DeepXDE: A Deep Learning Library for Solving Differential Equations , Lu Lu, Xuhui Meng, Zhiping Mao, George Em Karniadakis , SIAM Review, 2021. [paper](https://epubs.siam.org/doi/pdf/10.1137/19M1274067)[code](https://github.com/lululxvi/deepxde)
NVIDIA SimNet™: An AI-Accelerated Multi-Physics Simulation Framework , Oliver Hennigh, Susheela Narasimhan, Mohammad Amin Nabian, Akshay Subramaniam, Kaustubh Tangsali, Zhiwei Fang, Max Rietmann, Wonmin Byeon, Sanjay Choudhry , ICCS , 2021. [paper](https://link.springer.com/chapter/10.1007/978-3-030-77977-1_36)
SciANN: A Keras wrapper for scientific computations and physics-informed deep learning using artificial neural networks , Ehsan Haghighat, Ruben Juanes , arXiv preprint arXiv:2005.08803, 2020. [paper](https://www.sciencedirect.com/science/article/pii/S0045782520307374)[code](https://github.com/sciann/sciann)
Elvet -- a neural network-based differential equation and variational problem solver , Jack Y. Araz, Juan Carlos Criado, Michael Spannowsky , arXiv:2103.14575 hep-lat, physics:hep-ph, physics:hep-th, stat, 2021. [paper](https://arxiv.org/pdf/2103.14575)[code](https://gitlab.com/elvet/elvet)
TensorDiffEq: Scalable Multi-GPU Forward and Inverse Solvers for Physics Informed Neural Networks , Levi D. McClenny, Mulugeta A. Haile, Ulisses M. Braga-Neto , arXiv:2103.16034 physics, 2021. [paper](https://arxiv.org/pdf/2103.16034)[code](https://github.com/tensordiffeq/TensorDiffEq)
PyDEns: a Python Framework for Solving Differential Equations with Neural Networks , Alex Koryagin, er, Roman Khudorozkov, Sergey Tsimfer , arXiv:1909.11544 cs, stat, 2019. [paper]()
NeuroDiffEq: A Python package for solving differential equations with neural networks , Feiyu Chen, David Sondak, Pavlos Protopapas, Marios Mattheakis, Shuheng Liu, Devansh Agarwal, Marco Di Giovanni , Journal of Open Source Software, 2020. [paper](https://joss.theoj.org/papers/10.21105/joss.01931)[code](https://github.com/analysiscenter/pydens)
Universal Differential Equations for Scientific Machine Learning , Christopher Rackauckas, Yingbo Ma, Julius Martensen, Collin Warner, Kirill Zubov, Rohit Supekar, Dominic Skinner, Ali Ramadhan, Alan Edelman , arXiv:2001.04385 cs, math, q-bio, stat, 2020. [paper](https://arxiv.org/pdf/2001.04385.pdf)[code](https://github.com/ChrisRackauckas/universal_differential_equations)
NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators , Zongren Zou, Xuhui Meng, Apostolos F. Psaros, George Em Karniadakis , UNKNOWN_JOURNAL , 2022. [paper](http://arxiv.org/pdf/2208.11866.pdf)[code](https://github.com/Crunch-UQ4MI/neuraluq)
Papers on PINN Models
Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , M. Raissi, P. Perdikaris, G. E. Karniadakis , Journal of Computational Physics, 2019. [paper](https://www.sciencedirect.com/science/article/pii/S0021999118307125)
The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems , E Weinan, Bing Yu , Communications in Mathematics and Statistics, 2018. [paper](https://link.springer.com/article/10.1007/s40304-018-0127-z)
DGM: A deep learning algorithm for solving partial differential equations , Justin Sirignano, Konstantinos Spiliopoulos , Journal of Computational Physics, 2018. [paper](https://www.sciencedirect.com/science/article/pii/S0021999118305527)
SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs , Amuthan A. Ramabathiran, Ramach, Prabhu ran , Journal of Computational Physics, 2021. [paper](https://www.sciencedirect.com/science/article/pii/S0021999121004952)[code](https://github.com/nn4pde/SPINN)
Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative , Yinlin Ye, Yajing Li, Hongtao Fan, Xinyi Liu, Hongbing Zhang , arXiv:2108.07490 cs, math, 2021. [paper](http://arxiv.org/pdf/2108.07490.pdf)
NH-PINN: Neural homogenization based physics-informed neural network for multiscale problems , Wing Tat Leung, Guang Lin, Zecheng Zhang , arXiv:2108.12942 cs, math, 2021. [paper](http://arxiv.org/pdf/2108.12942.pdf)
Physics-Augmented Learning: A New Paradigm Beyond Physics-Informed Learning , Ziming Liu, Yunyue Chen, Yuanqi Du, Max Tegmark , arXiv:2109.13901 physics, 2021. [paper](http://arxiv.org/pdf/2109.13901.pdf)
Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method , Yuntian Chen, Dou Huang, Dongxiao Zhang, Junsheng Zeng, Nanzhe Wang, Haoran Zhang, Jinyue Yan , Journal of Computational Physics, 2021. [paper](https://linkinghub.elsevier.com/retrieve/pii/S0021999121005192)
Learning in Sinusoidal Spaces with Physics-Informed Neural Networks , Jian Cheng Wong, Chinchun Ooi, Abhishek Gupta, Yew-Soon Ong , arXiv:2109.09338 physics, 2021. [paper](http://arxiv.org/pdf/2109.09338.pdf)
HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks , Filipe de Avila Belbute-Peres, Yi-fan Chen, Fei Sha , NIPS , 2021. [paper](https://openreview.net/forum?id=LxUuRDUhRjM)
Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries , AliKashefi, TapanMukerji , Journal of Computational Physics, 2022. [paper](https://doi.org/10.1016/j.jcp.2022.111510)
Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems , HanGao, Matthew J.Zahr, Jian-XunWang , Computer Methods in Applied Mechanics and Engineering, 2022. [paper](https://doi.org/10.1016/j.cma.2021.114502)
ModalPINN : an extension of Physics-Informed Neural Networks with enforced truncated Fourier decomposition for periodic flow reconstruction using a limited number of imperfect sensors , * Ga´etan Raynaud , S´ebastien Houde, Fr´ed´erick P Gosselin*, Journal of Computational Physics, 2022. [paper](https://doi.org/10.1016/j.jcp.2022.111271)
∆-PINNs: physics-informed neural networks on complex geometries , Francisco Sahli Costabal, Simone Pezzuto, Paris Perdikaris , Arxiv , 2022. [paper](http://arxiv.org/pdf/2209.03984.pdf)
Robust Regression with Highly Corrupted Data via Physics Informed Neural Networks , Wei Peng, Wen Yao, Weien Zhou, Xiaoya Zhang, Weijie Yao , ArXiv , 2022. [paper](http://arxiv.org/pdf/2210.10646.pdf)[code](https://github.com/weipengOO98/robust_pinn)
Papers on Parallel PINN
Parallel Physics-Informed Neural Networks via Domain Decomposition , Khemraj Shukla, Ameya D. Jagtap, George Em Karniadakis , arXiv:2104.10013 cs, 2021. [paper](https://arxiv.org/pdf/2104.10013)
Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations , Ben Moseley, Andrew Markham, Tarje Nissen-Meyer , arXiv:2107.07871 physics, 2021. [paper](https://arxiv.org/pdf/2107.07871)
PPINN: Parareal physics-informed neural network for time-dependent PDEs , Xuhui Meng, Zhen Li, Dongkun Zhang, George Em Karniadakis , Computer Methods in Applied Mechanics and Engineering, 2020. [paper](https://arxiv.org/pdf/2104.10013)
When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization? , Zheyuan Hu, Ameya D. Jagtap, George Em Karniadakis, Kenji Kawaguchi , arXiv:2109.09444 cs, math, stat, 2021. [paper](http://arxiv.org/pdf/2109.09444.pdf)
Scaling physics-informed neural networks to large domains by using domain decomposition , Ben Moseley, Andrew Markham, Tarje Nissen-Meyer , NIPS , 2021. [paper](https://openreview.net/forum?id=o1WiAZiw_CE)
Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations , Ben Moseley, Andrew Markham, Tarje Nissen-Meyer , arXiv:2107.07871 physics, 2021. [paper](http://arxiv.org/pdf/2107.07871.pdf)
Improved Deep Neural Networks with Domain Decomposition in Solving Partial Differential Equations , Wei Wu, Xinlong Feng, Hui Xu , Journal of Scientific Computing, 2022. [paper](https://doi.org/10.1007/s10915-022-01980-y)
INN: Interfaced neural networks as an accessible meshless approach for solving interface PDE problems , Sidi Wu, Benzhuo Lu , Journal of Computational Physics, 2022. [paper](https://www.sciencedirect.com/science/article/pii/S0021999122006507)[code](https://github.com/bzlu-Group/INN)
Papers on PINN Accerleration
Self-adaptive loss balanced Physics-informed neural networks for the incompressible Navier-Stokes equations , Zixue Xiang, Wei Peng, Xiaohu Zheng, Xiaoyu Zhao, Wen Yao , arXiv:2104.06217 physics, 2021. [paper](https://arxiv.org/pdf/2104.06217)
A Dual-Dimer method for training physics-constrained neural networks with minimax architecture , Dehao Liu, Yan Wang , Neural Networks, 2021. [paper](https://www.sciencedirect.com/science/article/abs/pii/S0893608020304536)
DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation , Jungeun Kim, Kookjin Lee, Dongeun Lee, Sheo Yon Jin, Noseong Park , AAAI, 2021. [paper](https://www.aaai.org/AAAI21Papers/AAAI-4849.KimJ.pdf)
Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems , Jeremy Yu, Lu Lu, Xuhui Meng, George Em Karniadakis , Arxiv, 2021. [paper](https://arxiv.org/abs/2111.02801)
CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method , Pao-Hsiung Chiu, Jian Cheng Wong, Chinchun Ooi, My Ha Dao, Yew-Soon Ong , Arxiv, 2021. [paper](https://arxiv.org/abs/2110.15832)
A hybrid physics-informed neural network for nonlinear partial differential equation , Chunyue Lv, Lei Wang, Chenming Xie , Arxiv, 2021. [paper](https://arxiv.org/abs/2112.01696)
Multi-Objective Loss Balancing for Physics-Informed Deep Learning , Rafael Bischof, Michael Kraus , Arxiv, 2021. [paper](http://rgdoi.net/10.13140/RG.2.2.20057.24169)
A High-Efficient Hybrid Physics-Informed Neural Networks Based on Convolutional Neural Network , Zhiwei Fang , IEEE Transactions on Neural Networks and Learning Systems, 2021. [paper](https://ieeexplore.ieee.org/document/9403414)
A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks , Chenxi Wu,Min Zhu,Qinyang Tan,Yadhu Kartha,Lu Lu , arXiv:2207.10289 cs, 2022. [paper](https://arxiv.org/pdf/2207.10289.pdf)
Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations , Ramansh Sharma, Varun Shankar , NeurIPS, 2022. [paper](https://arxiv.org/abs/2205.09332)
Is L2 Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network , Chuwei Wang, Shanda Li, Di He, Liwei Wang , NeurIPS, 2022. [paper](https://arxiv.org/abs/2206.02016)
Papers on Model Transfer & Meta-Learning
A physics-aware learning architecture with input transfer networks for predictive modeling , Amir Behjat, Chen Zeng, Rahul Rai, Ion Matei, David Doermann, Souma Chowdhury , Applied Soft Computing, 2020. [paper](https://www.sciencedirect.com/science/article/abs/pii/S1568494620306037)
Transfer learning based multi-fidelity physics informed deep neural network , Souvik Chakraborty , Journal of Computational Physics, 2021. [paper](https://www.sciencedirect.com/science/article/pii/S0021999120307166)
Transfer learning enhanced physics informed neural network for phase-field modeling of fracture , Somdatta Goswami, Cosmin Anitescu, Souvik Chakraborty, Timon Rabczuk , Theoretical and Applied Fracture Mechanics, 2020. [paper](https://www.sciencedirect.com/science/article/abs/pii/S016784421930357X)
Meta-learning PINN loss functions , Apostolos F. Psaros, Kenji Kawaguchi, George Em Karniadakis , arXiv:2107.05544 cs, 2021. [paper](https://arxiv.org/pdf/2107.05544.pdf)
Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh , Tian Qin,Alex Beatson,Deniz Oktay,Nick McGreivy,Ryan P. Adams , arXiv:2211.01604 cs, 2022. [paper](https://arxiv.org/pdf/2211.01604.pdf)
Physics-Informed Neural Networks (PINNs) for Parameterized PDEs: A Metalearning Approach , Michael Penwarden, Sh Zhe, ian, Akil Narayan, Robert M. Kirby , Arxiv, 2021. [paper](https://arxiv.org/abs/2110.13361)
Papers on Probabilistic PINNs and Uncertainty Quantification
A physics-aware, probabilistic machine learning framework for coarse-graining high-dimensional systems in the Small Data regime , Constantin Grigo, Phaedon-Stelios Koutsourelakis , Journal of Computational Physics, 2019. [paper](https://www.sciencedirect.com/science/article/pii/S0021999119305261)
Adversarial uncertainty quantification in physics-informed neural networks , Yibo Yang, Paris Perdikaris , Journal of Computational Physics, 2019. [paper](https://www.sciencedirect.com/science/article/pii/S0021999119303584)
B-PINNs: Bayesian physics-informed neural networks for forward and inverse PDE problems with noisy data , Liu Yang, Xuhui Meng, George Em Karniadakis , Journal of Computational Physics, 2021. [paper](https://www.sciencedirect.com/science/article/pii/S0021999120306872)
PID-GAN: A GAN Framework based on a Physics-informed Discriminator for Uncertainty Quantification with Physics , Arka Daw, M. Maruf, Anuj Karpatne , arXiv:2106.02993 cs, stat, 2021. [paper](https://arxiv.org/pdf/2106.02993)
Quantifying Uncertainty in Physics-Informed Variational Autoencoders for Anomaly Detection , Marcus J. Neuer , ESTEP, 2020. [paper](https://link.springer.com/chapter/10.1007/978-3-030-69367-1_3)
A Physics-Data-Driven Bayesian Method for Heat Conduction Problems , Xinchao Jiang, Hu Wang, Yu li , arXiv:2109.00996 cs, math, 2021. [paper](http://arxiv.org/pdf/2109.00996.pdf)
Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networks , Yihang Gao, Michael K. Ng , arXiv:2108.13054 cs, math, 2021. [paper](http://arxiv.org/pdf/2108.13054.pdf)
Flow Field Tomography with Uncertainty Quantification using a Bayesian Physics-Informed Neural Network , Joseph P. Molnar, Samuel J. Grauer , arXiv:2108.09247 physics, 2021. [paper](http://arxiv.org/pdf/2108.09247.pdf)
Stochastic Physics-Informed Neural Networks (SPINN): A Moment-Matching Framework for Learning Hidden Physics within Stochastic Differential Equations , Jared O'Leary, Joel A. Paulson, Ali Mesbah , arXiv:2109.01621 cs, 2021. [paper](http://arxiv.org/pdf/2109.01621.pdf)
Spectral PINNs: Fast Uncertainty Propagation with Physics-Informed Neural Networks , Björn Lütjens, Catherine H. Crawford, Mark Veillette, Dava Newman , NIPS , 2021. [paper](https://openreview.net/forum?id=218sl_mPChc)
Robust Learning of Physics Informed Neural Networks , Ch Bajaj, rajit, Luke McLennan, Timothy Andeen, Avik Roy , Arxiv, 2021. [paper](https://arxiv.org/abs/2110.13330)
Bayesian Physics Informed Neural Networks for real-world nonlinear dynamical systems , Kevin Linka, Amelie Schäfer, Xuhui Meng, Zongren Zou, George EmKarniadakis, Ellen Kuhl , Computer Methods in Applied Mechanics and Engineering, 2022. [paper](https://doi.org/10.1016/j.cma.2022.115346)
Multi-output physics-informed neural networks for forward and inverse PDE problems with uncertainties , Mingyuan Yang, John T.Foster , Computer Methods in Applied Mechanics and Engineering, 2022. [paper](https://doi.org/10.1016/j.cma.2022.115041)
Papers on Applications
Physics-informed neural networks for high-speed flows , Zhiping Mao, Ameya D. Jagtap, George Em Karniadakis , Computer Methods in Applied Mechanics and Engineering, 2020. [paper](https://www.sciencedirect.com/science/article/abs/pii/S0045782519306814)
Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data , Luning Sun, Han Gao, Shaowu Pan, Jian-Xun Wang , Computer Methods in Applied Mechanics and Engineering, 2020. [paper](https://www.sciencedirect.com/science/article/abs/pii/S004578251930622X)
Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations , Maziar Raissi, Alireza Yazdani, George Em Karniadakis , Science, 2020. [paper](https://science.sciencemag.org/content/367/6481/1026.full)
NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations , Xiaowei Jin, Shengze Cai, Hui Li, George Em Karniadakis , Journal of Computational Physics, 2021. [paper](https://www.sciencedirect.com/science/article/pii/S0021999120307257)
A High-Efficient Hybrid Physics-Informed Neural Networks Based on Convolutional Neural Network , Zhiwei Fang , IEEE Transactions on Neural Networks and Learning Systems, 2021. [paper](https://ieeexplore.ieee.org/abstract/document/9403414)
A Study on a Feedforward Neural Network to Solve Partial Differential Equations in Hyperbolic-Transport Problems , Eduardo Abreu, Joao B. Florindo , ICCS, 2021. [paper](https://link.springer.com/chapter/10.1007/978-3-030-77964-1_31)
A Physics Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity , Mohammad Vahab, Ehsan Haghighat, Maryam Khaleghi, Nasser Khalili , arXiv:2108.07243 cs, 2021. [paper](http://arxiv.org/pdf/2108.07243.pdf)
Prediction of porous media fluid flow using physics informed neural networks , Muhammad M. Almajid, Moataz O. Abu-Alsaud , Journal of Petroleum Science and Engineering, 2021. [paper](https://linkinghub.elsevier.com/retrieve/pii/S0920410521008597)
Investigating a New Approach to Quasinormal Modes: Physics-Informed Neural Networks , Anele M. Ncube, Gerhard E. Harmsen, Alan S. Cornell , arXiv:2108.05867 gr-qc, 2021. [paper](http://arxiv.org/pdf/2108.05867.pdf)
Towards neural Earth system modelling by integrating artificial intelligence in Earth system science , Christopher Irrgang, Niklas Boers, Maike Sonnewald, Elizabeth A. Barnes, Christopher Kadow, Joanna Staneva, Jan Saynisch-Wagner , Nature Machine Intelligence, 2021. [paper](https://www.nature.com/articles/s42256-021-00374-3)
Physics-informed Neural Network for Nonlinear Dynamics in Fiber Optics , Xiaotian Jiang, Danshi Wang, Qirui Fan, Min Zhang, Chao Lu, Alan Pak Tao Lau , arXiv:2109.00526 physics, 2021. [paper](http://arxiv.org/pdf/2109.00526.pdf)
On Theory-training Neural Networks to Infer the Solution of Highly Coupled Differential Equations , M. Torabi Rad, A. Viardin, M. Apel , arXiv:2102.04890 physics, 2021. [paper](http://arxiv.org/pdf/2102.04890.pdf)
Theory-training deep neural networks for an alloy solidification benchmark problem , M. Torabi Rad, A. Viardin, G. J. Schmitz, M. Apel , arXiv:1912.09800 physics, 2019. [paper](http://arxiv.org/pdf/1912.09800.pdf)
Explicit physics-informed neural networks for nonlinear closure: The case of transport in tissues , Ehsan Taghizadeh, Helen M. Byrne, Brian D. Wood , Journal of Computational Physics, 2022. [paper](https://linkinghub.elsevier.com/retrieve/pii/S0021999121006768)
A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: comparison with finite element method , Shahed Rezaei, Ali Harandi, Ahmad Moeineddin, Bai-Xiang Xu, Stefanie Reese , arXiv:2206.13103 cs.CE, 2022. [paper](https://arxiv.org/abs/2206.13103)
A generalized framework for unsupervised learning and data recovery in computational fluid dynamics using discretized loss functions , Jot Singh Aulakh, Steven B. Beale, and Jon G. Pharoah , Physics of Fluids, 2022. [paper](https://doi.org/10.1063/5.0097480)
Physics-Informed Neural Networks for AC Optimal Power Flow , Rahul Nellikkath, Spyros Chatzivasileiadis , Electric Power Systems Research, 2022. [paper](https://doi.org/10.1016/j.epsr.2022.108412)
Physics-informed neural networks for the shallow-water equations on the sphere , Alex Bihlo, Roman O.Popovych , Journal of Computational Physics, 2022. [paper](https://doi.org/10.1016/j.jcp.2022.111024)
A Physics-Informed Machine Learning Approach for Estimating Lithium-Ion Battery Temperature , Gyouho Cho, Mengqi Wang, Youngki Kim, Jaerock Kwon, Wencong Su , IEEE Access, 2022. [paper](https://ieeexplore.ieee.org/document/9858911/)
Physically guided deep learning solver for time-dependent Fokker--Planck equation , Yang Zhang, Ka-Veng Yuen , International Journal of Non-Linear Mechanics, 2022. [paper](https://www.sciencedirect.com/science/article/pii/S0020746222001792)
A Physically Consistent Framework for Fatigue Life Prediction using Probabilistic Physics-Informed Neural Network , Taotao Zhou, Shan Jiang, Te Han, Shun-Peng Zhu, Yinan Cai , International Journal of Fatigue, 2022. [paper](https://www.sciencedirect.com/science/article/pii/S0142112322004844)
Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs , Siddhartha Mishra, Roberto Molinaro , IMA Journal of Numerical Analysis, 2021. [paper](https://academic.oup.com/imajna/advance-article-abstract/doi/10.1093/imanum/drab032/6297946)
Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs , Tim De Ryck, Siddhartha Mishra , arXiv:2106.14473 cs, math, 2021. [paper](https://arxiv.org/pdf/2106.14473.pdf)
Error Analysis of Deep Ritz Methods for Elliptic Equations , Yuling Jiao, Yanming Lai, Yisu Luo, Yang Wang, Yunfei Yang , arXiv:2107.14478 cs, math, 2021. [paper](https://arxiv.org/pdf/2107.14478.pdf)
Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces , George Stepaniants , arXiv:2108.11580 cs, math, stat, 2021. [paper](http://arxiv.org/pdf/2108.11580.pdf)
Simultaneous Neural Network Approximations in Sobolev Spaces , Sean Hon, Haizhao Yang , arXiv:2109.00161 cs, math, 2021. [paper](http://arxiv.org/pdf/2109.00161.pdf)
Characterizing possible failure modes in physics-informed neural networks , Aditi S. Krishnapriyan, Amir Gholami, Sh Zhe, ian, Robert M. Kirby, Michael W. Mahoney , arXiv:2109.01050 physics, 2021. [paper](http://arxiv.org/pdf/2109.01050.pdf)
Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks , Sifan Wang, Yujun Teng, Paris Perdikaris , SIAM Journal on Scientific Computing, 2021. [paper](https://epubs.siam.org/doi/10.1137/20M1318043)
Variational Physics Informed Neural Networks: the role of quadratures and test functions , Stefano Berrone, Claudio Canuto, Moreno Pintore , arXiv:2109.02035 cs, math, 2021. [paper](http://arxiv.org/pdf/2109.02035.pdf)
Convergence Analysis for the PINNs , Yuling Jiao, Yanming Lai, Dingwei Li, Xiliang Lu, Yang Wang, Jerry Zhijian Yang , arXiv:2109.01780 cs, math, 2021. [paper](http://arxiv.org/pdf/2109.01780.pdf)
Characterizing possible failure modes in physics-informed neural networks , Aditi Krishnapriyan, Amir Gholami, Sh Zhe, ian, Robert Kirby, Michael W. Mahoney , NIPS , 2021. [paper](https://openreview.net/forum?id=a2Gr9gNFD-J)
Convergence rate of DeepONets for learning operators arising from advection-diffusion equations , Beichuan Deng, Yeonjong Shin, Lu Lu, Zhongqiang Zhang, George Em Karniadakis , arXiv:2102.10621 math, 2021. [paper](https://arxiv.org/abs/2102.10621)
Estimates on the generalization error of physics-informed neural networks for approximating PDEs , Siddhartha Mishra, Roberto Molinaro , IMA Journal of Numerical Analysis, 2022. [paper](https://academic.oup.com/imajna/advance-article/doi/10.1093/imanum/drab093/6503953)
Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs) , Shamsulhaq Basir , arXiv:2209.09988v1cs , 2022 . [paper](https://arxiv.org/pdf/2209.09988.pdf)[code](https://github.com/shamsbasir/investigating_mitigating_failure_modes_in_pinns)
