@article{bauer1961absolute,
  title={Absolute norms and characteristic roots},
  author={Bauer, Friedrich L and Householder, Alston S},
  journal={Numerische Mathematik},
  volume={3},
  number={1},
  pages={241--246},
  year={1961},
  publisher={Springer}
}
@article{bauer1963optimally,
  title={Optimally scaled matrices},
  author={Bauer, Friedrich L},
  journal={Numerische Mathematik},
  volume={5},
  number={1},
  pages={73--87},
  year={1963},
  publisher={Springer}
}
@article{hornik1989multilayer,
  title={Multilayer feedforward networks are universal approximators},
  author={Hornik, Kurt and Stinchcombe, Maxwell and White, Halbert},
  journal={Neural networks},
  volume={2},
  number={5},
  pages={359--366},
  year={1989},
  publisher={Elsevier}
}
@article{braatz1994minimizing,
  title={Minimizing the Euclidean condition number},
  author={Braatz, Richard D and Morari, Manfred},
  journal={SIAM Journal on Control and Optimization},
  volume={32},
  number={6},
  pages={1763--1768},
  year={1994},
  publisher={SIAM}
}
@article{lagaris1998artificial,
  title={Artificial neural networks for solving ordinary and partial differential equations},
  author={Lagaris, Isaac E and Likas, Aristidis and Fotiadis, Dimitrios I},
  journal={IEEE transactions on neural networks},
  volume={9},
  number={5},
  pages={987--1000},
  year={1998},
  publisher={IEEE}
}
@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{chen2020neurodiffeq,
  title={NeuroDiffEq: A Python package for solving differential equations with neural networks},
  author={Chen, Feiyu and Sondak, David and Protopapas, Pavlos and Mattheakis, Marios and Liu, Shuheng and Agarwal, Devansh and Di Giovanni, Marco},
  journal={Journal of Open Source Software},
  volume={5},
  number={46},
  pages={1931},
  year={2020}
}
@article{flamant2020solving,
  title={Solving Differential Equations Using Neural Network Solution Bundles},
  author={Flamant, Cedric and Protopapas, Pavlos and Sondak, David},
  journal={arXiv preprint arXiv:2006.14372},
  year={2020}
}
@article{mcfall2009artificial,
  title={Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions},
  author={McFall, Kevin Stanley and Mahan, James Robert},
  journal={IEEE Transactions on Neural Networks},
  volume={20},
  number={8},
  pages={1221--1233},
  year={2009},
  publisher={IEEE}
}
@article{lagari2020systematic,
  title={Systematic construction of neural forms for solving partial differential equations inside rectangular domains, subject to initial, boundary and interface conditions},
  author={Lagari, Pola Lydia and Tsoukalas, Lefteri H and Safarkhani, Salar and Lagaris, Isaac E},
  journal={International Journal on Artificial Intelligence Tools},
  volume={29},
  number={05},
  pages={2050009},
  year={2020},
  publisher={World Scientific}
}
@article{sukumar2021exact,
  title={Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks},
  author={Sukumar, N and Srivastava, Ankit},
  journal={arXiv preprint arXiv:2104.08426},
  year={2021}
}
@article{lagaris2000neural,
  title={Neural-network methods for boundary value problems with irregular boundaries},
  author={Lagaris, Isaac E and Likas, Aristidis C and Papageorgiou, Dimitris G},
  journal={IEEE Transactions on Neural Networks},
  volume={11},
  number={5},
  pages={1041--1049},
  year={2000},
  publisher={IEEE}
}
@article{graf2021uncertainty,
  title={Uncertainty Quantification in Neural Differential Equations},
  author={Graf, Olga and Flores, Pablo and Protopapas, Pavlos and Pichara, Karim},
  journal={arXiv preprint arXiv:2111.04207},
  year={2021}
}
@article{guo2022energy,
  title={Energy-Based Error Bound of Physics-Informed Neural Network Solutions in Elasticity},
  author={Guo, Mengwu and Haghighat, Ehsan},
  journal={Journal of Engineering Mechanics},
  volume={148},
  number={8},
  pages={04022038},
  year={2022},
  publisher={American Society of Civil Engineers}
}
@article{mattheakis2020hamiltonian,
  title={Hamiltonian neural networks for solving differential equations},
  author={Mattheakis, Marios and Sondak, David and Dogra, Akshunna S and Protopapas, Pavlos},
  journal={arXiv preprint arXiv:2001.11107},
  year={2020}
}
@article{yu2017deep,
  title={The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems},
  author={Yu, Bing and others},
  journal={arXiv preprint arXiv:1710.00211},
  year={2017}
}
@article{sirignano2018dgm,
  title={DGM: A deep learning algorithm for solving partial differential equations},
  author={Sirignano, Justin and Spiliopoulos, Konstantinos},
  journal={Journal of computational physics},
  volume={375},
  pages={1339--1364},
  year={2018},
  publisher={Elsevier}
}
@article{krishnapriyan2021characterizing,
  title={Characterizing possible failure modes in physics-informed neural networks},
  author={Krishnapriyan, Aditi and Gholami, Amir and Zhe, Shandian and Kirby, Robert and Mahoney, Michael W},
  journal={Advances in Neural Information Processing Systems},
  volume={34},
  year={2021}
}
@article{DesaiShaan2021OTLo,
abstract = {Solving differential equations efficiently and accurately sits at the heart
of progress in many areas of scientific research, from classical dynamical
systems to quantum mechanics. There is a surge of interest in using
Physics-Informed Neural Networks (PINNs) to tackle such problems as they
provide numerous benefits over traditional numerical approaches. Despite their
potential benefits for solving differential equations, transfer learning has
been under explored. In this study, we present a general framework for transfer
learning PINNs that results in one-shot inference for linear systems of both
ordinary and partial differential equations. This means that highly accurate
solutions to many unknown differential equations can be obtained
instantaneously without retraining an entire network. We demonstrate the
efficacy of the proposed deep learning approach by solving several real-world
problems, such as first- and second-order linear ordinary equations, the
Poisson equation, and the time-dependent Schrodinger complex-value partial
differential equation.},
year = {2021},
title = {One-Shot Transfer Learning of Physics-Informed Neural Networks},
copyright = {http://arxiv.org/licenses/nonexclusive-distrib/1.0},
language = {eng},
author = {Desai, Shaan and Mattheakis, Marios and Joy, Hayden and Protopapas, Pavlos and Roberts, Stephen},
}
@article{parwani2021adversarial,
  title={Adversarial Sampling for Solving Differential Equations with Neural Networks},
  author={Parwani, Kshitij and Protopapas, Pavlos},
  journal={arXiv preprint arXiv:2111.12024},
  year={2021}
}
@article{mckeeman1962algorithm,
  title={Algorithm 145: Adaptive numerical integration by Simpson's rule},
  author={McKeeman, William Marshall},
  journal={Communications of the ACM},
  volume={5},
  number={12},
  pages={604},
  year={1962},
  publisher={ACM New York, NY, USA}
}
@article{de2021error,
  title={Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs},
  author={De Ryck, Tim and Mishra, Siddhartha},
  journal={arXiv preprint arXiv:2106.14473},
  year={2021}
}
@article{jenkins1970three,
  title={A three-stage variable-shift iteration for polynomial zeros and its relation to generalized Rayleigh iteration},
  author={Jenkins, Michael A and Traub, Joseph F},
  journal={Numerische Mathematik},
  volume={14},
  number={3},
  pages={252--263},
  year={1970},
  publisher={Springer}
}
@article{de2022errorkolmogorov,
  title={Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs},
  author={De Ryck, Tim and Mishra, Siddhartha},
  journal={Advances in Computational Mathematics},
  volume={48},
  number={6},
  pages={1--40},
  year={2022},
  publisher={Springer}
}
@article{de2022generic,
  title={Generic bounds on the approximation error for physics-informed (and) operator learning},
  author={De Ryck, Tim and Mishra, Siddhartha},
  journal={arXiv preprint arXiv:2205.11393},
  year={2022}
}
@article{de2022errorhyperbolic,
  title={Error analysis for deep neural network approximations of parametric hyperbolic conservation laws},
  author={De Ryck, Tim and Mishra, Siddhartha},
  journal={arXiv preprint arXiv:2207.07362},
  year={2022}
}