# Machine Learning for Partial Differential Equations

arXiv: https://arxiv.org/abs/2303.17078 · HTML: https://mlsanity.com/paper/2303.17078/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 141 references that are in the map.
- First posted on arXiv 30 Mar 2023.

## Builds on (most cited first, 30 of 46)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition](https://mlsanity.com/paper/1710.04340/) (2017)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Gauge Equivariant Mesh CNNs: Anisotropic convolutions on geometric graphs](https://mlsanity.com/paper/2003.05425/) (2020)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)

## Built on by (most cited first, 21 of 21)

- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions](https://mlsanity.com/paper/2306.00945/) (2023)
- [Neural SPH: Improved Neural Modeling of Lagrangian Fluid Dynamics](https://mlsanity.com/paper/2402.06275/) (2024)
- [Pseudo-Hamiltonian neural networks for learning partial differential equations](https://mlsanity.com/paper/2304.14374/) (2023)
- [Speeding up and reducing memory usage for scientific machine learning via mixed precision](https://mlsanity.com/paper/2401.16645/) (2024)
- [Nonlinear model reduction for operator learning](https://mlsanity.com/paper/2403.18735/) (2024)
- [BiLO: Bilevel Local Operator Learning for PDE Inverse Problems](https://mlsanity.com/paper/2404.17789/) (2024)
- [SIMSHIFT: A Benchmark for Adapting Neural Surrogates to Distribution Shifts](https://mlsanity.com/paper/2506.12007/) (2025)
- [An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning](https://mlsanity.com/paper/2408.13222/) (2024)
- [JAX-SPH: A Differentiable Smoothed Particle Hydrodynamics Framework](https://mlsanity.com/paper/2403.04750/) (2024)
- [Data-driven identification of latent port-Hamiltonian systems](https://mlsanity.com/paper/2408.08185/) (2024)
- [EquiNO: A Physics-Informed Neural Operator for Multiscale Simulations](https://mlsanity.com/paper/2504.07976/) (2025)
- [Gradient Flow Based Phase-Field Modeling Using Separable Neural Networks](https://mlsanity.com/paper/2405.06119/) (2024)
- [Towards scientific machine learning for granular material simulations -- challenges and opportunities](https://mlsanity.com/paper/2504.08766/) (2025)
- [Stable Long-Horizon PDE Forecasting via Latent Structured Spectral Propagators](https://mlsanity.com/paper/2605.10154/) (2026)
- [Reducing Spatial Discretization Error on Coarse CFD Simulations Using an OpenFOAM-Embedded Deep Learning Framework](https://mlsanity.com/paper/2405.07441/) (2024)
- [From Data to Laws: Neural Discovery of Conservation Laws Without False Positives](https://mlsanity.com/paper/2603.20474/) (2026)
- [Stability and Discretization Error of State Space Model Neural Operators](https://mlsanity.com/paper/2605.18905/) (2026)
- [Paving the Way for Point Cloud Video Representation Learning Using A PDE Model](https://mlsanity.com/paper/2606.01604/) (2026)

## Abstract (arXiv)

Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This review will examine several promising avenues of PDE research that are being advanced by machine learning, including: 1) the discovery of new governing PDEs and coarse-grained approximations for complex natural and engineered systems, 2) learning effective coordinate systems and reduced-order models to make PDEs more amenable to analysis, and 3) representing solution operators and improving traditional numerical algorithms. In each of these fields, we summarize key advances, ongoing challenges, and opportunities for further development.