# Physics-Guided Deep Learning for Dynamical Systems: A Survey

arXiv: https://arxiv.org/abs/2107.01272 · HTML: https://mlsanity.com/paper/2107.01272/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- Builds on 112 of its 236 references that are in the map.
- First posted on arXiv 2 Jul 2021.

## Builds on (most cited first, 30 of 112)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [End-to-End Object Detection with Transformers](https://mlsanity.com/paper/2005.12872/) (2020)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Understanding deep learning requires rethinking generalization](https://mlsanity.com/paper/1611.03530/) (2016)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Learning Transferable Features with Deep Adaptation Networks](https://mlsanity.com/paper/1502.02791/) (2015)
- [Distributionally Robust Neural Networks for Group Shifts: On the Importance of Regularization for Worst-Case Generalization](https://mlsanity.com/paper/1911.08731/) (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)
- [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)
- [Learning to Generalize: Meta-Learning for Domain Generalization](https://mlsanity.com/paper/1710.03463/) (2017)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Dynamic Routing Between Capsules](https://mlsanity.com/paper/1710.09829/) (2017)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [ConViT: Improving Vision Transformers with Soft Convolutional Inductive Biases](https://mlsanity.com/paper/2103.10697/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)

## Built on by (most cited first, 30 of 44)

- [Compute-Efficient Deep Learning: Algorithmic Trends and Opportunities](https://mlsanity.com/paper/2210.06640/) (2022)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [Stable Port-Hamiltonian Neural Networks](https://mlsanity.com/paper/2502.02480/) (2025)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Physics-Regulated Deep Reinforcement Learning: Invariant Embeddings](https://mlsanity.com/paper/2305.16614/) (2023)
- [PhyDA: Physics-Guided Diffusion Models for Data Assimilation in Atmospheric Systems](https://mlsanity.com/paper/2505.12882/) (2025)
- [Full Shot Predictions for the DIII-D Tokamak via Deep Recurrent Networks](https://mlsanity.com/paper/2404.12416/) (2024)
- [Learning Neural Hamiltonian Dynamics: A Methodological Overview](https://mlsanity.com/paper/2203.00128/) (2022)
- [Deep Grey-Box Modeling With Adaptive Data-Driven Models Toward Trustworthy Estimation of Theory-Driven Models](https://mlsanity.com/paper/2210.13103/) (2022)
- [Approaching epidemiological dynamics of COVID-19 with physics-informed neural networks](https://mlsanity.com/paper/2302.08796/) (2023)
- [Learning stochastic dynamics and predicting emergent behavior using transformers](https://mlsanity.com/paper/2202.08708/) (2022)
- [A Review on Organ Deformation Modeling Approaches for Reliable Surgical Navigation using Augmented Reality](https://mlsanity.com/paper/2408.02713/) (2024)
- [Symplectic Neural Flows for Modeling and Discovery](https://mlsanity.com/paper/2412.16787/) (2024)
- [Stochastic and Non-local Closure Modeling for Nonlinear Dynamical Systems via Latent Score-based Generative Models](https://mlsanity.com/paper/2506.20771/) (2025)
- [Out of Distribution Detection via Domain-Informed Gaussian Process State Space Models](https://mlsanity.com/paper/2309.06655/) (2023)
- [Learning Adaptive Hydrodynamic Models Using Neural ODEs in Complex Conditions](https://mlsanity.com/paper/2410.00490/) (2024)
- [Generalizable and Fast Surrogates: Model Predictive Control of Articulated Soft Robots using Physics-Informed Neural Networks](https://mlsanity.com/paper/2502.01916/) (2025)
- [Adaptive Symmetry Discovery for Dynamical System Identification](https://mlsanity.com/paper/2608.08091/) (2026)
- [CONFIDE: Contextual Finite Differences Modelling of PDEs](https://mlsanity.com/paper/2303.15827/) (2023)
- [Machine learning for phase-resolved reconstruction of nonlinear ocean wave surface elevations from sparse remote sensing data](https://mlsanity.com/paper/2305.11913/) (2023)
- [Mitigating distribution shift in machine learning-augmented hybrid simulation](https://mlsanity.com/paper/2401.09259/) (2024)
- [From Sparse Sensors to Continuous Fields: STRIDE for Spatiotemporal Reconstruction](https://mlsanity.com/paper/2602.04201/) (2026)
- [Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings](https://mlsanity.com/paper/2602.16193/) (2026)
- [PnP-Corrector: A Universal Correction Framework for Coupled Spatiotemporal Forecasting](https://mlsanity.com/paper/2605.08935/) (2026)
- [Latent Space Simulation for Carbon Capture Design Optimization](https://mlsanity.com/paper/2112.11656/) (2021)
- [PINF: Continuous Normalizing Flows for Physics-Constrained Deep Learning](https://mlsanity.com/paper/2309.15139/) (2023)
- [Data-driven discovery of self-similarity using neural networks](https://mlsanity.com/paper/2406.03896/) (2024)
- [Guiding Video Prediction with Explicit Procedural Knowledge](https://mlsanity.com/paper/2406.18220/) (2024)
- [Physics Informed Constrained Learning of Dynamics from Static Data](https://mlsanity.com/paper/2504.12675/) (2025)

## Abstract (arXiv)

Modeling complex physical dynamics is a fundamental task in science and engineering. Traditional physics-based models are sample efficient, and interpretable but often rely on rigid assumptions. Furthermore, direct numerical approximation is usually computationally intensive, requiring significant computational resources and expertise, and many real-world systems do not have fully-known governing laws. While deep learning (DL) provides novel alternatives for efficiently recognizing complex patterns and emulating nonlinear dynamics, its predictions do not necessarily obey the governing laws of physical systems, nor do they generalize well across different systems. Thus, the study of physics-guided DL emerged and has gained great progress. Physics-guided DL aims to take the best from both physics-based modeling and state-of-the-art DL models to better solve scientific problems. In this paper, we provide a structured overview of existing methodologies of integrating prior physical knowledge or physics-based modeling into DL, with a special emphasis on learning dynamical systems. We also discuss the fundamental challenges and emerging opportunities in the area.