# Learning to Discretize: Solving 1D Scalar Conservation Laws via Deep Reinforcement Learning

arXiv: https://arxiv.org/abs/1905.11079 · HTML: https://mlsanity.com/paper/1905.11079/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2019.

## Built on by (most cited first, 15 of 15)

- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [AI Poincar'e 2.0: Machine Learning Conservation Laws from Differential Equations](https://mlsanity.com/paper/2203.12610/) (2022)
- [Deep Reinforcement Learning for Adaptive Mesh Refinement](https://mlsanity.com/paper/2209.12351/) (2022)
- [Approximating Numerical Fluxes Using Fourier Neural Operators for Hyperbolic Conservation Laws](https://mlsanity.com/paper/2401.01783/) (2024)
- [A Thermodynamics-informed Active Learning Approach to Perception and Reasoning about Fluids](https://mlsanity.com/paper/2203.05775/) (2022)
- [GoRINNs: Godunov-Riemann Informed Neural Networks for Learning Hyperbolic Conservation Laws](https://mlsanity.com/paper/2410.22193/) (2024)
- [Least-Squares ReLU Neural Network (LSNN) Method For Scalar Nonlinear Hyperbolic Conservation Law](https://mlsanity.com/paper/2105.11627/) (2021)
- [Bayesian Learning of Coupled Biogeochemical-Physical Models](https://mlsanity.com/paper/2211.06714/) (2022)
- [Neural Operators Learn the Local Physics of Magnetohydrodynamics](https://mlsanity.com/paper/2404.16015/) (2024)
- [Cell-average based neural network method for hyperbolic and parabolic partial differential equations](https://mlsanity.com/paper/2107.00813/) (2021)
- [Backpropagation through Time and Space: Learning Numerical Methods with Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2203.08937/) (2022)
- [Neural Partial Differential Equations with Functional Convolution](https://mlsanity.com/paper/2303.07194/) (2023)
- [A third-order finite difference weighted essentially non-oscillatory scheme with shallow neural network](https://mlsanity.com/paper/2407.06333/) (2024)
- [Multi-Agent Learning of Numerical Methods for Hyperbolic PDEs with Factored Dec-MDP](https://mlsanity.com/paper/2205.15716/) (2022)
- [Conservative approximation-based feedforward neural network for WENO schemes](https://mlsanity.com/paper/2507.06190/) (2025)

## Abstract (arXiv)

Conservation laws are considered to be fundamental laws of nature. It has broad applications in many fields, including physics, chemistry, biology, geology, and engineering. Solving the differential equations associated with conservation laws is a major branch in computational mathematics. The recent success of machine learning, especially deep learning in areas such as computer vision and natural language processing, has attracted a lot of attention from the community of computational mathematics and inspired many intriguing works in combining machine learning with traditional methods. In this paper, we are the first to view numerical PDE solvers as an MDP and to use (deep) RL to learn new solvers. As proof of concept, we focus on 1-dimensional scalar conservation laws. We deploy the machinery of deep reinforcement learning to train a policy network that can decide on how the numerical solutions should be approximated in a sequential and spatial-temporal adaptive manner. We will show that the problem of solving conservation laws can be naturally viewed as a sequential decision-making process, and the numerical schemes learned in such a way can easily enforce long-term accuracy. Furthermore, the learned policy network is carefully designed to determine a good local discrete approximation based on the current state of the solution, which essentially makes the proposed method a meta-learning approach. In other words, the proposed method is capable of learning how to discretize for a given situation mimicking human experts. Finally, we will provide details on how the policy network is trained, how well it performs compared with some state-of-the-art numerical solvers such as WENO schemes, and supervised learning based approach L3D and PINN, and how well it generalizes.