# Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications

arXiv: https://arxiv.org/abs/2211.08064 · HTML: https://mlsanity.com/paper/2211.08064/

## Key facts

- Cited by 92 papers in the map (data through 14 Oct 2026).
- Builds on 210 of its 348 references that are in the map.
- First posted on arXiv 15 Nov 2022.

## Builds on (most cited first, 30 of 210)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Distilling the Knowledge in a Neural Network](https://mlsanity.com/paper/1503.02531/) (2015)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [When Does Label Smoothing Help?](https://mlsanity.com/paper/1906.02629/) (2019)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [A Survey on Multi-Task Learning](https://mlsanity.com/paper/1707.08114/) (2017)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)

## Built on by (most cited first, 30 of 92)

- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [FINER: Flexible spectral-bias tuning in Implicit NEural Representation by Variable-periodic Activation Functions](https://mlsanity.com/paper/2312.02434/) (2023)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [What Has a Foundation Model Found? Using Inductive Bias to Probe for World Models](https://mlsanity.com/paper/2507.06952/) (2025)
- [Physics-Informed Computer Vision: A Review and Perspectives](https://mlsanity.com/paper/2305.18035/) (2023)
- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)
- [RoPINN: Region Optimized Physics-Informed Neural Networks](https://mlsanity.com/paper/2405.14369/) (2024)
- [A Survey on Physics Informed Reinforcement Learning: Review and Open Problems](https://mlsanity.com/paper/2309.01909/) (2023)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Large language models for artificial general intelligence (AGI): A survey of foundational principles and approaches](https://mlsanity.com/paper/2501.03151/) (2025)
- [MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networks](https://mlsanity.com/paper/2306.02816/) (2023)
- [NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data](https://mlsanity.com/paper/2305.18694/) (2023)
- [Accelerating PDE Data Generation via Differential Operator Action in Solution Space](https://mlsanity.com/paper/2402.05957/) (2024)
- [Preconditioning for Physics-Informed Neural Networks](https://mlsanity.com/paper/2402.00531/) (2024)
- [DeepPhysiNet: Bridging Deep Learning and Atmospheric Physics for Accurate and Continuous Weather Modeling](https://mlsanity.com/paper/2401.04125/) (2024)
- [Accelerating Data Generation for Neural Operators via Krylov Subspace Recycling](https://mlsanity.com/paper/2401.09516/) (2024)
- [FINER++: Building a Family of Variable-periodic Functions for Activating Implicit Neural Representation](https://mlsanity.com/paper/2407.19434/) (2024)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [PIETRA: Physics-Informed Evidential Learning for Traversing Out-of-Distribution Terrain](https://mlsanity.com/paper/2409.03005/) (2024)
- [NeuralClothSim: Neural Deformation Fields Meet the Thin Shell Theory](https://mlsanity.com/paper/2308.12970/) (2023)
- [Inverse Design in Nanophotonics via Representation Learning](https://mlsanity.com/paper/2507.00546/) (2025)
- [Physics-Encoded Graph Neural Networks for Deformation Prediction under Contact](https://mlsanity.com/paper/2402.03466/) (2024)
- [ProPINN: Demystifying Propagation Failures in Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.00803/) (2025)
- [Sobolev Training for Operator Learning](https://mlsanity.com/paper/2402.09084/) (2024)
- [Deep adaptive sampling for surrogate modeling without labeled data](https://mlsanity.com/paper/2402.11283/) (2024)
- [PINNs for Medical Image Analysis: A Survey](https://mlsanity.com/paper/2408.01026/) (2024)
- [DAC-MR: Data Augmentation Consistency Based Meta-Regularization for Meta-Learning](https://mlsanity.com/paper/2305.07892/) (2023)
- [The Hard-Constraint PINNs for Interface Optimal Control Problems](https://mlsanity.com/paper/2308.06709/) (2023)

## Abstract (arXiv)

Recent advances of data-driven machine learning have revolutionized fields like computer vision, reinforcement learning, and many scientific and engineering domains. In many real-world and scientific problems, systems that generate data are governed by physical laws. Recent work shows that it provides potential benefits for machine learning models by incorporating the physical prior and collected data, which makes the intersection of machine learning and physics become a prevailing paradigm. By integrating the data and mathematical physics models seamlessly, it can guide the machine learning model towards solutions that are physically plausible, improving accuracy and efficiency even in uncertain and high-dimensional contexts. In this survey, we present this learning paradigm called Physics-Informed Machine Learning (PIML) which is to build a model that leverages empirical data and available physical prior knowledge to improve performance on a set of tasks that involve a physical mechanism. We systematically review the recent development of physics-informed machine learning from three perspectives of machine learning tasks, representation of physical prior, and methods for incorporating physical prior. We also propose several important open research problems based on the current trends in the field. We argue that encoding different forms of physical prior into model architectures, optimizers, inference algorithms, and significant domain-specific applications like inverse engineering design and robotic control is far from being fully explored in the field of physics-informed machine learning. We believe that the interdisciplinary research of physics-informed machine learning will significantly propel research progress, foster the creation of more effective machine learning models, and also offer invaluable assistance in addressing long-standing problems in related disciplines.