綜述類文獻
1.深度學習在計算物理領域中的應用："Deep learning: a tool for computational nuclear physics." Negoita, Gianina Alina, et al. arXiv preprint arXiv:1803.03215 (2018).[paper]
總結了深度學習在核物理方面的應用，不包括解決微分方程的相關內容。
2.自動微分理論的綜述： "Automatic differentiation in machine learning: a survey." Baydin, Atilim Gunes, et al.Journal of machine learning research 18 (2018).[paper]
自動微分是基於梯度的機器學習方法的基礎。
3.知信機器學習（Informed ML）綜述： "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]
自然科學、社會科學、專家知識、社會知識如何整合到機器學習中的分類與調查。
4.機器學習在復雜系統領域內的應用綜述： "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]
機器學習在復雜系統、非線性動力系統中的應用。
5.回顧了將物理學嵌入機器學習的一些普遍趨勢：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]
本文回顧了將物理嵌入機器學習的一些流行趨勢，介紹了當前的一些能力和局限性，並討論了物理的各種應用-正問題和反問題的知情學習，包括發現隱藏的物理和解決高維度問題。
6.混合學習方法綜述： "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]
這篇文章有點奇奇怪怪的。
7.用深度學習解決微分方程的三個熱點方向： "Three Ways to Solve Partial Differential Equations with Neural Networks -- A Review." Blechschmidt, Jan , and O. G. Ernst .(2021).[paper][code]
三個熱點方向：物理信息神經網絡，基於Feynman-Kac公式的方法以及基於倒向隨機微分方程解的方法。
8.PINN方法的總結和展望：基於物理信息的神經網絡：最新進展與展望
語言簡潔，但內容全面。
理論研究類文獻
1.使用人工神經網絡解決微分方程（1998）：Artificial neural networks for solving ordinary and partial differential equations | IEEE Journals & Magazine | IEEE Xplore
2.動力學方程的流體動力學近似，並證明簡化模型在從流體動力學極限到自由分子流動的廣泛Knudsen數范圍內實現了均勻的精度：Uniformly Accurate Machine Learning Based Hydrodynamic Models for Kinetic Equations
3.求解偏微分方程的數值問題被轉換為無約束最小化問題（1994）：Neural‐network‐based approximations for solving partial differential equations - Dissanayake - 1994 - Communications in Numerical Methods in Engineering - Wiley Online Library
4.神經網絡的萬能近似定理：ニューラルネットワークの萬能近似定理
Physics-Informed Neural Network 相關文獻
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的提出和應用：A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE[paper][code]
使用多個神經網絡來實現PINN在低保真數據集上的訓練。
5.將高斯過程和Physics-informed ML結合的方法：(PDF) Physics-Informed Learning Machines for Partial Differential Equations: Gaussian Processes Versus Neural Networks (researchgate.net)
6.SPINN的提出和應用: 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的提出和應用: 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的提出和應用: Learning parameterized differential equations with physics-informed hypernetworks, Filipe de Avila Belbute-Peres, Yi-fan Chen, Fei Sha, NIPS, 2021. [paper]
13.基於正交基構建的損失函數： “NeuralPDE: Automating Physics-Informed Neural Networks (PINNs) with Error Approximations.” ArXiv abs/2107.09443 (2021): n. pag.[paper][code]（Julia實現）
PINN並行化的研究
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]
模型遷移和元學習
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和不確定性量化
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的應用
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]
純數據驅動解微分方程相關文獻
1.使用稀疏回歸方法求解微分方程：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.將深度神經網絡和稀疏回歸方法結合的DL-PDE算法：DL-PDE: Deep-learning based data-driven discovery of partial differential equations from discrete and noisy data
4.數據驅動的離散化，這是一種基於已知基礎方程的實際解來學習偏微分方程的優化近似的方法：Learning data-driven discretizations for partial differential equations
5.用 DNN 參數化 Green 函數，並根據 Green 的方法獲取神經算子來近似解：MOD-Net: A Machine Learning Approach via ModelOperator-Data Network for Solving PDEs
解決高維度微分方程相關文獻：
1.針對高維橢圓形偏微分方程和倒向隨機微分方程的深度學習數值方法：Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
2.深度伽遼金方法和一類准線性拋物線偏微分方程的神經網絡近似能力的定理的證明：DGM: A deep learning algorithm for solving partial differential equations
3.用於數值解決變分問題，特別是由偏微分方程引起的問題：The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems
解決非線性偏微分方程算法相關文獻：
1.依賴高斯過程解決非線性偏微分方程，在小樣本下給定一定形式的偏微分方程，學習偏微分方程未知的參數：Hidden physics models: Machine learning of nonlinear partial differential equations - ScienceDirect
2.依賴高斯過程解決分數階、積分微分算子：Machine learning of linear differential equations using Gaussian processes - ScienceDirect
3.直接從噪聲數據中識別非線性偏微分方程形式的時空動力學模型：Robust and optimal sparse regression for nonlinear PDE models
應用類文獻
1.可以有效地學習具有深度網絡的機械系統（即系統動力學）運動方程的深度拉格朗日網絡：Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning[code][code]
2.用於欠驅動系統能量控制端到端學習的深拉格朗日網絡：Deep Lagrangian Networks for end-to-end learning of energy-based control for under-actuated systems
3.流體力學中的流動可視化問題：Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
4.基於pytorch和tensorflow實現PINN的Library：DeepXDE: A Deep Learning Library for Solving Differential Equations
合計 94 篇
開源項目：
GitHub - idrl-lab/PINNpapers: Must-read Papers on Physics-Informed Neural Networks.
不定期更新！