The following is a list of resources on PINNS, taken from https://github.com/idrl-lab/PINNpapers # PINNpapers Contributed by [IDRL lab](https://github.com/idrl-lab). - [PINNpapers](#pinnpapers) - [Introduction](#introduction) - [Software](#software) - [Papers on PINN Models](#papers-on-pinn-models) - [Papers on Parallel PINN](#papers-on-parallel-pinn) - [Papers on PINN Accerleration](#papers-on-pinn-accerleration) - [Papers on Model Transfer & Meta-Learning](#papers-on-model-transfer--meta-learning) - [Papers on Probabilistic PINNs and Uncertainty Quantification](#papers-on-probabilistic-pinns-and-uncertainty-quantification) - [Papers on Applications](#papers-on-applications) - [Papers on PINN Analysis](#papers-on-pinn-analysis) ## Software 1. **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)] 2. **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)] 3. **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)] 4. **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)] 5. **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)] 6. **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]()] 7. **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)] 8. **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)] 9. **NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations**, *Kirill Zubov, Zoe McCarthy, Yingbo Ma, Francesco Calisto, Valerio Pagliarino, Simone Azeglio, Luca Bottero, Emmanuel Luján, Valentin Sulzer, Ashutosh Bharambe, N Vinchhi, , Kaushik Balakrishnan, Devesh Upadhyay, Chris Rackauckas*, arXiv:2107.09443 [cs], 2021. [[paper](https://arxiv.org/pdf/2107.09443)][[code](https://github.com/SciML/NeuralPDE.jl)] 10. **IDRLnet: A Physics-Informed Neural Network Library**, *Wei Peng, Jun Zhang, Weien Zhou, Xiaoyu Zhao, Wen Yao, Xiaoqian Chen*, arXiv:2107.04320 [cs, math], 2021. [[paper](https://arxiv.org/pdf/2107.04320.pdf)][[code](https://github.com/idrl-lab/idrlnet)] 11. **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 1. **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)] 2. **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)] 3. **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)] 4. **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)] 5. **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)] 6. **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)] 7. **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)] 8. **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)] 9. **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)] 10. **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)] 11. **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)] 12. **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)] 13. **PhyGNNet: Solving spatiotemporal PDEs with Physics-informed Graph Neural Network**, *Longxiang Jiang, Liyuan Wang, Xinkun Chu, Yonghao Xiao and Hao Zhang*, arXiv:2208.04319 [cs.NE], 2022. [[paper](https://arxiv.org/abs/2208.04319)] 14. **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)] 15. **∆-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)] 16. **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 1. **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)] 2. **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)] 3. **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)] 4. **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)] 5. **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)] 6. **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)] 7. **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)] 8. **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 1. **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)] 2. **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)] 3. **Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations**, *Pongpisit Thanasutives, Masayuki Numao, Ken-ichi Fukui*, arXiv:2104.14320 [cs, math], 2021. [[paper](https://arxiv.org/pdf/2104.14320)] 4. **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)] 5. **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)] 6. **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)] 7. **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)] 8. **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)] 9. **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)] 10. **RPINNs: Rectified-physics informed neural networks for solving stationary partial differential equations**, *Pai Peng, Jiangong Pan, Hui Xu, Xinlong Feng*, Computers & Fluids, 2022. [[paper](https://doi.org/10.1016/j.compfluid.2022.105583)] 11. **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)] 12. **A Novel Adaptive Causal Sampling Method for Physics-Informed Neural Networks**, *Jia Guo, Haifeng Wang, Chenping Hou*, arXiv:2210.12914 [cs], 2022. [[paper](https://arxiv.org/pdf/2210.12914.pdf)] 13. **Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations**, *Ramansh Sharma, Varun Shankar*, NeurIPS, 2022. [[paper](https://arxiv.org/abs/2205.09332)] 14. **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 1. **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)] 2. **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)] 3. **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)] 4. **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)] 5. **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)] 6. **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 1. **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)] 2. **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)] 3. **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)] 4. **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)] 5. **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)] 6. **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)] 7. **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)] 8. **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)] 9. **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)] 10. **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)] 11. **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)] 12. **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)] 13. **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 1. **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)] 2. **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)] 3. **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)] 4. **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)] 5. **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)] 6. **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)] 7. **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)] 8. **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)] 9. **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)] 10. **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)] 11. **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)] 12. **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)] 13. **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)] 14. **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)] 15. **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)] 16. **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)] 17. **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)] 18. **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)] 19. **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/)] 20. **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)] 21. **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)] 22. **Inverse modeling of nonisothermal multiphase poromechanics using physics-informed neural networks**, *Danial Amini, Ehsan Haghighat, Ruben Juanes*, **Arxiv**, 2022. [[paper](http://arxiv.org/pdf/2209.03276.pdf)][[code](https://github.com/sciann/sciann-applications/tree/master/SciANN-PoroElasticity))] ## Papers on PINN Analysis 1. **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)] 2. **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)] 3. **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)] 4. **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)] 5. **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)] 6. **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)] 7. **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)] 8. **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)] 9. **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)] 10. **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)] 11. **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)] 12. **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)] 13. **Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)**, *Shamsulhaq Basir*, **arXiv:2209.09988v1[cs]**, **2022**. [[paper](https://arxiv.org/pdf/2209.09988.pdf)][[code](https://github.com/shamsbasir/investigating_mitigating_failure_modes_in_pinns)]