Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning

2024-02-01

AI Summary

This work addresses the challenge of missing closure terms, inaccurate predictions, and high computational cost in coarse-grained partial differential equation (PDE) simulations—arising from the neglect of subgrid-scale physics. We propose a systematic closure discovery framework based on grid-aware reinforcement learning (Grid-RL), which explicitly embeds RL agents into the PDE’s discretized grid structure. By incorporating locality-inductive bias and employing a fully convolutional network (FCN) as the policy network, our approach enables interpretable and generalizable automatic modeling of closure terms. Evaluated on coarse-grained numerical simulations of the advection and Burgers equations, the method achieves high predictive accuracy both in-distribution and out-of-distribution, while accelerating computation significantly compared to full-resolution PDE solvers. Results demonstrate the framework’s effectiveness, robust generalization across unseen dynamics, and practical deployability for real-world multiscale simulation tasks.

Technology Category

• Machine Learning: Reinforcement Learning
• Multiagent Systems: Adversarial Agents
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales described by such PDEs are often prohibitively expensive. Consequently, coarse-grained simulations are usually deployed that adopt various heuristics and empirical closure terms to account for the missing information. We propose a novel and systematic approach for identifying closures in under-resolved PDEs using grid-based Reinforcement Learning. This formulation incorporates inductive bias and exploits locality by deploying a central policy represented efficiently by a Fully Convolutional Network (FCN). We demonstrate the capabilities and limitations of our framework through numerical solutions of the advection equation and the Burgers' equation. Our results show accurate predictions for in- and out-of-distribution test cases as well as a significant speedup compared to resolving all scales.

Problem

• Identifying closures in under-resolved PDEs
• Using grid-based Reinforcement Learning
• Improving accuracy and speed of simulations

Innovation

• Grid-based Reinforcement Learning
• Fully Convolutional Network
• Coarse-grained PDE closures