综述类文献
1.深度学习在计算物理领域中的应用："Deep learning: a tool for computational nuclear physics." Negoita, Gianina Alina, et al. arXiv preprint arXiv:1803.03215 (2018).[paper]
总结了深度学习在核物理方面的应用，不包括解决微分方程的相关内容。
2.自动微分理论的综述： "Automatic differentiation in machine learning: a survey." Baydin, Atilim Gunes, et al.Journal of machine learning research 18 (2018).[paper]
自动微分是基于梯度的机器学习方法的基础。
3.知信机器学习（Informed ML）综述： "Informed Machine Learning--A Taxonomy and Survey of Integrating Knowledge into Learning Systems." von Rueden, Laura, et al. arXiv preprint arXiv:1903.12394 (2019).[paper]
自然科学、社会科学、专家知识、社会知识如何整合到机器学习中的分类与调查。
4.机器学习在复杂系统领域内的应用综述： "Introduction to Focus Issue: When machine learning meets complex systems: Networks, chaos, and nonlinear dynamics."Tang, Yang, et al. Chaos: An Interdisciplinary Journal of Nonlinear Science 30.6 (2020): 063151.[paper]
机器学习在复杂系统、非线性动力系统中的应用。
5.回顾了将物理学嵌入机器学习的一些普遍趋势：Physics-informed machine learning. Karniadakis, G.E., Kevrekidis, I.G., Lu, L. et al. Nat Rev Phys 3, 422–440 (2021). https://doi.org/10.1038/s42254-021-00314-5 [paper]
本文回顾了将物理嵌入机器学习的一些流行趋势，介绍了当前的一些能力和局限性，并讨论了物理的各种应用-正问题和反问题的知情学习，包括发现隐藏的物理和解决高维度问题。
6.混合学习方法综述： "Driven by Data or Derived Through Physics? A Review of Hybrid Physics Guided Machine Learning Techniques With Cyber-Physical System (CPS) Focus," R. Rai and C. K. Sahu,in IEEE Access, vol. 8, pp. 71050-71073, 2020, doi: 10.1109/ACCESS.2020.2987324.[paper]
这篇文章有点奇奇怪怪的。
7.用深度学习解决微分方程的三个热点方向： "Three Ways to Solve Partial Differential Equations with Neural Networks -- A Review." Blechschmidt, Jan , and O. G. Ernst .(2021).[paper][code]
三个热点方向：物理信息神经网络，基于Feynman-Kac公式的方法以及基于倒向随机微分方程解的方法。
8.PINN方法的总结和展望：基于物理信息的神经网络：最新进展与展望
语言简洁，但内容全面。
理论研究类文献
1.使用人工神经网络解决微分方程（1998）：Artificial neural networks for solving ordinary and partial differential equations | IEEE Journals & Magazine | IEEE Xplore
2.动力学方程的流体动力学近似，并证明简化模型在从流体动力学极限到自由分子流动的广泛Knudsen数范围内实现了均匀的精度：Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations
3.求解偏微分方程的数值问题被转换为无约束最小化问题（1994）：Neural‐network‐based approximations for solving partial differential equations - Dissanayake - 1994 - Communications in Numerical Methods in Engineering - Wiley Online Library
4.神经网络的万能近似定理：ニューラルネットワークの万能近似定理
Physics-Informed Neural Network 相关文献
PINN模型的研究：
1.PINN的提出：Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations [paper][code]
2.PINN的提出：Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations[paper][code]
3.PINN的提出：Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[paper]
4.mPINN的提出和应用：A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE[paper][code]
使用多个神经网络来实现PINN在低保真数据集上的训练。
5.将高斯过程和Physics-informed ML结合的方法：(PDF) Physics-Informed Learning Machines for Partial Differential Equations: Gaussian Processes Versus Neural Networks (researchgate.net)
6.SPINN的提出和应用: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs, Amuthan A. Ramabathiran, Ramach, Prabhu ran, Journal of Computational Physics, 2021. [paper][code]
7.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]
8.NH-PINN的提出和应用: Neural homogenization based physics-informed neural network for multiscale problems, Wing Tat Leung, Guang Lin, Zecheng Zhang, arXiv:2108.12942 [cs, math], 2021. [paper]
9.Physics-Augmented Learning: A New Paradigm Beyond Physics-Informed Learning, Ziming Liu, Yunyue Chen, Yuanqi Du, Max Tegmark, arXiv:2109.13901 [physics], 2021. [paper]
10.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]
11.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]
12.HyperPINN的提出和应用: Learning parameterized differential equations with physics-informed hypernetworks, Filipe de Avila Belbute-Peres, Yi-fan Chen, Fei Sha, NIPS, 2021. [paper]
13.基于正交基构建的损失函数： “NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations.” ArXiv abs/2107.09443 (2021): n. pag.[paper][code]（Julia实现）
PINN并行化的研究
13.Parallel Physics-Informed Neural Networks via Domain Decomposition, Khemraj Shukla, Ameya D. Jagtap, George Em Karniadakis, arXiv:2104.10013 [cs], 2021. [paper]
14.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]
15.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]
16.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]
17.Scaling physics-informed neural networks to large domains by using domain decomposition, Ben Moseley, Andrew Markham, Tarje Nissen-Meyer, NIPS, 2021. [paper]
18.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]
PINN加速的研究
19.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]
20.A Dual-Dimer method for training physics-constrained neural networks with minimax architecture, Dehao Liu, Yan Wang, Neural Networks, 2021. [paper]
21.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]
22.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]
23.Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems, Jeremy Yu, Lu Lu, Xuhui Meng, George Em Karniadakis, Arxiv, 2021. [paper]
24.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]
25.A hybrid physics-informed neural network for nonlinear partial differential equation, Chunyue Lv, Lei Wang, Chenming Xie, Arxiv, 2021. [paper]
26.Multi-Objective Loss Balancing for Physics-Informed Deep Learning, Rafael Bischof, Michael Kraus, Arxiv, 2021. [paper]
模型迁移和元学习
27.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]
28.Transfer learning based multi-fidelity physics informed deep neural network, Souvik Chakraborty, Journal of Computational Physics, 2021. [paper]
29.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]
30.Meta-learning PINN loss functions, Apostolos F. Psaros, Kenji Kawaguchi, George Em Karniadakis, arXiv:2107.05544 [cs], 2021. [paper]
31.Physics-Informed Neural Networks (PINNs) for Parameterized PDEs: A Metalearning Approach, Michael Penwarden, Sh Zhe, ian, Akil Narayan, Robert M. Kirby, Arxiv, 2021. [paper]
概率PINN和不确定性量化
32.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]
33.Adversarial uncertainty quantification in physics-informed neural networks, Yibo Yang, Paris Perdikaris, Journal of Computational Physics, 2019. [paper]
34.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]
35.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]
36.Quantifying Uncertainty in Physics-Informed Variational Autoencoders for Anomaly Detection, Marcus J. Neuer, ESTEP, 2020. [paper]
37.A Physics-Data-Driven Bayesian Method for Heat Conduction Problems, Xinchao Jiang, Hu Wang, Yu li, arXiv:2109.00996 [cs, math], 2021. [paper]
38.Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networks, Yihang Gao, Michael K. Ng, arXiv:2108.13054 [cs, math], 2021. [paper]
39.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]
40.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]
41.Spectral PINNs: Fast Uncertainty Propagation with Physics-Informed Neural Networks, Björn Lütjens, Catherine H. Crawford, Mark Veillette, Dava Newman, NIPS, 2021. [paper]
42.Robust Learning of Physics Informed Neural Networks, Ch Bajaj, rajit, Luke McLennan, Timothy Andeen, Avik Roy, Arxiv, 2021. [paper]
PINN的应用
43.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]
44.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]
45.Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, Maziar Raissi, Alireza Yazdani, George Em Karniadakis, Science, 2020. [paper]
46.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]
47.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]
48.A Study on a Feedforward Neural Network to Solve Partial Differential Equations in Hyperbolic-Transport Problems, Eduardo Abreu, Joao B. Florindo, ICCS, 2021. [paper]
49.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]
50.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]
51.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]
52.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]
53.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]
54.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]
55.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]
56.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]
PINN分析
57.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]
58.Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs, Tim De Ryck, Siddhartha Mishra, arXiv:2106.14473 [cs, math], 2021. [paper]
59.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]
60.Learning Partial Differential Equations in Reproducing Kernel Hilbert Spaces, George Stepaniants, arXiv:2108.11580 [cs, math, stat], 2021. [paper]
61.Simultaneous Neural Network Approximations in Sobolev Spaces, Sean Hon, Haizhao Yang, arXiv:2109.00161 [cs, math], 2021. [paper]
62.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]
63.Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks, Sifan Wang, Yujun Teng, Paris Perdikaris, SIAM Journal on Scientific Computing, 2021. [paper]
64.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]
65.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]
66.Characterizing possible failure modes in physics-informed neural networks, Aditi Krishnapriyan, Amir Gholami, Sh Zhe, ian, Robert Kirby, Michael W. Mahoney, NIPS, 2021. [paper]
67.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]
68.Estimates on the generalization error of physics-informed neural networks for approximating PDEs, Siddhartha Mishra, Roberto Molinaro, IMA Journal of Numerical Analysis, 2022. [paper]
纯数据驱动解微分方程相关文献
1.使用稀疏回归方法求解微分方程：Data-driven discovery of partial differential equations (science.org)
2.以逼近映射的方式求解微分方程(DeepONet)：Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators | Nature Machine Intelligence
3.将深度神经网络和稀疏回归方法结合的DL-PDE算法：DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data
4.数据驱动的离散化，这是一种基于已知基础方程的实际解来学习偏微分方程的优化近似的方法：Learning data-driven discretizations for partial differential equations
5.用 DNN 参数化 Green 函数，并根据 Green 的方法获取神经算子来近似解：MOD-Net: A Machine Learning Approach via ModelOperator-Data Network for Solving PDEs
解决高维度微分方程相关文献：
1.针对高维椭圆形偏微分方程和倒向随机微分方程的深度学习数值方法：Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
2.深度伽辽金方法和一类准线性抛物线偏微分方程的神经网络近似能力的定理的证明：DGM: A deep learning algorithm for solving partial differential equations
3.用于数值解决变分问题，特别是由偏微分方程引起的问题：The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems
解决非线性偏微分方程算法相关文献：
1.依赖高斯过程解决非线性偏微分方程，在小样本下给定一定形式的偏微分方程，学习偏微分方程未知的参数：Hidden physics models: Machine learning of nonlinear partial differential equations - ScienceDirect
2.依赖高斯过程解决分数阶、积分微分算子：Machine learning of linear differential equations using Gaussian processes - ScienceDirect
3.直接从噪声数据中识别非线性偏微分方程形式的时空动力学模型：Robust and optimal sparse regression for nonlinear PDE models
应用类文献
1.可以有效地学习具有深度网络的机械系统（即系统动力学）运动方程的深度拉格朗日网络：Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning[code][code]
2.用于欠驱动系统能量控制端到端学习的深拉格朗日网络：Deep Lagrangian Networks for end-to-end learning of energy-based control for under-actuated systems
3.流体力学中的流动可视化问题：Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
4.基于pytorch和tensorflow实现PINN的Library：DeepXDE: A Deep Learning Library for Solving Differential Equations
合计 94 篇
开源项目：
GitHub - idrl-lab/PINNpapers: Must-read Papers on Physics-Informed Neural Networks.
不定期更新！