# Encoding physics to learn reaction-diffusion processes

arXiv: https://arxiv.org/abs/2106.04781 · HTML: https://mlsanity.com/paper/2106.04781/

## Key facts

- Cited by 61 papers in the map (data through 14 Oct 2026).
- Builds on 57 of its 114 references that are in the map.
- First posted on arXiv 9 Jun 2021.

## Builds on (most cited first, 30 of 57)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [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)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [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)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [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)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Deep Spatio-Temporal Residual Networks for Citywide Crowd Flows Prediction](https://mlsanity.com/paper/1610.00081/) (2016)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Deep Learning: A Critical Appraisal](https://mlsanity.com/paper/1801.00631/) (2018)
- [FractalNet: Ultra-Deep Neural Networks without Residuals](https://mlsanity.com/paper/1605.07648/) (2016)
- [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)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)

## Built on by (most cited first, 30 of 61)

- [PhyMPGN: Physics-encoded Message Passing Graph Network for spatiotemporal PDE systems](https://mlsanity.com/paper/2410.01337/) (2024)
- [P2C2Net: PDE-Preserved Coarse Correction Network for efficient prediction of spatiotemporal dynamics](https://mlsanity.com/paper/2411.00040/) (2024)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [PiRD: Physics-informed Residual Diffusion for Flow Field Reconstruction](https://mlsanity.com/paper/2404.08412/) (2024)
- [Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations](https://mlsanity.com/paper/2304.04234/) (2023)
- [PGODE: Towards High-quality System Dynamics Modeling](https://mlsanity.com/paper/2311.06554/) (2023)
- [Observation-Guided Meteorological Field Downscaling at Station Scale: A Benchmark and a New Method](https://mlsanity.com/paper/2401.11960/) (2024)
- [Spectral Informed Neural Network: An Efficient and Low-Memory PINN](https://mlsanity.com/paper/2408.16414/) (2024)
- [DCEM: A deep complementary energy method for solid mechanics](https://mlsanity.com/paper/2302.01538/) (2023)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Advancing Generalizability by the Baldwin Effect](https://mlsanity.com/paper/2312.03243/) (2023)
- [PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems](https://mlsanity.com/paper/2505.01736/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [PAPM: A Physics-aware Proxy Model for Process Systems](https://mlsanity.com/paper/2407.05232/) (2024)
- [Conservation-informed Graph Learning for Spatiotemporal Dynamics Prediction](https://mlsanity.com/paper/2412.20962/) (2024)
- [Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularization](https://mlsanity.com/paper/2402.15636/) (2024)
- [MultiPDENet: PDE-embedded Learning with Multi-time-stepping for Accelerated Flow Simulation](https://mlsanity.com/paper/2501.15987/) (2025)
- [PINP: Physics-Informed Neural Predictor with latent estimation of fluid flows](https://mlsanity.com/paper/2504.06070/) (2025)
- [Out-of-distribution generalization of deep-learning surrogates for 2D PDE-generated dynamics in the small-data regime](https://mlsanity.com/paper/2601.08404/) (2026)
- [Closed-form Solutions: A New Perspective on Solving Differential Equations](https://mlsanity.com/paper/2405.14620/) (2024)
- [Implicit factorized transformer approach to fast prediction of turbulent channel flows](https://mlsanity.com/paper/2412.18840/) (2024)
- [Towards Generalizable PDE Dynamics Forecasting via Physics-Guided Invariant Learning](https://mlsanity.com/paper/2509.24332/) (2025)
- [DSAF: A Dual-Stage Adaptive Framework for Numerical Weather Prediction Downscaling](https://mlsanity.com/paper/2312.12476/) (2023)
- [A Physics-embedded Deep Learning Framework for Cloth Simulation](https://mlsanity.com/paper/2403.12820/) (2024)
- [Physics-embedded Fourier Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2407.11158/) (2024)
- [Spatiotemporal Graph Learning with Direct Volumetric Information Passing and Feature Enhancement](https://mlsanity.com/paper/2409.18013/) (2024)
- [PIMRL: Physics-Informed Multi-Scale Recurrent Learning for Burst-Sampled Spatiotemporal Dynamics](https://mlsanity.com/paper/2503.10253/) (2025)
- [Spectral-inspired Operator Learning with Limited Data and Unknown Physics](https://mlsanity.com/paper/2505.21573/) (2025)
- [PMNO: A novel physics guided multi-step neural operator predictor for partial differential equations](https://mlsanity.com/paper/2506.01598/) (2025)
- [Learnable-Differentiable Finite Volume Solver for Accelerated Simulation of Flows](https://mlsanity.com/paper/2507.01975/) (2025)
- [Efficient High-Accuracy PDEs Solver with the Linear Attention Neural Operator](https://mlsanity.com/paper/2510.16816/) (2025)

## Abstract (arXiv)

Modeling complex spatiotemporal dynamical systems, such as the reaction-diffusion processes, have largely relied on partial differential equations (PDEs). However, due to insufficient prior knowledge on some under-explored dynamical systems, such as those in chemistry, biology, geology, physics and ecology, and the lack of explicit PDE formulation used for describing the nonlinear process of the system variables, to predict the evolution of such a system remains a challenging task. Unifying measurement data and our limited prior physics knowledge via machine learning provides us with a new path to solving this problem. Existing physics-informed learning paradigms impose physics laws through soft penalty constraints, whose solution quality largely depends on a trial-and-error proper setting of hyperparameters. Since the core of such methods is still rooted in black-box neural networks, the resulting model generally lacks interpretability and suffers from critical issues of extrapolation and generalization. To this end, we propose a deep learning framework that forcibly encodes given physics structure to facilitate the learning of the spatiotemporal dynamics in sparse data regimes. We show how the proposed approach can be applied to a variety of problems regarding the PDE system, including forward and inverse analysis, data-driven modeling, and discovery of PDEs. The resultant learning paradigm that encodes physics shows high accuracy, robustness, interpretability and generalizability demonstrated via extensive numerical experiments.