@article{doi:10.1126/sciadv.abi8605,
author = {Sifan Wang  and Hanwen Wang  and Paris Perdikaris },
title = {Learning the solution operator of parametric partial differential equations with physics-informed DeepONets},
journal = {Science Advances},
volume = {7},
number = {40},
pages = {eabi8605},
year = {2021},
doi = {10.1126/sciadv.abi8605},
URL = {https://www.science.org/doi/abs/10.1126/sciadv.abi8605},
eprint = {https://www.science.org/doi/pdf/10.1126/sciadv.abi8605},
abstract = {Enabling the rapid emulation of parametric differential equations with physics-informed deep operator networks. Partial differential equations (PDEs) play a central role in the mathematical analysis and modeling of complex dynamic processes across all corners of science and engineering. Their solution often requires laborious analytical or computational tools, associated with a cost that is markedly amplified when different scenarios need to be investigated, for example, corresponding to different initial or boundary conditions, different inputs, etc. In this work, we introduce physics-informed DeepONets, a deep learning framework for learning the solution operator of arbitrary PDEs, even in the absence of any paired input-output training data. We illustrate the effectiveness of the proposed framework in rapidly predicting the solution of various types of parametric PDEs up to three orders of magnitude faster compared to conventional PDE solvers, setting a previously unexplored paradigm for modeling and simulation of nonlinear and nonequilibrium processes in science and engineering.}}

@INPROCEEDINGS{10204903,
  author={Bai, Yikun and Schmitzer, Bernhard and Thorpe, Matthew and Kolouri, Soheil},
  booktitle={2023 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)}, 
  title={Sliced Optimal Partial Transport}, 
  year={2023},
  volume={},
  number={},
  pages={13681-13690},
  keywords={Point cloud compression;Computer vision;Image color analysis;Optimized production technology;Machine learning;Particle measurements;Linear programming;Machine learning (other than deep learning)},
  doi={10.1109/CVPR52729.2023.01315}}

@article{Schiebinger2019,
title = "Optimal-Transport Analysis of Single-Cell Gene Expression Identifies Developmental Trajectories in Reprogramming",
author = "Geoffrey Schiebinger and Jian Shu and Marcin Tabaka and Brian Cleary and Vidya Subramanian and Aryeh Solomon and Joshua Gould and Siyan Liu and Stacie Lin and Peter Berube and Lia Lee and Jenny Chen and Justin Brumbaugh and Philippe Rigollet and Konrad Hochedlinger and Rudolf Jaenisch and Aviv Regev and Lander, {Eric S.}",
year = "2019",
day = "7",
doi = "10.1016/j.cell.2019.01.006",
language = "English",
volume = "176",
pages = "928--943",
journal = "Cell",
issn = "0092-8674",
publisher = "Cell Press",
number = "4",
}

@article{10.1007/s10851-022-01121-y,
author = {Lacombe, Julien and Digne, Julie and Courty, Nicolas and Bonneel, Nicolas},
title = {Learning to Generate Wasserstein Barycenters},
year = {2022},
issue_date = {Apr 2023},
publisher = {Kluwer Academic Publishers},
address = {USA},
volume = {65},
number = {2},
issn = {0924-9907},
url = {https://doi.org/10.1007/s10851-022-01121-y},
doi = {10.1007/s10851-022-01121-y},
abstract = {Optimal transport is a notoriously difficult problem to solve numerically, with current approaches often remaining intractable for very large-scale applications such as those encountered in machine learning. Wasserstein barycenters—the problem of finding measures in-between given input measures in the optimal transport sense—are even more computationally demanding as it requires to solve an optimization problem involving optimal transport distances. By training a deep convolutional neural network, we improve by a factor of 80 the computational speed of Wasserstein barycenters over the fastest state-of-the-art approach on the GPU, resulting in milliseconds computational times on 512\texttimes{}512 regular grids. We show that our network, trained on Wasserstein barycenters of pairs of measures, generalizes well to the problem of finding Wasserstein barycenters of more than two measures. We demonstrate the efficiency of our approach for computing barycenters of sketches and transferring colors between multiple images.},
journal = {J. Math. Imaging Vis.},
month = {oct},
pages = {354–370},
numpages = {17},
keywords = {Color transfer, Convolutional neural network, Optimal transport, Wasserstein barycenter}
}

@ARTICLE{7053911,
  author={Su, Zhengyu and Wang, Yalin and Shi, Rui and Zeng, Wei and Sun, Jian and Luo, Feng and Gu, Xianfeng},
  journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, 
  title={Optimal Mass Transport for Shape Matching and Comparison}, 
  year={2015},
  volume={37},
  number={11},
  pages={2246-2259},
  keywords={Shape;Extraterrestrial measurements;Three-dimensional displays;Conformal mapping;Surface morphology;Space vehicles;optimal mass transport;shape representation;surface matching;shape space;Optimal mass transport;shape representation;surface matching;shape space},
  doi={10.1109/TPAMI.2015.2408346}}

@inproceedings{xu-etal-2021-vocabulary,
    title = "Vocabulary Learning via Optimal Transport for Neural Machine Translation",
    author = "Xu, Jingjing  and
      Zhou, Hao  and
      Gan, Chun  and
      Zheng, Zaixiang  and
      Li, Lei",
    editor = "Zong, Chengqing  and
      Xia, Fei  and
      Li, Wenjie  and
      Navigli, Roberto",
    booktitle = "Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)",
    month = aug,
    year = "2021",
    address = "Online",
    publisher = "Association for Computational Linguistics",
    url = "https://aclanthology.org/2021.acl-long.571",
    doi = "10.18653/v1/2021.acl-long.571",
    pages = "7361--7373",
    abstract = "The choice of token vocabulary affects the performance of machine translation. This paper aims to figure out what is a good vocabulary and whether we can find the optimal vocabulary without trial training. To answer these questions, we first provide an alternative understanding of vocabulary from the perspective of information theory. It motivates us to formulate the quest of vocabularization {--} finding the best token dictionary with a proper size {--} as an optimal transport (OT) problem. We propose VOLT, a simple and efficient solution without trial training. Empirical results show that VOLT beats widely-used vocabularies in diverse scenarios, including WMT-14 English-German translation, TED bilingual translation, and TED multilingual translation. For example, VOLT achieves 70{\%} vocabulary size reduction and 0.5 BLEU gain on English-German translation. Also, compared to BPE-search, VOLT reduces the search time from 384 GPU hours to 30 GPU hours on English-German translation. Codes are available at \url{https://github.com/Jingjing-NLP/VOLT}.",
}

@article{FU2023112346,
	abstract = {Mean-field games (MFGs) have shown strong modeling capabilities for large systems in various fields, driving growth in computational methods for mean-field game problems. However, high order methods have not been thoroughly investigated. In this work, we explore applying general high-order numerical schemes with finite element methods in the space-time domain for computing the optimal transport (OT), mean-field planning (MFP), and potential MFG problems. We conduct several experiments to validate the convergence rate of the high order method numerically. Those numerical experiments also demonstrate the efficiency and effectiveness of our approach.},
	author = {Guosheng Fu and Siting Liu and Stanley Osher and Wuchen Li},
	doi = {https://doi.org/10.1016/j.jcp.2023.112346},
	issn = {0021-9991},
	journal = {Journal of Computational Physics},
	keywords = {High order computation, Optimal transport, Mean-field planning, Mean-field games},
	pages = {112346},
	title = {High order computation of optimal transport, mean field planning, and potential mean field games},
	url = {https://www.sciencedirect.com/science/article/pii/S0021999123004412},
	volume = {491},
	year = {2023},
	bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0021999123004412},
	bdsk-url-2 = {https://doi.org/10.1016/j.jcp.2023.112346}}

@article{Caffarelli_1996,
 ISSN = {0003486X},
 URL = {http://www.jstor.org/stable/2118564},
 author = {Caffarelli, Luis A.},
 journal = {Annals of Mathematics},
 number = {3},
 pages = {453--496},
 publisher = {Annals of Mathematics},
 title = {Boundary Regularity of Maps with Convex Potentials--II},
 urldate = {2023-11-22},
 volume = {144},
 year = {1996}
}

@article{hutter2021minimax,
  title={Minimax estimation of smooth optimal transport maps},
  author={H{\"u}tter, Jan-Christian and Rigollet, Philippe},
  journal={The Annals of Statistics},
  volume={49},
  number={2},
  pages={1166--1194},
  year={2021}
}

@InProceedings{amos2023meta,
      title={Meta Optimal Transport}, 
      author={Brandon Amos and Samuel Cohen and Giulia Luise and Ievgen Redko},
      year={2023},
      booktitle = {ICML}
}

@article{LacombeDigneCourtyBonneel_2023,
	abstract = {Optimal transport is a notoriously difficult problem to solve numerically, with current approaches often remaining intractable for very large-scale applications such as those encountered in machine learning. Wasserstein barycenters---the problem of finding measures in-between given input measures in the optimal transport sense---are even more computationally demanding as it requires to solve an optimization problem involving optimal transport distances. By training a deep convolutional neural network, we improve by a factor of 80 the computational speed of Wasserstein barycenters over the fastest state-of-the-art approach on the GPU, resulting in milliseconds computational times on {\$}{\$}512{$\backslash$}times 512{\$}{\$}regular grids. We show that our network, trained on Wasserstein barycenters of pairs of measures, generalizes well to the problem of finding Wasserstein barycenters of more than two measures. We demonstrate the efficiency of our approach for computing barycenters of sketches and transferring colors between multiple images.},
	author = {Lacombe, Julien and Digne, Julie and Courty, Nicolas and Bonneel, Nicolas},
	date = {2023/04/01},
	date-added = {2023-11-21 23:32:58 -0800},
	date-modified = {2023-11-21 23:32:58 -0800},
	doi = {10.1007/s10851-022-01121-y},
	id = {Lacombe2023},
	isbn = {1573-7683},
	journal = {Journal of Mathematical Imaging and Vision},
	number = {2},
	pages = {354--370},
	title = {Learning to Generate Wasserstein Barycenters},
	url = {https://doi.org/10.1007/s10851-022-01121-y},
	volume = {65},
	year = {2023},
	bdsk-url-1 = {https://doi.org/10.1007/s10851-022-01121-y}}

@misc{tong2023improving,
      title={Improving and generalizing flow-based generative models with minibatch optimal transport}, 
      author={Alexander Tong and Nikolay Malkin and Guillaume Huguet and Yanlei Zhang and Jarrid Rector-Brooks and Kilian Fatras and Guy Wolf and Yoshua Bengio},
      year={2023},
      eprint={2302.00482},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@InProceedings{pooladian2023multisample,
      title={Multisample Flow Matching: Straightening Flows with Minibatch Couplings}, 
      author={Aram-Alexandre Pooladian and Heli Ben-Hamu and Carles Domingo-Enrich and Brandon Amos and Yaron Lipman and Ricky T. Q. Chen},
      year={2023},
      booktitle = {ICML}
}

@InProceedings{LiuGongLiu_2023,
  title = 	 {Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow},
  author =       {Liu, Xingchao and Gong, Chengyue and Liu, Qiang},
  booktitle = 	 {ICLR},
  year = 	 {2023}
}

@article{doi:10.1080/25742558.2020.1750861,
	abstract = {Given sufficiently many components, it is often cited that finite mixture models can approximate any other probability density function (pdf) to an arbitrary degree of accuracy. Unfortunately, the nature of this approximation result is often left unclear. We prove that finite mixture models constructed from pdfs in C0 can be used to conduct approximation of various classes of approximands in a number of different modes. That is, we prove approximands in C0 can be uniformly approximated, approximands in Cb can be uniformly approximated on compact sets, and approximands in Lp can be approximated with respect to the Lp, for p∈1,∞. Furthermore, we also prove that measurable functions can be approximated, almost everywhere. },
	author = {T. Tin Nguyen and Hien D. Nguyen and Faicel Chamroukhi and Geoffrey J. McLachlan},
	doi = {10.1080/25742558.2020.1750861},
	editor = {Lishan Liu},
	eprint = {https://doi.org/10.1080/25742558.2020.1750861},
	journal = {Cogent Mathematics \& Statistics},
	number = {1},
	pages = {1750861},
	publisher = {Cogent OA},
	title = {Approximation by finite mixtures of continuous density functions that vanish at infinity},
	url = {https://doi.org/10.1080/25742558.2020.1750861},
	volume = {7},
	year = {2020},
	bdsk-url-1 = {https://doi.org/10.1080/25742558.2020.1750861}}

@article{Gershman2014AmortizedII,
  title={Amortized Inference in Probabilistic Reasoning},
  author={Gershman, Samuel J. and Goodman, Noah D.},
  journal={Cognitive Science},
  year={2014},
  volume={36}
}

@InProceedings{pmlr-v80-dvurechensky18a,
  title = 	 {Computational Optimal Transport: Complexity by Accelerated Gradient Descent Is Better Than by Sinkhorn’s Algorithm},
  author =       {Dvurechensky, Pavel and Gasnikov, Alexander and Kroshnin, Alexey},
  booktitle = 	 {Proceedings of the 35th International Conference on Machine Learning},
  pages = 	 {1367--1376},
  year = 	 {2018},
  editor = 	 {Dy, Jennifer and Krause, Andreas},
  volume = 	 {80},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {10--15 Jul},
  publisher =    {PMLR}
}

@book{LuenbergerYe2015,
author = {Luenberger, David G. and Ye, Yinyu},
title = {Linear and Nonlinear Programming},
year = {2015},
isbn = {3319188410},
publisher = {Springer Publishing Company, Incorporated}
}

@article{BertsekaCastanon1989s,
title = "The auction algorithm for the transportation problem",
author = "Bertsekas, {Dimitri P.} and Castanon, {David A.}",
year = "1989",
month = dec,
doi = "10.1007/BF02216923",
language = "English (US)",
volume = "20",
pages = "67--96",
journal = "Annals of Operations Research",
issn = "0254-5330",
publisher = "Springer Netherlands",
number = "1",
}

@article{Kuhn1955,
author = {Kuhn, H. W.},
title = {The Hungarian method for the assignment problem},
journal = {Naval Research Logistics Quarterly},
volume = {2},
number = {1-2},
pages = {83-97},
doi = {https://doi.org/10.1002/nav.3800020109},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nav.3800020109},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nav.3800020109},
abstract = {Abstract Assuming that numerical scores are available for the performance of each of n persons on each of n jobs, the “assignment problem” is the quest for an assignment of persons to jobs so that the sum of the n scores so obtained is as large as possible. It is shown that ideas latent in the work of two Hungarian mathematicians may be exploited to yield a new method of solving this problem.},
year = {1955}
}

@article{KlattTamelingMunk2020,
author = {Klatt, Marcel and Tameling, Carla and Munk, Axel},
title = {Empirical Regularized Optimal Transport: Statistical Theory and Applications},
journal = {SIAM Journal on Mathematics of Data Science},
volume = {2},
number = {2},
pages = {419-443},
year = {2020},
doi = {10.1137/19M1278788},
}

@ARTICLE{9366690,
  author={Chen, Xiaohui and Yang, Yun},
  journal={IEEE Transactions on Information Theory}, 
  title={Cutoff for Exact Recovery of Gaussian Mixture Models}, 
  year={2021},
  volume={67},
  number={6},
  pages={4223-4238},
  doi={10.1109/TIT.2021.3063155}}
  
 @ARTICLE{392253,
  author={Tianping Chen and Hong Chen},
  journal={IEEE Transactions on Neural Networks}, 
  title={Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems}, 
  year={1995},
  volume={6},
  number={4},
  pages={911-917},
  doi={10.1109/72.392253}}
  
@article{BenamouCarlierCuturiNennaPerye2015,
	author = {Benamou, Jean-David and Carlier, Guillaume and Cuturi, Marco and Nenna, Luca and Peyr{\'e}, Gabriel},
	doi = {10.1137/141000439},
	eprint = {https://doi.org/10.1137/141000439},
	journal = {SIAM Journal on Scientific Computing},
	number = {2},
	pages = {A1111-A1138},
	title = {Iterative Bregman Projections for Regularized Transportation Problems},
	url = {https://doi.org/10.1137/141000439},
	volume = {37},
	year = {2015},
	bdsk-url-1 = {https://doi.org/10.1137/141000439}}

@inproceedings{KingmaB14,
  author    = {Kingma, Diederik P.  and
               Ba, Jimmy},
  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}
}

@article{YU2022114823,
	author = {Yu, Jeremy and Lu, Lu and Meng, Xuhui and Karniadakis, George Em},
	doi = {https://doi.org/10.1016/j.cma.2022.114823},
	issn = {0045-7825},
	journal = {Computer Methods in Applied Mechanics and Engineering},
	keywords = {Deep learning, Partial differential equations, Physics-informed neural networks, Gradient-enhanced, Residual-based adaptive refinement},
	pages = {114823},
	title = {Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems},
	url = {https://www.sciencedirect.com/science/article/pii/S0045782522001438},
	volume = {393},
	year = {2022},
	bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0045782522001438},
	bdsk-url-2 = {https://doi.org/10.1016/j.cma.2022.114823}}

@book{sabtanbrogio2015_OT,
	author = {Santambrogio, Filippo},
	biburl = {https://www.bibsonomy.org/bibtex/281ed66090d576f173cc088fb4596d431/kirk86},
	date-added = {2021-08-25 21:53:44 -0500},
	date-modified = {2021-08-25 21:53:44 -0500},
	edition = {1},
	keywords = {book optimal-transport},
	publisher = {Birkh\"auser Basel},
	title = {Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs and Modeling},
	url = {https://www.math.u-psud.fr/~filippo/OTAM-cvgmt.pdf},
	year = 2015,
	bdsk-url-1 = {https://www.math.u-psud.fr/~filippo/OTAM-cvgmt.pdf}}

@InProceedings{ZSSR,
               author = {Shocher, Assaf and Cohen, Nadav Cohen and Irani, Michal},
               title = {``Zero-Shot" Super-Resolution using Deep Internal Learning},
               booktitle = {The IEEE Conference on Computer Vision and Pattern Recognition (CVPR)},
               month = {June},
               year = {2018}
              }
              
@misc{Kovachki_neuraloperator,
  doi = {10.48550/ARXIV.2108.08481},
  
  url = {https://arxiv.org/abs/2108.08481},
  
  author = {Kovachki, Nikola and Li, Zongyi and Liu, Burigede and Azizzadenesheli, Kamyar and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
  
  keywords = {Machine Learning (cs.LG), Numerical Analysis (math.NA), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Mathematics, FOS: Mathematics},
  
  title = {Neural Operator: Learning Maps Between Function Spaces},
  
  publisher = {arXiv},
  
  year = {2021},
  
  copyright = {arXiv.org perpetual, non-exclusive license}
}

@article{COTFNT,
year = {2019},
volume = {11},
journal = {Foundations and Trends in Machine Learning},
title = {Computational Optimal Transport},
number = {5-6},
pages = {355--607},
author = {Gabriel Peyr\'e and Marco Cuturi}
}

@INPROCEEDINGS{9483194,
  author={Inoue, Daisuke and Ito, Yuji and Yoshida, Hiroaki},
  booktitle={2021 American Control Conference (ACC)}, 
  title={Optimal Transport-based Coverage Control for Swarm Robot Systems: Generalization of the Voronoi Tessellation-based Method},
  year={2021},
  volume={},
  number={},
  pages={3032-3037},
  doi={10.23919/ACC50511.2021.9483194}}

@article{Solomon_2015,
author = {Solomon, Justin and de Goes, Fernando and Peyr\'{e}, Gabriel and Cuturi, Marco and Butscher, Adrian and Nguyen, Andy and Du, Tao and Guibas, Leonidas},
title = {Convolutional Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains},
year = {2015},
issue_date = {August 2015},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {34},
number = {4},
issn = {0730-0301},
url = {https://doi.org/10.1145/2766963},
doi = {10.1145/2766963},
journal = {ACM Trans. Graph.},
month = {jul},
articleno = {66},
numpages = {11},
keywords = {optimal transportation, wasserstein distances, entropy, displacement interpolation}
}


@inproceedings{Kusner_2015ICML,
author = {Kusner, Matt J. and Sun, Yu and Kolkin, Nicholas I. and Weinberger, Kilian Q.},
title = {From Word Embeddings to Document Distances},
year = {2015},
publisher = {JMLR.org},
booktitle = {Proceedings of the 32nd International Conference on International Conference on Machine Learning - Volume 37},
pages = {957–966},
numpages = {10},
location = {Lille, France},
series = {ICML'15}
}

@inproceedings{Altschuler_2017,
author = {Altschuler, Jason and Weed, Jonathan and Rigollet, Philippe},
title = {Near-Linear Time Approximation Algorithms for Optimal Transport via Sinkhorn Iteration},
year = {2017},
isbn = {9781510860964},
publisher = {Curran Associates Inc.},
address = {Red Hook, NY, USA},
abstract = {Computing optimal transport distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. Despite the recent introduction of several algorithms with good empirical performance, it is unknown whether general optimal transport distances can be approximated in near-linear time. This paper demonstrates that this ambitious goal is in fact achieved by Cuturi's Sinkhorn Distances. This result relies on a new analysis of Sinkhorn iterations, which also directly suggests a new greedy coordinate descent algorithm GREENKHORN with the same theoretical guarantees. Numerical simulations illustrate that GREENKHORN significantly outperforms the classical SINKHORN algorithm in practice.},
booktitle = {Proceedings of the 31st International Conference on Neural Information Processing Systems},
pages = {1961–1971},
numpages = {11},
location = {Long Beach, California, USA},
series = {NIPS'17}
}

@article{Sinkhorn_1964,
	author = {Richard Sinkhorn},
	doi = {10.1214/aoms/1177703591},
	journal = {The Annals of Mathematical Statistics},
	number = {2},
	pages = {876 -- 879},
	publisher = {Institute of Mathematical Statistics},
	title = {{A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices}},
	url = {https://doi.org/10.1214/aoms/1177703591},
	volume = {35},
	year = {1964},
	bdsk-url-1 = {https://doi.org/10.1214/aoms/1177703591}}

@inproceedings{NIPS2013_Cuturi,
	author = {Cuturi, Marco},
	booktitle = {Advances in Neural Information Processing Systems},
	editor = {C.J. Burges and L. Bottou and M. Welling and Z. Ghahramani and K.Q. Weinberger},
	publisher = {Curran Associates, Inc.},
	title = {Sinkhorn Distances: Lightspeed Computation of Optimal Transport},
	url = {https://proceedings.neurips.cc/paper/2013/file/af21d0c97db2e27e13572cbf59eb343d-Paper.pdf},
	volume = {26},
	year = {2013},
	bdsk-url-1 = {https://proceedings.neurips.cc/paper/2013/file/af21d0c97db2e27e13572cbf59eb343d-Paper.pdf}}

@inproceedings{Genevay2016,
author = {Genevay, Aude and Cuturi, Marco and Peyr\'{e}, Gabriel and Bach, Francis},
title = {Stochastic Optimization for Large-Scale Optimal Transport},
year = {2016},
isbn = {9781510838819},
publisher = {Curran Associates Inc.},
address = {Red Hook, NY, USA},
booktitle = {Proceedings of the 30th International Conference on Neural Information Processing Systems},
pages = {3440–3448},
numpages = {9},
location = {Barcelona, Spain},
series = {NIPS'16}
}

@INPROCEEDINGS{8619816,
  author={Krishnan, Vishaal and Martínez, Sonia},
  booktitle={2018 IEEE Conference on Decision and Control (CDC)}, 
  title={Distributed Optimal Transport for the Deployment of Swarms}, 
  year={2018},
  volume={},
  number={},
  pages={4583-4588},
  doi={10.1109/CDC.2018.8619816}}
  
@article{ChenGeorgiouPavon2021,
	author = {Chen, Yongxin and Georgiou, Tryphon T. and Pavon, Michele},
	doi = {10.1146/annurev-control-070220-100858},
	eprint = {https://doi.org/10.1146/annurev-control-070220-100858},
	journal = {Annual Review of Control, Robotics, and Autonomous Systems},
	number = {1},
	pages = {89-113},
	title = {Optimal Transport in Systems and Control},
	url = {https://doi.org/10.1146/annurev-control-070220-100858},
	volume = {4},
	year = {2021},
	bdsk-url-1 = {https://doi.org/10.1146/annurev-control-070220-100858}}

@InProceedings{finlay2020train,
  author = {Finlay, Chris and Jacobsen, Jörn-Henrik and Nurbekyan, Levon and Oberman, Adam M},
  booktitle = {ICML},
  title = {"How to train your neural ODE: the world of Jacobian and kinetic regularization"},
  year = 2020
}


@ARTICLE{7586038,
  author={Courty, Nicolas and Flamary, Rémi and Tuia, Devis and Rakotomamonjy, Alain},
  journal={IEEE Transactions on Pattern Analysis and Machine Intelligence}, 
  title={Optimal Transport for Domain Adaptation}, 
  year={2017},
  volume={39},
  number={9},
  pages={1853-1865},
  doi={10.1109/TPAMI.2016.2615921}}
  
@InProceedings{arjovsky17a,
  title = 	 {{W}asserstein Generative Adversarial Networks},
  author =       {Martin Arjovsky and Soumith Chintala and L{\'e}on Bottou},
  booktitle = 	 {Proceedings of the 34th International Conference on Machine Learning},
  pages = 	 {214--223},
  year = 	 {2017},
  editor = 	 {Precup, Doina and Teh, Yee Whye},
  volume = 	 {70},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {06--11 Aug},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v70/arjovsky17a/arjovsky17a.pdf},
  url = 	 {https://proceedings.mlr.press/v70/arjovsky17a.html},
  abstract = 	 {We introduce a new algorithm named WGAN, an alternative to traditional GAN training. In this new model, we show that we can improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches. Furthermore, we show that the corresponding optimization problem is sound, and provide extensive theoretical work highlighting the deep connections to different distances between distributions.}
}

@article{flamary2021pot,
author = { Rémi Flamary and Nicolas Courty and Alexandre Gramfort and   Mokhtar Z. Alaya and Aurélie Boisbunon and Stanislas Chambon and Laetitia
  Chapel and Adrien Corenflos and Kilian Fatras and Nemo Fournier and Léo
  Gautheron and Nathalie T.H. Gayraud and Hicham Janati and Alain Rakotomamonjy
  and Ievgen Redko and Antoine Rolet and Antony Schutz and Vivien Seguy and
  Danica J. Sutherland and Romain Tavenard and Alexander Tong and Titouan
  Vayer},
title = {POT: Python Optimal Transport},
journal = { Journal of Machine Learning Research},
volume = { 22},
number = { 78},
pages = { 1-8},
year = {2021}
}

@misc{Li_FourierNeuralOperator,
  doi = {10.48550/ARXIV.2010.08895},
  
  url = {https://arxiv.org/abs/2010.08895},
  
  author = {Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
  
  keywords = {Machine Learning (cs.LG), Numerical Analysis (math.NA), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Mathematics, FOS: Mathematics},
  
  title = {Fourier Neural Operator for Parametric Partial Differential Equations},
  
  publisher = {arXiv},
  
  year = {2020},
  
  copyright = {arXiv.org perpetual, non-exclusive license}
}


@article{LuJinPangZhangKarniadakis2021_DeepONet,
title = {Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators},
journal = {Nat Mach Intell},
volume = {3},
pages = {218-229},
year = {2021},
author = {Lu, Lu and Jin, Pengzhan and Pang, Guofei and Zhang, Zhongqiang and Karniadakis, George Em}
}

@misc{LiuMaChenZhaZhou2021,
  doi = {10.48550/ARXIV.2102.02992},
  
  url = {https://arxiv.org/abs/2102.02992},
  
  author = {Liu, Shu and Ma, Shaojun and Chen, Yongxin and Zha, Hongyuan and Zhou, Haomin},
  
  keywords = {Machine Learning (cs.LG), FOS: Computer and information sciences, FOS: Computer and information sciences},
  
  title = {Learning High Dimensional Wasserstein Geodesics},
  
  publisher = {arXiv},
  
  year = {2021},
  
  copyright = {arXiv.org perpetual, non-exclusive license}
}

@article{RAISSI2019686,
title = {Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations},
journal = {Journal of Computational Physics},
volume = {378},
pages = {686-707},
year = {2019},
issn = {0021-9991},
doi = {https://doi.org/10.1016/j.jcp.2018.10.045},
url = {https://www.sciencedirect.com/science/article/pii/S0021999118307125},
author = {M. Raissi and P. Perdikaris and G.E. Karniadakis},
keywords = {Data-driven scientific computing, Machine learning, Predictive modeling, Runge–Kutta methods, Nonlinear dynamics},
abstract = {We introduce physics-informed neural networks – neural networks that are trained to solve supervised learning tasks while respecting any given laws of physics described by general nonlinear partial differential equations. In this work, we present our developments in the context of solving two main classes of problems: data-driven solution and data-driven discovery of partial differential equations. Depending on the nature and arrangement of the available data, we devise two distinct types of algorithms, namely continuous time and discrete time models. The first type of models forms a new family of data-efficient spatio-temporal function approximators, while the latter type allows the use of arbitrarily accurate implicit Runge–Kutta time stepping schemes with unlimited number of stages. The effectiveness of the proposed framework is demonstrated through a collection of classical problems in fluids, quantum mechanics, reaction–diffusion systems, and the propagation of nonlinear shallow-water waves.}
}

@article{Benamou2000ACF,
  title={A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem},
  author={Jean-David Benamou and Yann Brenier},
  journal={Numerische Mathematik},
  year={2000},
  volume={84},
  pages={375-393}
}

@book{Villani2003_topics-in-ot,
	abstract = {This is the first comprehensive introduction to the theory of mass transportation with its many - and sometimes unexpected - applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook. In 1781, Gaspard Monge defined the problem of "optimal transportation" (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is intended for graduate students and researchers, covering both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis. (4e de couverture)},
	address = {Providence, Rhode Island},
	author = {Villani, C\'edric},
	booktitle = {Topics in optimal transportation / C{\~A}{\copyright}dric Villani},
	date-added = {2021-08-25 10:01:11 -0500},
	date-modified = {2021-08-25 10:01:11 -0500},
	isbn = {0-8218-3312-X},
	publisher = {American mathematical society},
	series = {Graduate studies in mathematics},
	title = {Topics in optimal transportation},
	year = {2003}}
	
	
@book{BurageBuragoIvanov2001_MetricGeometry,
	address = {Providence, Rhode Island},
	author = {Burage, Dmitri and Burago, Yuri and Ivanov, Sergei},
	publisher = {American mathematical society},
	series = {Graduate studies in mathematics},
	title = {A Course in Metric Geometry},
	year = {2001}}
	

@book{AmbrosioGigliSavare2008_GradientFlows,
	author = {Ambrosio, Luigi and Gigli, Nicola and Savar\'e, Giuseppe},
	date-added = {2021-08-25 21:53:44 -0500},
	date-modified = {2021-08-25 21:53:44 -0500},
	edition = {Second},
	publisher = {Birkh\"auser Baseluser Basel},
	series = {Lectures in Mathematics ETH Z\"urich},
	title = {Gradient Flows in Metric Spaces and in the Space of Probability Measures},
	year = {2008}}
	

@article{MCCANN1997153,
	author = {Robert J. McCann},
	journal = {Advances in Mathematics},
	number = {1},
	pages = {153-179},
	title = {A Convexity Principle for Interacting Gases},
	url = {https://www.sciencedirect.com/science/article/pii/S0001870897916340},
	volume = {128},
	year = {1997},
	bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0001870897916340},
	bdsk-url-2 = {https://doi.org/10.1006/aima.1997.1634}}


@article{Brenier1991,
	author = {Brenier, Yann},
	doi = {https://doi.org/10.1002/cpa.3160440402},
	eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.3160440402},
	journal = {Communications on Pure and Applied Mathematics},
	number = {4},
	pages = {375-417},
	title = {Polar factorization and monotone rearrangement of vector-valued functions},
	url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160440402},
	volume = {44},
	year = {1991},
	bdsk-url-1 = {https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160440402},
	bdsk-url-2 = {https://doi.org/10.1002/cpa.3160440402}}


@article{ChenGeorgiouPavon2016,
	author = {Chen, Yongxin and Georgiou, Tryphon T. and Pavon, Michele},
	date = {2016/05/01},
	date-added = {2022-05-09 22:52:02 -0500},
	date-modified = {2022-05-09 22:52:02 -0500},
	doi = {10.1007/s10957-015-0803-z},
	id = {Chen2016},
	isbn = {1573-2878},
	journal = {Journal of Optimization Theory and Applications},
	number = {2},
	pages = {671--691},
	title = {On the Relation Between Optimal Transport and Schr{\"o}dinger Bridges: A Stochastic Control Viewpoint},
	url = {https://doi.org/10.1007/s10957-015-0803-z},
	volume = {169},
	year = {2016},
	bdsk-url-1 = {https://doi.org/10.1007/s10957-015-0803-z}}

@book{evans2010,
  address = {Providence, R.I.},
  author = {Evans, Lawrence C.},
  isbn = {9780821849743 0821849743},
  publisher = {American Mathematical Society},
  title = {Partial Differential Equations},
  year = 2010
}

@article{Mikami2004,
	author = {Mikami, Toshio},
	date = {2004/06/01},
	doi = {10.1007/s00440-004-0340-4},
	id = {Mikami2004},
	isbn = {1432-2064},
	journal = {Probability Theory and Related Fields},
	number = {2},
	pages = {245--260},
	title = {Monge's problem with a quadratic cost by the zero-noise limit of h-path processes},
	url = {https://doi.org/10.1007/s00440-004-0340-4},
	volume = {129},
	year = {2004},
	bdsk-url-1 = {https://doi.org/10.1007/s00440-004-0340-4}}

@misc{LiKovachkiAzizzadenesheliLiuBhattacharyaStuartAnandkumar,
  doi = {10.48550/ARXIV.2003.03485},
  
  url = {https://arxiv.org/abs/2003.03485},
  
  author = {Li, Zongyi and Kovachki, Nikola and Azizzadenesheli, Kamyar and Liu, Burigede and Bhattacharya, Kaushik and Stuart, Andrew and Anandkumar, Anima},
  
  keywords = {Machine Learning (cs.LG), Numerical Analysis (math.NA), Machine Learning (stat.ML), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Mathematics, FOS: Mathematics},
  
  title = {Neural Operator: Graph Kernel Network for Partial Differential Equations},
  
  publisher = {arXiv},
  
  year = {2020},
  
  copyright = {arXiv.org perpetual, non-exclusive license}
}

@misc{https://doi.org/10.48550/arxiv.2111.03794,
  doi = {10.48550/ARXIV.2111.03794},
  
  url = {https://arxiv.org/abs/2111.03794},
  
  author = {Li, Zongyi and Zheng, Hongkai and Kovachki, Nikola and Jin, David and Chen, Haoxuan and Liu, Burigede and Azizzadenesheli, Kamyar and Anandkumar, Anima},
  
  keywords = {Machine Learning (cs.LG), Numerical Analysis (math.NA), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Mathematics, FOS: Mathematics},
  
  title = {Physics-Informed Neural Operator for Learning Partial Differential Equations},
  
  publisher = {arXiv},
  
  year = {2021},
  
  copyright = {arXiv.org perpetual, non-exclusive license}
}

@misc{https://doi.org/10.48550/arxiv.2202.08942,
  doi = {10.48550/ARXIV.2202.08942},
  
  url = {https://arxiv.org/abs/2202.08942},
  
  author = {Tan, Lesley and Chen, Liang},
  
  keywords = {Machine Learning (cs.LG), Numerical Analysis (math.NA), FOS: Computer and information sciences, FOS: Computer and information sciences, FOS: Mathematics, FOS: Mathematics},
  
  title = {Enhanced DeepONet for Modeling Partial Differential Operators Considering Multiple Input Functions},
  
  publisher = {arXiv},
  
  year = {2022},
  
  copyright = {arXiv.org perpetual, non-exclusive license}
}



@article{DBLP:journals/corr/abs-2009-01717,
  author    = {Rick Groenendijk and
               Sezer Karaoglu and
               Theo Gevers and
               Thomas Mensink},
  title     = {Multi-Loss Weighting with Coefficient of Variations},
  journal   = {CoRR},
  volume    = {abs/2009.01717},
  year      = {2020},
  url       = {https://arxiv.org/abs/2009.01717},
  eprinttype = {arXiv},
  eprint    = {2009.01717},
  timestamp = {Thu, 14 Oct 2021 09:16:37 +0200},
  biburl    = {https://dblp.org/rec/journals/corr/abs-2009-01717.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@article{Wang_2021,
   title={On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks},
   volume={384},
   ISSN={0045-7825},
   url={http://dx.doi.org/10.1016/j.cma.2021.113938},
   DOI={10.1016/j.cma.2021.113938},
   journal={Computer Methods in Applied Mechanics and Engineering},
   publisher={Elsevier BV},
   author={Wang, Sifan and Wang, Hanwen and Perdikaris, Paris},
   year={2021},
   month=oct, pages={113938} }


@misc{tancik2020fourier,
      title={Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains}, 
      author={Matthew Tancik and Pratul P. Srinivasan and Ben Mildenhall and Sara Fridovich-Keil and Nithin Raghavan and Utkarsh Singhal and Ravi Ramamoorthi and Jonathan T. Barron and Ren Ng},
      year={2020},
      eprint={2006.10739},
      archivePrefix={arXiv},
      primaryClass={cs.CV}
}