# Learning Physical Models that Can Respect Conservation Laws

arXiv: https://arxiv.org/abs/2302.11002 · HTML: https://mlsanity.com/paper/2302.11002/

## Key facts

- Cited by 59 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 62 references that are in the map.
- First posted on arXiv 21 Feb 2023.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (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)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Neural Conservation Laws: A Divergence-Free Perspective](https://mlsanity.com/paper/2210.01741/) (2022)
- [Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows](https://mlsanity.com/paper/2005.13420/) (2020)
- [Learning differentiable solvers for systems with hard constraints](https://mlsanity.com/paper/2207.08675/) (2022)
- [Learning continuous models for continuous physics](https://mlsanity.com/paper/2202.08494/) (2022)
- [Adaptive Self-supervision Algorithms for Physics-informed Neural Networks](https://mlsanity.com/paper/2207.04084/) (2022)
- [Guiding continuous operator learning through Physics-based boundary constraints](https://mlsanity.com/paper/2212.07477/) (2022)
- [ResNet After All? Neural ODEs and Their Numerical Solution](https://mlsanity.com/paper/2007.15386/) (2020)
- [Exact conservation laws for neural network integrators of dynamical systems](https://mlsanity.com/paper/2209.11661/) (2022)
- [Using Conservation Laws to Infer Deep Learning Model Accuracy of Richtmyer-meshkov Instabilities](https://mlsanity.com/paper/2208.11477/) (2022)

## Built on by (most cited first, 30 of 59)

- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [PreDiff: Precipitation Nowcasting with Latent Diffusion Models](https://mlsanity.com/paper/2307.10422/) (2023)
- [AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformers](https://mlsanity.com/paper/2502.09692/) (2025)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [Gradient-Free Generation for Hard-Constrained Systems](https://mlsanity.com/paper/2412.01786/) (2024)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Limits of Deep Learning: Sequence Modeling through the Lens of Complexity Theory](https://mlsanity.com/paper/2405.16674/) (2024)
- [Graph Neural PDE Solvers with Conservation and Similarity-Equivariance](https://mlsanity.com/paper/2405.16183/) (2024)
- [Renormalizing Diffusion Models](https://mlsanity.com/paper/2308.12355/) (2023)
- [Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs](https://mlsanity.com/paper/2403.10642/) (2024)
- [AhmedML: High-Fidelity Computational Fluid Dynamics Dataset for Incompressible, Low-Speed Bluff Body Aerodynamics](https://mlsanity.com/paper/2407.20801/) (2024)
- [Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation](https://mlsanity.com/paper/2505.22391/) (2025)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Neural equilibria for long-term prediction of nonlinear conservation laws](https://mlsanity.com/paper/2501.06933/) (2025)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [Learning Physics for Unveiling Hidden Earthquake Ground Motions via Conditional Generative Modeling](https://mlsanity.com/paper/2407.15089/) (2024)
- [Project and Generate: Divergence-Free Neural Operators for Incompressible Flows](https://mlsanity.com/paper/2603.24500/) (2026)
- [Physics-informed Discretization-independent Deep Compositional Operator Network](https://mlsanity.com/paper/2404.13646/) (2024)
- [Strictly Constrained Generative Modeling via Split Augmented Langevin Sampling](https://mlsanity.com/paper/2505.18017/) (2025)
- [Gaussian process learning of nonlinear dynamics](https://mlsanity.com/paper/2312.12193/) (2023)
- [Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations](https://mlsanity.com/paper/2401.02363/) (2024)
- [Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics](https://mlsanity.com/paper/2406.15509/) (2024)
- [BeamVQ: Beam Search with Vector Quantization to Mitigate Data Scarcity in Physical Spatiotemporal Forecasting](https://mlsanity.com/paper/2502.18925/) (2025)
- [Residuals Are Not Enough: Limits of Physics-Informed Pre-Training for Scientific Foundation Models](https://mlsanity.com/paper/2503.19081/) (2025)
- [Smooth and Sparse Latent Dynamics in Operator Learning with Jerk Regularization](https://mlsanity.com/paper/2402.15636/) (2024)
- [BeamVQ: Aligning Space-Time Forecasting Model via Self-training on Physics-aware Metrics](https://mlsanity.com/paper/2405.17051/) (2024)
- [Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Data](https://mlsanity.com/paper/2406.02581/) (2024)
- [RAIN: Reinforcement Algorithms for Improving Numerical Weather and Climate Models](https://mlsanity.com/paper/2408.16118/) (2024)

## Abstract (arXiv)

Recent work in scientific machine learning (SciML) has focused on incorporating partial differential equation (PDE) information into the learning process. Much of this work has focused on relatively "easy" PDE operators (e.g., elliptic and parabolic), with less emphasis on relatively "hard" PDE operators (e.g., hyperbolic). Within numerical PDEs, the latter problem class requires control of a type of volume element or conservation constraint, which is known to be challenging. Delivering on the promise of SciML requires seamlessly incorporating both types of problems into the learning process. To address this issue, we propose ProbConserv, a framework for incorporating conservation constraints into a generic SciML architecture. To do so, ProbConserv combines the integral form of a conservation law with a Bayesian update. We provide a detailed analysis of ProbConserv on learning with the Generalized Porous Medium Equation (GPME), a widely-applicable parameterized family of PDEs that illustrates the qualitative properties of both easier and harder PDEs. ProbConserv is effective for easy GPME variants, performing well with state-of-the-art competitors; and for harder GPME variants it outperforms other approaches that do not guarantee volume conservation. ProbConserv seamlessly enforces physical conservation constraints, maintains probabilistic uncertainty quantification (UQ), and deals well with shocks and heteroscedasticities. In each case, it achieves superior predictive performance on downstream tasks.