  publisher={Aston University}
}

@INPROCEEDINGS{374138,
  author={Nix, D.A. and Weigend, A.S.},
  booktitle={Proceedings of 1994 IEEE International Conference on Neural Networks (ICNN'94)}, 
  title={Estimating the mean and variance of the target probability distribution}, 
  year={1994},
  volume={1},
  number={},
  pages={55-60 vol.1},
  keywords={Probability distribution;Noise level;Feedforward systems;Computer science;Cognitive science;Computer errors;Measurement uncertainty;Cost function;Equations;Error correction},
  doi={10.1109/ICNN.1994.374138}}

@article{DBLP:journals/corr/abs-1912-00042,
  author       = {Christina Winkler and
                  Daniel E. Worrall and
                  Emiel Hoogeboom and
                  Max Welling},
  title        = {Learning Likelihoods with Conditional Normalizing Flows},
  journal      = {CoRR},
  volume       = {abs/1912.00042},
  year         = {2019},
  url          = {http://arxiv.org/abs/1912.00042},
  eprinttype    = {arXiv},
  eprint       = {1912.00042},
  timestamp    = {Thu, 02 Jan 2020 18:08:18 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/abs-1912-00042.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{10.1007/BF02573985,
author = {Chazelle, Bernard},
title = {An optimal convex hull algorithm in any fixed dimension},
year = {1993},
issue_date = {December  1993},
publisher = {Springer-Verlag},
address = {Berlin, Heidelberg},
volume = {10},
number = {4},
issn = {0179-5376},
url = {https://doi.org/10.1007/BF02573985},
doi = {10.1007/BF02573985},
abstract = {We present a deterministic algorithm for computing the convex hull ofn points inEd in optimalO(n logn+n d/2 ) time. Optimal solutions were previously known only in even dimension and in dimension 3. A by-product of our result is an algorithm for computing the Voronoi diagram ofn points ind-space in optimalO(n logn+n d/2 ) time.},
journal = {Discrete Comput. Geom.},
month = dec,
pages = {377–409},
numpages = {33}
}

@inproceedings{10.1145/777792.777823,
author = {Attali, Dominique and Boissonnat, Jean-Daniel and Lieutier, Andr\'{e}},
title = {Complexity of the delaunay triangulation of points on surfaces the smooth case},
year = {2003},
isbn = {1581136633},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
url = {https://doi.org/10.1145/777792.777823},
doi = {10.1145/777792.777823},
abstract = {It is well known that the complexity of the Delaunay triangulation of N points in R 3, i.e. the number of its faces, can be O (N2). The case of points distributed on a surface is of great practical importance in reverse engineering since most surface reconstruction algorithms first construct the Delaunay triangulation of a set of points measured on a surface.In this paper, we bound the complexity of the Delaunay triangulation of points distributed on generic smooth surfaces of R 3. Under a mild uniform sampling condition, we show that the complexity of the 3D Delaunay triangulation of the points is O(N log N).},
booktitle = {Proceedings of the Nineteenth Annual Symposium on Computational Geometry},
pages = {201–210},
numpages = {10},
keywords = {Delaunay triangulation, Voronoi diagram, complexity, generic surfaces, reconstruction, smooth surfaces},
location = {San Diego, California, USA},
series = {SCG '03}
}

@inproceedings{10.5555/1283383.1283502,
author = {Amenta, Nina and Attali, Dominique and Devillers, Olivier},
title = {Complexity of Delaunay triangulation for points on lower-dimensional polyhedra},
year = {2007},
isbn = {9780898716245},
publisher = {Society for Industrial and Applied Mathematics},
address = {USA},
abstract = {We show that the Delaunay triangulation of a set of points distributed nearly uniformly on a polyhedron (not necessarily convex) of dimension p in d-dimensional space is O(n(d-1)/p). For all 2 ≤ p ≤ d - 1, this improves on the well-known worst-case bound of O(n⌈d/2⌉).},
booktitle = {Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms},
pages = {1106–1113},
numpages = {8},
location = {New Orleans, Louisiana},
series = {SODA '07}
}

@article{liu1989limited,
  title={On the limited memory BFGS method for large scale optimization},
  author={Liu, Dong C and Nocedal, Jorge},
  journal={Mathematical programming},
  volume={45},
  number={1},
  pages={503--528},
  year={1989},
  publisher={Springer}
}

% ============================================================================
% Reasoning Shortcuts
% ============================================================================

@inproceedings{marconato2023neuro,
  title={Neuro Symbolic Continual Learning: Knowledge, Reasoning Shortcuts and Concept Rehearsal},
  author={Marconato, Emanuele and Bontempo, Gianpaolo and Ficarra, Elisa and Calderara, Simone and Passerini, Andrea and Teso, Stefano},
  booktitle={ICML},
  year={2023}
}

@article{cerutti2022handling,
  title={Handling epistemic and aleatory uncertainties in probabilistic circuits},
  author={Cerutti, Federico and Kaplan, Lance M and Kimmig, Angelika and {\c{S}}ensoy, Murat},
  journal={Machine Learning},
  pages={1--43},
  year={2022},
  publisher={Springer}
}

@inproceedings{marconato2023not,
  title={Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts},
  author={Marconato, Emanuele and Teso, Stefano and Vergari, Antonio and Passerini, Andrea},
  booktitle={NeurIPS},
  year={2023}
}

@inproceedings{wang2023learning,
  title={On Learning Latent Models with Multi-Instance Weak Supervision},
  author={Wang, Kaifu and Tsamoura, Efi and Roth, Dan},
  booktitle={NeurIPS},
  year={2023}
}

@article{dai2020abductive,
  title={Abductive knowledge induction from raw data},
  author={Dai, Wang-Zhou and Muggleton, Stephen H},
  journal={arXiv preprint arXiv:2010.03514},
  year={2020}
}

@article{tao2024deciphering, title={Deciphering Raw Data in Neuro-Symbolic Learning with Provable Guarantees}, volume={38}, url={https://ojs.aaai.org/index.php/AAAI/article/view/29455}, DOI={10.1609/aaai.v38i14.29455}, 
number={14}, journal={Proceedings of the AAAI Conference on Artificial Intelligence}, author={Tao, Lue and Huang, Yu-Xuan and Dai, Wang-Zhou and Jiang, Yuan}, year={2024}, month={Mar.}, pages={15310-15318} }

@inproceedings{li2023learning,
  title={Learning with Logical Constraints but without Shortcut Satisfaction},
  author={Li, Zenan and Liu, Zehua and Yao, Yuan and Xu, Jingwei and Chen, Taolue and Ma, Xiaoxing and Jian, L and others},
  booktitle={ICLR},
  year={2023}
}

@inproceedings{wang2023grounding,
  title={Grounding Neural Inference with Satisfiability Modulo Theories},
  author={Wang, Zifan and Vijayakumar, Saranya and Lu, Kaiji and Ganesh, Vijay and Jha, Somesh and Fredrikson, Matt},
  booktitle={Thirty-seventh Conference on Neural Information Processing Systems},
  year={2023}
}


% ============================================================================
% Experiments: Data Sets, Models, Tools
% ============================================================================
 
@article{lecun1998mnist,
  title={The MNIST database of handwritten digits},
  author={LeCun, Yann},
  journal={http://yann. lecun. com/exdb/mnist/},
  year={1998}
}

@inproceedings{johnson2017clevr,
  title={Clevr: A diagnostic dataset for compositional language and elementary visual reasoning},
  author={Johnson, Justin and Hariharan, Bharath and Van Der Maaten, Laurens and Fei-Fei, Li and Lawrence Zitnick, C and Girshick, Ross},
  booktitle={CVPR},
  year={2017}
}

@inproceedings{li2022qlevr,
  title={QLEVR: A Diagnostic Dataset for Quantificational Language and Elementary Visual Reasoning},
  author={Li, Zechen and Søgaard, Anders},
  booktitle={Findings of NAACL},
  year={2022}
}

@article{lecun1995convolutional,
  title={Convolutional networks for images, speech, and time series},
  author={LeCun, Yann and Bengio, Yoshua},
  journal={The Handbook of Brain Theory and Neural Networks},
  year={1995}
}

@manual{blender,
  title = {Blender - a 3D modelling and rendering package},
  author = {{Blender Online Community}},
  organization = {Blender Foundation},
  address = {Stichting Blender Foundation, Amsterdam},
  year = {2018},
  url = {http://www.blender.org},
}


@InProceedings{xu2020boia,
author = {Xu, Yiran and Yang, Xiaoyin and Gong, Lihang and Lin, Hsuan-Chu and Wu, Tz-Ying and Li, Yunsheng and Vasconcelos, Nuno},
title = {Explainable Object-Induced Action Decision for Autonomous Vehicles},
booktitle = {IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)},
month = {June},
year = {2020}}

@INPROCEEDINGS{deng2009imagenet,
  author={Deng, Jia and Dong, Wei and Socher, Richard and Li, Li-Jia and Kai Li and Li Fei-Fei},
  booktitle={2009 IEEE Conference on Computer Vision and Pattern Recognition}, 
  title={ImageNet: A large-scale hierarchical image database}, 
  year={2009},
  volume={},
  number={},
  pages={248-255},
  keywords={Large-scale systems;Image databases;Explosions;Internet;Robustness;Information retrieval;Image retrieval;Multimedia databases;Ontologies;Spine},
  doi={10.1109/CVPR.2009.5206848}
}



@article{sawada2022concept,
  title={Concept bottleneck model with additional unsupervised concepts},
  author={Sawada, Yoshihide and Nakamura, Keigo},
  journal={IEEE Access},
  volume={10},
  pages={41758--41765},
  year={2022},
  publisher={IEEE}
}

@software{LTNtorch,
  author = {Tommaso Carraro},
  title = {{LTNtorch: PyTorch implementation of Logic Tensor Networks}},
  month = {mar},
  year = {2022},
  publisher = {Zenodo},
  version = {1.0.0},
  doi = {10.5281/zenodo.6394282},
  url = {https://doi.org/10.5281/zenodo.6394282}
}

@article{muller_kandisnky_2021,
   title={Kandinsky Patterns},
   volume={300},
   ISSN={0004-3702},
   url={http://dx.doi.org/10.1016/j.artint.2021.103546},
   DOI={10.1016/j.artint.2021.103546},
   journal={Artificial Intelligence},
   publisher={Elsevier BV},
   author={Müller, Heimo and Holzinger, Andreas},
   year={2021},
   month=nov, pages={103546} 
}

@misc{he2015resnet,
      title={Deep Residual Learning for Image Recognition}, 
      author={Kaiming He and Xiangyu Zhang and Shaoqing Ren and Jian Sun},
      year={2015},
      eprint={1512.03385},
      archivePrefix={arXiv},
      primaryClass={cs.CV}
}

% ============================================================================
% Neuro-symbolic Integration
% ============================================================================

@article{garcez2019neural,
  title={Neural-symbolic computing: An effective methodology for principled integration of machine learning and reasoning},
  author={Garcez, A and Gori, M and Lamb, LC and Serafini, L and Spranger, M and Tran, SN},
  journal={Journal of Applied Logics},
  volume={6},
  number={4},
  pages={611--632},
  year={2019}
}

@inproceedings{de2021statistical,
  title={From statistical relational to neural-symbolic artificial intelligence},
  author={De Raedt, Luc and Duman{\v{c}}i{\'c}, Sebastijan and Manhaeve, Robin and Marra, Giuseppe},
  booktitle={Proceedings of the Twenty-Ninth International Conference on International Joint Conferences on Artificial Intelligence},
  pages={4943--4950},
  year={2021}
}

@article{garcez2022neural,
  title={Neural-symbolic learning and reasoning: A survey and interpretation},
  author={Garcez, Artur d’Avila and Bader, Sebastian and Bowman, Howard and Lamb, Luis C and de Penning, Leo and Illuminoo, BV and Poon, Hoifung and Gerson Zaverucha, COPPE},
  journal={Neuro-Symbolic Artificial Intelligence: The State of the Art},
  volume={342},
  pages={1},
  year={2022},
  publisher={IOS Press}
}

@article{dash2022review,
  title={A review of some techniques for inclusion of domain-knowledge into deep neural networks},
  author={Dash, Tirtharaj and Chitlangia, Sharad and Ahuja, Aditya and Srinivasan, Ashwin},
  journal={Scientific Reports},
  volume={12},
  number={1},
  pages={1--15},
  year={2022},
  publisher={Nature Publishing Group}
}

@article{giunchiglia2022deep,
  title={Deep Learning with Logical Constraints},
  author={Giunchiglia, Eleonora and Stoian, Mihaela Catalina and Lukasiewicz, Thomas},
  journal={arXiv preprint arXiv:2205.00523},
  year={2022}
}


% ============================================================================
% Neuro-Symbolic Architectures
% ============================================================================

% GS-MLNs
@article{lippi2009prediction,
  title={Prediction of protein $\beta$-residue contacts by Markov logic networks with grounding-specific weights},
  author={Lippi, Marco and Frasconi, Paolo},
  journal={Bioinformatics},
  year={2009},
  publisher={Oxford University Press}
}

% SBR
@article{diligenti2012bridging,
  title={Bridging logic and kernel machines},
  author={Diligenti, Michelangelo and Gori, Marco and Maggini, Marco and Rigutini, Leonardo},
  journal={Machine learning},
  volume={86},
  number={1},
  pages={57--88},
  year={2012},
  publisher={Springer}
}

% SBR
@article{diligenti2017semantic,
  title={Semantic-based regularization for learning and inference},
  author={Diligenti, Michelangelo and Gori, Marco and Sacca, Claudio},
  journal={Artificial Intelligence},
  year={2017},
  publisher={Elsevier}
}

% NeuPSL
@article{pryor2022neupsl,
  title={Neupsl: Neural probabilistic soft logic},
  author={Pryor, Connor and Dickens, Charles and Augustine, Eriq and Albalak, Alon and Wang, William and Getoor, Lise},
  journal={arXiv preprint arXiv:2205.14268},
  year={2022}
}

% LTNs
@inproceedings{donadello2017logic,
  title={Logic tensor networks for semantic image interpretation},
  author={Donadello, Ivan and Serafini, Luciano and Garcez, Artur D'Avila},
  booktitle={IJCAI},
  year={2017}
}

@inproceedings{hou2019learning,
  title={Learning a unified classifier incrementally via rebalancing},
  author={Hou, Saihui and Pan, Xinyu and Loy, Chen Change and Wang, Zilei and Lin, Dahua},
  booktitle={Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition},
  pages={831--839},
  year={2019}
}

@article{badreddine2022ltn,
	doi = {10.1016/j.artint.2021.103649},
  
	url = {https://doi.org/10.1016%2Fj.artint.2021.103649},
  
	year = 2022,
	month = {feb},
  
	publisher = {Elsevier {BV}
},
  
	volume = {303},
  
	pages = {103649},
  
	author = {Samy Badreddine and Artur d{\textquotesingle}Avila Garcez and Luciano Serafini and Michael Spranger},
  
	title = {Logic Tensor Networks},
  
	journal = {Artificial Intelligence}
}

% SL


@inproceedings{AhmedKLR23,
  author= {Ahmed, Kareem and Chang, Kai-Wei and Van den Broeck, Guy},
  title= {A Pseudo-Semantic Loss for Deep Generative Models with Logical Constraints},
  booktitle= {Knowledge and Logical Reasoning in the Era of Data-driven Learning Workshop},
  month= {July},
  year= {2023}
}

@article{lu2021physics,
  title={Physics-informed neural networks with hard constraints for inverse design},
  author={Lu, Lu and Pestourie, Raphael and Yao, Wenjie and Wang, Zhicheng and Verdugo, Francesc and Johnson, Steven G},
  journal={SIAM Journal on Scientific Computing},
  volume={43},
  number={6},
  pages={B1105--B1132},
  year={2021},
  publisher={SIAM}
}


% DeepProbLog
@article{manhaeve2018deepproblog,
  title={{DeepProbLog: Neural Probabilistic Logic Programming}},
  author={Manhaeve, Robin and Dumancic, Sebastijan and Kimmig, Angelika and Demeester, Thomas and De Raedt, Luc},
  journal={NeurIPS},
  year={2018}
}

% DeepProbLog Journal
@article{manhaeve2021neural,
  title={Neural probabilistic logic programming in DeepProbLog},
  author={Manhaeve, Robin and Duman{\v{c}}i{\'c}, Sebastijan and Kimmig, Angelika and Demeester, Thomas and De Raedt, Luc},
  journal={Artificial Intelligence},
  volume={298},
  pages={103504},
  year={2021},
  publisher={Elsevier}
}

% DL2
@inproceedings{fischer2019dl2,
  title={Dl2: Training and querying neural networks with logic},
  author={Fischer, Marc and Balunovic, Mislav and Drachsler-Cohen, Dana and Gehr, Timon and Zhang, Ce and Vechev, Martin},
  booktitle={International Conference on Machine Learning},
  pages={1931--1941},
  year={2019},
  organization={PMLR}
}

% C-HNN
@article{giunchiglia2020coherent,
  title={Coherent Hierarchical Multi-label Classification Networks},
  author={Giunchiglia, Eleonora and Lukasiewicz, Thomas},
  journal={NeurIPS},
  year={2020}
}

% NeurASP
@inproceedings{yang2020neurasp,
  title={{NeurASP: Embracing neural networks into answer set programming}},
  author={Yang, Zhun and Ishay, Adam and Lee, Joohyung},
  booktitle={IJCAI},
  year={2020}
}

@inproceedings{rocktaschel2016learning,
  title={Learning knowledge base inference with neural theorem provers},
  author={Rockt{\"a}schel, Tim and Riedel, Sebastian},
  booktitle={Proceedings of the 5th workshop on automated knowledge base construction},
  pages={45--50},
  year={2016}
}

% Scallop
@article{huang2021scallop,
  title={Scallop: From Probabilistic Deductive Databases to Scalable Differentiable Reasoning},
  author={Huang, Jiani and Li, Ziyang and Chen, Binghong and Samel, Karan and Naik, Mayur and Song, Le and Si, Xujie},
  journal={NeurIPS},
  year={2021}
}

% DPLA*
@inproceedings{manhaeve2021approximate,
  title={Approximate Inference for Neural Probabilistic Logic Programming},
  author={Manhaeve, Robin and Marra, Giuseppe and De Raedt, Luc},
  booktitle={KR},
  year={2021}
}

% N-MLNs
@inproceedings{marra2021neural,
  title={Neural markov logic networks},
  author={Marra, Giuseppe and Ku{\v{z}}elka, Ond{\v{r}}ej},
  booktitle={Uncertainty in Artificial Intelligence},
  year={2021}
}

@article{giunchiglia2024ccn,
  title={CCN+: A Neuro-symbolic Framework for Deep Learning with Requirements},
  author={Giunchiglia, Eleonora and Tatomir, Alex and Stoian, Mihaela C{\u{a}}t{\u{a}}lina and Lukasiewicz, Thomas},
  journal={International Journal of Approximate Reasoning},
  pages={109124},
  year={2024},
  publisher={Elsevier}
}

% NeSyEnt
@inproceedings{ahmed2022neuro,
  title={Neuro-symbolic entropy regularization},
  author={Ahmed, Kareem and Wang, Eric and Chang, Kai-Wei and Van den Broeck, Guy},
  booktitle={UAI},
  year={2022}
}

% DeepStochLog
@inproceedings{winters2022deepstochlog,
  title={{DeepStochLog: Neural Stochastic Logic Programming}},
  author={Winters, Thomas and Marra, Giuseppe and Manhaeve, Robin and De Raedt, Luc},
  booktitle={AAAI},
  year={2022}
}

% MultiPlexNet
@inproceedings{hoernle2022multiplexnet,
  title={Multiplexnet: Towards fully satisfied logical constraints in neural networks},
  author={Hoernle, Nick and Karampatsis, Rafael Michael and Belle, Vaishak and Gal, Kobi},
  booktitle={AAAI},
  year={2022}
}

% CAN
@article{di2020efficient,
  title={Efficient generation of structured objects with constrained adversarial networks},
  author={Di Liello, Luca and Ardino, Pierfrancesco and Gobbi, Jacopo and Morettin, Paolo and Teso, Stefano and Passerini, Andrea},
  journal={Advances in neural information processing systems},
  volume={33},
  pages={14663--14674},
  year={2020}
}

% VAEL
@article{misino2022vael,
  title={{VAEL: Bridging Variational Autoencoders and Probabilistic Logic Programming}},
  author={Misino, Eleonora and Marra, Giuseppe and Sansone, Emanuele},
  journal={NeurIPS},
  year={2022}
}

% SPL
@inproceedings{ahmed2022semantic,
  title={{Semantic Probabilistic Layers for Neuro-Symbolic Learning}},
  author={Ahmed, Kareem and Teso, Stefano and Chang, Kai-Wei and Van den Broeck, Guy and Vergari, Antonio},
  booktitle={NeurIPS},
  year={2022}
}

% A-NeSI
@article{van2022anesi,
  title={A-NeSI: A Scalable Approximate Method for Probabilistic Neurosymbolic Inference},
  author={van Krieken, Emile and Thanapalasingam, Thiviyan and Tomczak, Jakub M and van Harmelen, Frank and Teije, Annette ten},
  journal={arXiv preprint arXiv:2212.12393},
  year={2022}
}

% Expressiveness
@inproceedings{van2024Independence,
  title={On the Independence Assumption in Neurosymbolic Learning
},
  author={van Krieken, Emile and Minervini, Pasquale and M. Ponti, Edoardo and Vergari, Antonio},
  booktitle={International Conference on Machine Learning},
  year={2024},
  organization={PMLR}
}

% Issues with T-norms
@article{giannini2018convex,
  title={On a convex logic fragment for learning and reasoning},
  author={Giannini, Francesco and Diligenti, Michelangelo and Gori, Marco and Maggini, Marco},
  journal={IEEE Transactions on Fuzzy Systems},
  year={2018}
}

% Issues with T-norms
@article{van2022analyzing,
  title={Analyzing differentiable fuzzy logic operators},
  author={van Krieken, Emile and Acar, Erman and van Harmelen, Frank},
  journal={Artificial Intelligence},
  year={2022},
  publisher={Elsevier}
}

% LTN
@article{badreddine2022logic,
  title={Logic tensor networks},
  author={Badreddine, Samy and Garcez, Artur d'Avila and Serafini, Luciano and Spranger, Michael},
  journal={Artificial Intelligence},
  volume={303},
  pages={103649},
  year={2022},
  publisher={Elsevier}
}

%DSL 
@article{daniele2022deep,
  title={Deep symbolic learning: Discovering symbols and rules from perceptions},
  author={Daniele, Alessandro and Campari, Tommaso and Malhotra, Sagar and Serafini, Luciano},
  journal={arXiv preprint arXiv:2208.11561},
  year={2022}
}

% ROAP
@inproceedings{tang2023perception,
  title={From perception to programs: regularize, overparameterize, and amortize},
  author={Tang, Hao and Ellis, Kevin},
  booktitle={International Conference on Machine Learning},
  pages={33616--33631},
  year={2023},
  organization={PMLR}
}

## ANNEALING

@inproceedings{li2020closed,
  title={Closed loop neural-symbolic learning via integrating neural perception, grammar parsing, and symbolic reasoning},
  author={Li, Qing and Huang, Siyuan and Hong, Yining and Chen, Yixin and Wu, Ying Nian and Zhu, Song-Chun},
  booktitle={International Conference on Machine Learning},
  pages={5884--5894},
  year={2020},
  organization={PMLR}
}

@inproceedings{li2022softened,
  title={Softened Symbol Grounding for Neuro-symbolic Systems},
  author={Li, Zenan and Yao, Yuan and Chen, Taolue and Xu, Jingwei and Cao, Chun and Ma, Xiaoxing and Jian, L and others},
  booktitle={The Eleventh International Conference on Learning Representations},
  year={2022}
}

% LABEL SMOOTHING

@article{muller2019does,
  title={When does label smoothing help?},
  author={M{\"u}ller, Rafael and Kornblith, Simon and Hinton, Geoffrey E},
  journal={Advances in neural information processing systems},
  volume={32},
  year={2019}
}

## OVERCONFIDENCE

@inproceedings{wei2022mitigating,
  title={Mitigating neural network overconfidence with logit normalization},
  author={Wei, Hongxin and Xie, Renchunzi and Cheng, Hao and Feng, Lei and An, Bo and Li, Yixuan},
  booktitle={International Conference on Machine Learning},
  pages={23631--23644},
  year={2022},
  organization={PMLR}
}

% other 

@article{carratino2022mixup,
  title={On mixup regularization},
  author={Carratino, Luigi and Ciss{\'e}, Moustapha and Jenatton, Rodolphe and Vert, Jean-Philippe},
  journal={The Journal of Machine Learning Research},
  volume={23},
  number={1},
  pages={14632--14662},
  year={2022},
  publisher={JMLRORG}
}

% focal loss

@article{mukhoti2020calibrating,
  title={Calibrating deep neural networks using focal loss},
  author={Mukhoti, Jishnu and Kulharia, Viveka and Sanyal, Amartya and Golodetz, Stuart and Torr, Philip and Dokania, Puneet},
  journal={Advances in Neural Information Processing Systems},
  volume={33},
  pages={15288--15299},
  year={2020}
}


% ============================================================================
% Bayesian inference
% ============================================================================

@article{daxberger2021laplace,
  title={Laplace redux-effortless bayesian deep learning},
  author={Daxberger, Erik and Kristiadi, Agustinus and Immer, Alexander and Eschenhagen, Runa and Bauer, Matthias and Hennig, Philipp},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={20089--20103},
  year={2021}
}

@inproceedings{deepesembles2017,
 author = {Lakshminarayanan, Balaji and Pritzel, Alexander and Blundell, Charles},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {I. Guyon and U. Von Luxburg and S. Bengio and H. Wallach and R. Fergus and S. Vishwanathan and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles},
 volume = {30},
 year = {2017}
}

@inproceedings{gal2016dropout,
  title={Dropout as a bayesian approximation: Representing model uncertainty in deep learning},
  author={Gal, Yarin and Ghahramani, Zoubin},
  booktitle={international conference on machine learning},
  pages={1050--1059},
  year={2016},
  organization={PMLR}
}


@inproceedings{
    wild2023rigorous,
    title={A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methods},
    author={Veit David Wild and Sahra Ghalebikesabi and Dino Sejdinovic and Jeremias Knoblauch},
    booktitle={Thirty-seventh Conference on Neural Information Processing Systems},
    year={2023},
    url={https://openreview.net/forum?id=eTHawKFT4h}
}

@article{Jospin_2022,
   title={Hands-On Bayesian Neural Networks—A Tutorial for Deep Learning Users},
   volume={17},
   ISSN={1556-6048},
   url={http://dx.doi.org/10.1109/MCI.2022.3155327},
   DOI={10.1109/mci.2022.3155327},
   number={2},
   journal={IEEE Computational Intelligence Magazine},
   publisher={Institute of Electrical and Electronics Engineers (IEEE)},
   author={Jospin, Laurent Valentin and Laga, Hamid and Boussaid, Farid and Buntine, Wray and Bennamoun, Mohammed},
   year={2022},
   month=may, pages={29–48} }

@article{wilson2020bayesian,
  title={Bayesian deep learning and a probabilistic perspective of generalization},
  author={Wilson, Andrew G and Izmailov, Pavel},
  journal={Advances in neural information processing systems},
  volume={33},
  pages={4697--4708},
  year={2020}
}

% ============================================================================
% Circuits and Knowledge Compilation
% ============================================================================

@article{darwiche2002knowledge,
  title={A knowledge compilation map},
  author={Darwiche, Adnan and Marquis, Pierre},
  journal={Journal of Artificial Intelligence Research},
  volume={17},
  pages={229--264},
  year={2002}
}

@article{de2015probabilistic,
  title={Probabilistic (logic) programming concepts},
  author={De Raedt, Luc and Kimmig, Angelika},
  journal={Machine Learning},
  year={2015}
}

@article{choi2020probabilistic,
  title={Probabilistic Circuits: A Unifying Framework for Tractable Probabilistic Models},
  author={Choi, YooJung and Vergari, Antonio and Van den Broeck, Guy},
  year={2020}
}

@article{vergari2021compositional,
  title={A Compositional Atlas of Tractable Circuit Operations for Probabilistic Inference},
  author={Vergari, Antonio and Choi, YooJung and Liu, Anji and Teso, Stefano and Van den Broeck, Guy},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  year={2021}
}


% ============================================================================
% Concept Semantics
% ============================================================================

@article{marconato2023interpretability,
  title={Interpretability is in the Mind of the Beholder: A Causal Framework for Human-interpretable Representation Learning},
  author={Marconato, Emanuele and Passerini, Andrea and Teso, Stefano},
  journal={Entropy},
  volume={25},
  number={12},
  pages={1574},
  year={2023},
  publisher={Multidisciplinary Digital Publishing Institute}
}


% ============================================================================
% Concept-based Models
% ============================================================================

@inproceedings{alvarez2018towards,
  title={Towards robust interpretability with self-explaining neural networks},
  author={Alvarez-Melis, David and Jaakkola, Tommi S},
  booktitle={NeurIPS},
  year={2018}
}

@article{chen2019looks,
  title={This Looks Like That: Deep Learning for Interpretable Image Recognition},
  author={Chen, Chaofan and Li, Oscar and Tao, Daniel and Barnett, Alina and Rudin, Cynthia and Su, Jonathan K},
  journal={NeurIPS},
  year={2019}
}

@article{chen2020concept,
  title={Concept whitening for interpretable image recognition},
  author={Chen, Zhi and Bei, Yijie and Rudin, Cynthia},
  journal={Nature Machine Intelligence},
  year={2020}
}

@article{marconato2022glancenets,
  title={GlanceNets: Interpretabile, Leak-proof Concept-based Models},
  author={Marconato, Emanuele and Passerini, Andrea and Teso, Stefano},
  journal={NeurIPS},
  year={2022}
}

@article{zarlenga2022concept,
  title={Concept embedding models},
  author={Zarlenga, Mateo Espinosa and Barbiero, Pietro and Ciravegna, Gabriele and Marra, Giuseppe and Giannini, Francesco and Diligenti, Michelangelo and Shams, Zohreh and Precioso, Frederic and Melacci, Stefano and Weller, Adrian and others},
  journal={arXiv preprint arXiv:2209.09056},
  year={2022}
}


% ============================================================================
% Disentanglement
% ============================================================================

@article{bengio2013representation,
  title={Representation learning: A review and new perspectives},
  author={Bengio, Yoshua and Courville, Aaron and Vincent, Pascal},
  journal={IEEE transactions on pattern analysis and machine intelligence},
  volume={35},
  number={8},
  pages={1798--1828},
  year={2013},
  publisher={IEEE}
}

@article{scholkopf2021toward,
  title={Toward causal representation learning},
  author={Sch{\"o}lkopf, Bernhard and Locatello, Francesco and Bauer, Stefan and Ke, Nan Rosemary and Kalchbrenner, Nal and Goyal, Anirudh and Bengio, Yoshua},
  journal={Proceedings of the IEEE},
  year={2021}
}

@inproceedings{lippe2023biscuit,
  title={Biscuit: Causal representation learning from binary interactions},
  author={Lippe, Phillip and Magliacane, Sara and L{\"o}we, Sindy and Asano, Yuki M and Cohen, Taco and Gavves, Efstratios},
  booktitle={Uncertainty in Artificial Intelligence},
  pages={1263--1273},
  year={2023},
  organization={PMLR}
}

@article{liang2023causal,
  title={Causal Component Analysis},
  author={Liang, Wendong and Keki{\'c}, Armin and von K{\"u}gelgen, Julius and Buchholz, Simon and Besserve, Michel and Gresele, Luigi and Sch{\"o}lkopf, Bernhard},
  journal={arXiv preprint arXiv:2305.17225},
  year={2023}
}

@article{gresele2021independent,
  title={Independent mechanism analysis, a new concept?},
  author={Gresele, Luigi and Von K{\"u}gelgen, Julius and Stimper, Vincent and Sch{\"o}lkopf, Bernhard and Besserve, Michel},
  journal={Advances in neural information processing systems},
  volume={34},
  pages={28233--28248},
  year={2021}
}


@inproceedings{suter2019robustly,
  title={Robustly disentangled causal mechanisms: Validating deep representations for interventional robustness},
  author={Suter, Raphael and Miladinovic, Djordje and Sch{\"o}lkopf, Bernhard and Bauer, Stefan},
  booktitle={International Conference on Machine Learning},
  pages={6056--6065},
  year={2019},
  organization={PMLR}
}

@inproceedings{locatello2019challenging,
  title={Challenging common assumptions in the unsupervised learning of disentangled representations},
  author={Locatello, Francesco and Bauer, Stefan and Lucic, Mario and Raetsch, Gunnar and Gelly, Sylvain and Sch{\"o}lkopf, Bernhard and Bachem, Olivier},
  booktitle={ICML},
  year={2019}
}

@article{buchholz2023learning,
  title={Learning Linear Causal Representations from Interventions under General Nonlinear Mixing},
  author={Buchholz, Simon and Rajendran, Goutham and Rosenfeld, Elan and Aragam, Bryon and Sch{\"o}lkopf, Bernhard and Ravikumar, Pradeep},
  journal={arXiv preprint arXiv:2306.02235},
  year={2023}
}

@article{lachapelle2024nonparametric,
  title={Nonparametric Partial Disentanglement via Mechanism Sparsity: Sparse Actions, Interventions and Sparse Temporal Dependencies},
  author={Lachapelle, S{\'e}bastien and L{\'o}pez, Pau Rodr{\'\i}guez and Sharma, Yash and Everett, Katie and Priol, R{\'e}mi Le and Lacoste, Alexandre and Lacoste-Julien, Simon},
  journal={arXiv preprint arXiv:2401.04890},
  year={2024}
}

% ============================================================================
% Causality
% ============================================================================

@book{pearl2009causality,
  title={Causality},
  author={Pearl, Judea},
  year={2009},
  publisher={Cambridge university press}
}

@book{peters2017elements,
  title={Elements of causal inference: foundations and learning algorithms},
  author={Peters, Jonas and Janzing, Dominik and Sch{\"o}lkopf, Bernhard},
  year={2017}
}
 

% ============================================================================
% Misc
% ============================================================================

@article{gruber1995toward,
  title={Toward principles for the design of ontologies used for knowledge sharing?},
  author={Gruber, Thomas R},
  journal={International journal of human-computer studies},
  volume={43},
  number={5-6},
  pages={907--928},
  year={1995},
  publisher={Elsevier}
}

@inproceedings{radford2021learning,
  title={Learning transferable visual models from natural language supervision},
  author={Radford, Alec and Kim, Jong Wook and Hallacy, Chris and Ramesh, Aditya and Goh, Gabriel and Agarwal, Sandhini and Sastry, Girish and Askell, Amanda and Mishkin, Pamela and Clark, Jack and others},
  booktitle={ICML},
  year={2021}
}

@inproceedings{parascandolo2020learning,
  title={Learning explanations that are hard to vary},
  author={Parascandolo, Giambattista and Neitz, Alexander and ORVIETO, ANTONIO and Gresele, Luigi and Sch{\"o}lkopf, Bernhard},
  booktitle={International Conference on Learning Representations},
  year={2020}
}

@inproceedings{marconato2022catastrophic,
  title={Catastrophic Forgetting in Continual Concept Bottleneck Models},
  author={Marconato, Emanuele and Bontempo, Gianpaolo and Teso, Stefano and Ficarra, Elisa and Calderara, Simone and Passerini, Andrea},
  booktitle={Image Analysis and Processing. ICIAP 2022 Workshops: ICIAP International Workshops, Lecce, Italy, May 23--27, 2022, Revised Selected Papers, Part II},
  pages={539--547},
  year={2022},
  organization={Springer}
}

@inproceedings{KingmaB14@adam,
  author    = {Diederik P. Kingma and
               Jimmy Ba},
  editor    = {Yoshua Bengio and
               Yann LeCun},
  title     = {Adam: {A} Method for Stochastic Optimization},
  booktitle = {3rd International Conference on Learning Representations, {ICLR} 2015,
               San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings},
  year      = {2015},
  url       = {http://arxiv.org/abs/1412.6980},
  timestamp = {Thu, 25 Jul 2019 14:25:37 +0200},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@article{paszke2019pytorch,
  title={Pytorch: An imperative style, high-performance deep learning library},
  author={Paszke, Adam and Gross, Sam and Massa, Francisco and Lerer, Adam and Bradbury, James and Chanan, Gregory and Killeen, Trevor and Lin, Zeming and Gimelshein, Natalia and Antiga, Luca and others},
  journal={Advances in neural information processing systems},
  volume={32},
  year={2019}
}

@article{bontempelli2022concept,
  title={Concept-level debugging of part-prototype networks},
  author={Bontempelli, Andrea and Teso, Stefano and Giunchiglia, Fausto and Passerini, Andrea},
  journal={arXiv preprint arXiv:2205.15769},
  year={2022}
}


% ============================================================================
% Misc
% ============================================================================

@inproceedings{xiao2020noise,
  title={Noise or Signal: The Role of Image Backgrounds in Object Recognition},
  author={Xiao, Kai Yuanqing and Engstrom, Logan and Ilyas, Andrew and Madry, Aleksander},
  booktitle={ICLR},
  year={2020}
}


@inproceedings{xie2022neuro,
  title={Neuro-symbolic verification of deep neural networks},
  author={Xie, Xuan and Kersting, Kristian and Neider, Daniel},
  booktitle={IJCAI},
  year={2022}
}

@article{fredrikson2023learning,
  title={Learning Modulo Theories},
  author={Fredrikson, Matt and Lu, Kaiji and Vijayakumar, Saranya and Jha, Somesh and Ganesh, Vijay and Wang, Zifan},
  journal={arXiv preprint arXiv:2301.11435},
  year={2023}
}

@article{hinton1993autoencoders,
  title={Autoencoders, minimum description length and Helmholtz free energy},
  author={Hinton, Geoffrey E and Zemel, Richard},
  journal={Advances in neural information processing systems},
  volume={6},
  year={1993}
}

@inproceedings{kingma2014auto,
  title={Auto-encoding variational bayes},
  author={Kingma, Diederik P and Welling, Max},
  booktitle={International conference on machine learning},
  year={2014}
}

@inproceedings{rezende2014stochastic,
  title={Stochastic backpropagation and approximate inference in deep generative models},
  author={Rezende, Danilo Jimenez and Mohamed, Shakir and Wierstra, Daan},
  booktitle={International conference on machine learning},
  year={2014}
}

@inproceedings{ghosh2020variational,
  title={From Variational to Deterministic Autoencoders},
  author={Ghosh, Partha and Sajjadi, Mehdi SM and Vergari, Antonio and Black, Michael J and Sch{\"o}lkopf, Bernhard},
  booktitle={ICLR},
  year={2020}
}

@book{cover1999elements,
  title={Elements of information theory},
  author={Cover, Thomas M},
  year={1999},
  publisher={John Wiley \& Sons}
}

@article{de2023differentiable,
  title={Differentiable sampling of categorical distributions using the catlog-derivative trick},
  author={De Smet, Lennert and Sansone, Emanuele and Zuidberg Dos Martires, Pedro},
  journal={Advances in Neural Information Processing Systems},
  volume={36},
  year={2023}
}

@article{zeng2020probabilistic,
  title={Probabilistic inference with algebraic constraints: Theoretical limits and practical approximations},
  author={Zeng, Zhe and Morettin, Paolo and Yan, Fanqi and Vergari, Antonio and Van den Broeck, Guy},
  journal={Advances in Neural Information Processing Systems},
  volume={33},
  pages={11564--11575},
  year={2020}
}


@inproceedings{zeng2020scaling,
  title={Scaling up hybrid probabilistic inference with logical and arithmetic constraints via message passing},
  author={Zeng, Zhe and Morettin, Paolo and Yan, Fanqi and Vergari, Antonio and Van den Broeck, Guy},
  booktitle={International Conference on Machine Learning},
  pages={10990--11000},
  year={2020},
  organization={PMLR}
}


@inproceedings{defast,
  title={A Fast Convoluted Story: Scaling Probabilistic Inference for Integer Arithmetics},
  author={De Smet, Lennert and Dos Martires, Pedro Zuidberg},
  booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
  year={2024}
}


@article{giunchiglia2023road,
  title={ROAD-R: The autonomous driving dataset with logical requirements},
  author={Giunchiglia, Eleonora and Stoian, Mihaela C{\u{a}}t{\u{a}}lina and Khan, Salman and Cuzzolin, Fabio and Lukasiewicz, Thomas},
  journal={Machine Learning},
  pages={1--31},
  year={2023},
  publisher={Springer}
}



@article{badue2021self,
  title={Self-driving cars: A survey},
  author={Badue, Claudine and Guidolini, R{\^a}nik and Carneiro, Raphael Vivacqua and Azevedo, Pedro and Cardoso, Vinicius B and Forechi, Avelino and Jesus, Luan and Berriel, Rodrigo and Paixao, Thiago M and Mutz, Filipe and others},
  journal={Expert Systems with Applications},
  volume={165},
  pages={113816},
  year={2021},
  publisher={Elsevier}
}

@inproceedings{maiettini2019weakly,
  title={A weakly supervised strategy for learning object detection on a humanoid robot},
  author={Maiettini, Elisa and Pasquale, Giulia and Tikhanoff, Vadim and Rosasco, Lorenzo and Natale, Lorenzo},
  booktitle={2019 IEEE-RAS 19th International Conference on Humanoid Robots (Humanoids)},
  pages={194--201},
  year={2019},
  organization={IEEE}
}

@book{quinonero2008dataset,
  title={Dataset shift in machine learning},
  author={Quinonero-Candela, Joaquin and Sugiyama, Masashi and Schwaighofer, Anton and Lawrence, Neil D},
  year={2008},
  publisher={{MIT Press}}
}

@article{kim2023probabilistic,
  title={Probabilistic Concept Bottleneck Models},
  author={Kim, Eunji and Jung, Dahuin and Park, Sangha and Kim, Siwon and Yoon, Sungroh},
  journal={arXiv preprint arXiv:2306.01574},
  year={2023}
}

@article{abdar2021review,
  title={A review of uncertainty quantification in deep learning: Techniques, applications and challenges},
  author={Abdar, Moloud and Pourpanah, Farhad and Hussain, Sadiq and Rezazadegan, Dana and Liu, Li and Ghavamzadeh, Mohammad and Fieguth, Paul and Cao, Xiaochun and Khosravi, Abbas and Acharya, U Rajendra and others},
  journal={Information fusion},
  volume={76},
  pages={243--297},
  year={2021},
  publisher={Elsevier}
}

@article{poeta2023concept,
  title={Concept-based Explainable Artificial Intelligence: A Survey},
  author={Poeta, Eleonora and Ciravegna, Gabriele and Pastor, Eliana and Cerquitelli, Tania and Baralis, Elena},
  journal={arXiv preprint arXiv:2312.12936},
  year={2023}
}



@inproceedings{sansone2023learning,
  title={Learning Symbolic Representations Through Joint GEnerative and DIscriminative Training},
  author={Sansone, Emanuele and Manhaeve, Robin},
  booktitle={International Joint Conference on Artificial Intelligence 2023 Workshop on Knowledge-Based Compositional Generalization},
  year={2023}
}

@book{koller2009probabilistic,
  title={Probabilistic graphical models: principles and techniques},
  author={Koller, Daphne and Friedman, Nir},
  year={2009},
  publisher={MIT press}
}

@article{caruana1997multitask,
  title={Multitask learning},
  author={Caruana, Rich},
  journal={Machine learning},
  volume={28},
  pages={41--75},
  year={1997},
  publisher={Springer}
}

@inproceedings{khemakhem2020variational,
  title={{Variational autoencoders and nonlinear ICA: A unifying framework}},
  author={Khemakhem, Ilyes and Kingma, Diederik and Monti, Ricardo and Hyvarinen, Aapo},
  booktitle={International Conference on Artificial Intelligence and Statistics},
  pages={2207--2217},
  year={2020},
  organization={PMLR}
}

# Faster R-CNN

@inproceedings{ren2015fasterrcnn,
 author = {Ren, Shaoqing and He, Kaiming and Girshick, Ross and Sun, Jian},
 booktitle = {Advances in Neural Information Processing Systems},
 editor = {C. Cortes and N. Lawrence and D. Lee and M. Sugiyama and R. Garnett},
 pages = {},
 publisher = {Curran Associates, Inc.},
 title = {Faster R-CNN: Towards Real-Time Object Detection with Region Proposal Networks},
 url = {https://proceedings.neurips.cc/paper_files/paper/2015/file/14bfa6bb14875e45bba028a21ed38046-Paper.pdf},
 volume = {28},
 year = {2015}
}

@book{neal2012bayesian,
  title={Bayesian learning for neural networks},
  author={Neal, Radford M},
  volume={118},
  year={2012},
  publisher={Springer Science \& Business Media}
}

@article{kendall2017uncertainties,
  title={What uncertainties do we need in bayesian deep learning for computer vision?},
  author={Kendall, Alex and Gal, Yarin},
  journal={Advances in neural information processing systems},
  volume={30},
  year={2017}
}

@article{wang2020survey,
  title={A survey on Bayesian deep learning},
  author={Wang, Hao and Yeung, Dit-Yan},
  journal={ACM computing surveys (csur)},
  volume={53},
  number={5},
  pages={1--37},
  year={2020},
  publisher={ACM New York, NY, USA}
}

@book{settles2012active,
  title={Active Learning},
  author={Settles, B.},
  isbn={9781608457250},
  series={Synthesis lectures on artificial intelligence and machine learning},
  url={https://books.google.it/books?id=z7toC3z_QjQC},
  year={2012},
  publisher={Morgan \& Claypool}
}

@inproceedings{skryagin2022neural,
  title={Neural-probabilistic answer set programming},
  author={Skryagin, Arseny and Stammer, Wolfgang and Ochs, Daniel and Dhami, Devendra Singh and Kersting, Kristian},
  booktitle={Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning},
  volume={19},
  number={1},
  pages={463--473},
  year={2022}
}

% DEs are not nec calibrated
@article{rahaman2021uncertainty,
  title={Uncertainty quantification and deep ensembles},
  author={Rahaman, Rahul and Thiery, Alexandre},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={20063--20075},
  year={2021}
}

@article{abe2022deep,
  title={Deep ensembles work, but are they necessary?},
  author={Abe, Taiga and Buchanan, Estefany Kelly and Pleiss, Geoff and Zemel, Richard and Cunningham, John P},
  journal={Advances in Neural Information Processing Systems},
  volume={35},
  pages={33646--33660},
  year={2022}
}

@inproceedings{stammer2021right,
  title={Right for the right concept: Revising neuro-symbolic concepts by interacting with their explanations},
  author={Stammer, Wolfgang and Schramowski, Patrick and Kersting, Kristian},
  booktitle={Proceedings of the IEEE/CVF conference on computer vision and pattern recognition},
  pages={3619--3629},
  year={2021}
}

@article{osawa2019practical,
  title={Practical deep learning with Bayesian principles},
  author={Osawa, Kazuki and Swaroop, Siddharth and Khan, Mohammad Emtiyaz E and Jain, Anirudh and Eschenhagen, Runa and Turner, Richard E and Yokota, Rio},
  journal={Advances in neural information processing systems},
  volume={32},
  year={2019}
}

@article{jospin2022hands,
  title={Hands-on Bayesian neural networks—A tutorial for deep learning users},
  author={Jospin, Laurent Valentin and Laga, Hamid and Boussaid, Farid and Buntine, Wray and Bennamoun, Mohammed},
  journal={IEEE Computational Intelligence Magazine},
  volume={17},
  number={2},
  pages={29--48},
  year={2022},
  publisher={IEEE}
}

@article{marconato2024bears,
  title={{BEARS Make Neuro-Symbolic Models Aware of their Reasoning Shortcuts}},
  author={Marconato, Emanuele and Bortolotti, Samuele and van Krieken, Emile and Vergari, Antonio and Passerini, Andrea and Teso, Stefano},
  journal={Uncertainty in AI},
  year={2024}
}

@article{bortolotti2024benchmark,
  title={A Benchmark Suite for Systematically Evaluating Reasoning Shortcuts}, 
  author={Samuele Bortolotti and Emanuele Marconato and Tommaso Carraro and Paolo Morettin and Emile van Krieken and Antonio Vergari and Stefano Teso and Andrea Passerini},
  year={2024},
  journal={NeurIPS Dataset and Benchmarks},
}

@inproceedings{
wild2023a,
title={A Rigorous Link between Deep Ensembles and (Variational) Bayesian Methods},
author={Veit David Wild and Sahra Ghalebikesabi and Dino Sejdinovic and Jeremias Knoblauch},
booktitle={Thirty-seventh Conference on Neural Information Processing Systems},
year={2023},
url={https://openreview.net/forum?id=eTHawKFT4h}
}

@article{d2021repulsive,
  title={Repulsive deep ensembles are bayesian},
  author={D'Angelo, Francesco and Fortuin, Vincent},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  pages={3451--3465},
  year={2021}
}

@article{SEIDEL1995115,
title = {The upper bound theorem for polytopes: an easy proof of its asymptotic version},
journal = {Computational Geometry},
volume = {5},
number = {2},
pages = {115-116},
year = {1995},
issn = {0925-7721},
doi = {https://doi.org/10.1016/0925-7721(95)00013-Y},
url = {https://www.sciencedirect.com/science/article/pii/092577219500013Y},
author = {Raimund Seidel},
abstract = {Since at least half of the d edges incident to a vertex v of a simple d-polytope P either all point “up” or all point “down,” v must be the unique “bottom” or “top” vertex of a face of P of dimension at least d2. Thus the number of P's vertices is at most twice the number of such high-dimensional faces, which is at most ∑d/2 ≤ i ≤ d(nd−v) = 0(n⌊d/2⌋), if P has n facets. This, in a nutshell, provides a proof of the asymptotic version of the famous upper bound theorem: A convex d-polytope with n facets (or, dually, with n vertices) has O(n[d2]) faces, when d is constant.}
}

@article{baldoni2011integrate,
  title={How to integrate a polynomial over a simplex},
  author={Baldoni, Velleda and Berline, Nicole and De Loera, Jesus and K{\"o}ppe, Matthias and Vergne, Mich{\`e}le},
  journal={Mathematics of Computation},
  volume={80},
  number={273},
  pages={297--325},
  year={2011}
}

@inproceedings{DBLP:conf/nips/DurkanB0P19,
  author       = {Conor Durkan and
                  Artur Bekasov and
                  Iain Murray and
                  George Papamakarios},
  editor       = {Hanna M. Wallach and
                  Hugo Larochelle and
                  Alina Beygelzimer and
                  Florence d'Alch{\'{e}}{-}Buc and
                  Emily B. Fox and
                  Roman Garnett},
  title        = {Neural Spline Flows},
  booktitle    = {Advances in Neural Information Processing Systems 32: Annual Conference
                  on Neural Information Processing Systems 2019, NeurIPS 2019, December
                  8-14, 2019, Vancouver, BC, Canada},
  pages        = {7509--7520},
  year         = {2019},
  url          = {https://proceedings.neurips.cc/paper/2019/hash/7ac71d433f282034e088473244df8c02-Abstract.html},
  timestamp    = {Mon, 16 May 2022 15:41:51 +0200},
  biburl       = {https://dblp.org/rec/conf/nips/DurkanB0P19.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{smith1980practical,
  title={A practical guide to splines (Carl de Boor)},
  author={Smith, Philip W},
  journal={SIAM Review},
  volume={22},
  number={4},
  pages={520},
  year={1980},
  publisher={Society for Industrial and Applied Mathematics}
}

@inproceedings{DBLP:conf/l4dc/PfrommerGZS22,
  author       = {Samuel Pfrommer and
                  Tanmay Gautam and
                  Alec Zhou and
                  Somayeh Sojoudi},
  editor       = {Roya Firoozi and
                  Negar Mehr and
                  Esen Yel and
                  Rika Antonova and
                  Jeannette Bohg and
                  Mac Schwager and
                  Mykel J. Kochenderfer},
  title        = {Safe Reinforcement Learning with Chance-constrained Model Predictive
                  Control},
  booktitle    = {Learning for Dynamics and Control Conference, {L4DC} 2022, 23-24 June
                  2022, Stanford University, Stanford, CA, {USA}},
  series       = {Proceedings of Machine Learning Research},
  volume       = {168},
  pages        = {291--303},
  publisher    = {{PMLR}},
  year         = {2022},
  url          = {https://proceedings.mlr.press/v168/pfrommer22a.html},
  timestamp    = {Fri, 20 May 2022 16:11:24 +0200},
  biburl       = {https://dblp.org/rec/conf/l4dc/PfrommerGZS22.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}



@article{DBLP:journals/corr/abs-2401-03799,
  author       = {Kai Ren and
                  Colin Chen and
                  Hyeontae Sung and
                  Heejin Ahn and
                  Ian M. Mitchell and
                  Maryam Kamgarpour},
  title        = {Safe Chance-constrained Model Predictive Control under Gaussian Mixture
                  Model Uncertainty},
  journal      = {CoRR},
  volume       = {abs/2401.03799},
  year         = {2024},
  url          = {https://doi.org/10.48550/arXiv.2401.03799},
  doi          = {10.48550/ARXIV.2401.03799},
  eprinttype    = {arXiv},
  eprint       = {2401.03799},
  timestamp    = {Wed, 24 Jan 2024 16:19:32 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2401-03799.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}


@article{PhysRevLett.126.098302,
  title = {Enforcing Analytic Constraints in Neural Networks Emulating Physical Systems},
  author = {Beucler, Tom and Pritchard, Michael and Rasp, Stephan and Ott, Jordan and Baldi, Pierre and Gentine, Pierre},
  journal = {Phys. Rev. Lett.},
  volume = {126},
  issue = {9},
  pages = {098302},
  numpages = {7},
  year = {2021},
  month = {Mar},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevLett.126.098302},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.126.098302}
}

@article{DBLP:journals/jmlr/HarderHRYSSWR23,
  author       = {Paula Harder and
                  Alex Hern{\'{a}}ndez{-}Garc{\'{\i}}a and
                  Venkatesh Ramesh and
                  Qidong Yang and
                  Prasanna Sattegeri and
                  Daniela Szwarcman and
                  Campbell D. Watson and
                  David Rolnick},
  title        = {Hard-Constrained Deep Learning for Climate Downscaling},
  journal      = {J. Mach. Learn. Res.},
  volume       = {24},
  pages        = {365:1--365:40},
  year         = {2023},
  url          = {http://jmlr.org/papers/v24/23-0158.html},
  timestamp    = {Thu, 12 Sep 2024 13:29:32 +0200},
  biburl       = {https://dblp.org/rec/journals/jmlr/HarderHRYSSWR23.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}

@article{willard2020integrating,
  title={Integrating physics-based modeling with machine learning: A survey},
  author={Willard, Jared and Jia, Xiaowei and Xu, Shaoming and Steinbach, Michael and Kumar, Vipin},
  journal={arXiv preprint arXiv:2003.04919},
  volume={1},
  number={1},
  pages={1--34},
  year={2020}
}

@inproceedings{beucler2020towards,
  title={Towards physically-consistent, data-driven models of convection},
  author={Beucler, Tom and Pritchard, Michael and Gentine, Pierre and Rasp, Stephan},
  booktitle={Igarss 2020-2020 ieee international geoscience and remote sensing symposium},
  pages={3987--3990},
  year={2020},
  organization={IEEE}
}

@article{DBLP:journals/corr/abs-2502-18237,
  author       = {Mihaela Catalina Stoian and
                  Eleonora Giunchiglia},
  title        = {Beyond the convexity assumption: Realistic tabular data generation
                  under quantifier-free real linear constraints},
  journal      = {CoRR},
  volume       = {abs/2502.18237},
  year         = {2025},
  url          = {https://doi.org/10.48550/arXiv.2502.18237},
  doi          = {10.48550/ARXIV.2502.18237},
  eprinttype    = {arXiv},
  eprint       = {2502.18237},
  timestamp    = {Thu, 20 Mar 2025 21:38:06 +0100},
  biburl       = {https://dblp.org/rec/journals/corr/abs-2502-18237.bib},
  bibsource    = {dblp computer science bibliography, https://dblp.org}
}