Closure Discovery for Coarse-Grained Partial Differential Equations Using Grid-based Reinforcement Learning
This addresses the challenge of reliable predictions for critical phenomena like weather and epidemics by providing a systematic method for closure discovery, though it is incremental as it builds on existing coarse-graining techniques.
The paper tackles the problem of expensive simulations for PDEs by proposing a grid-based reinforcement learning approach to discover closure terms for coarse-grained PDEs, achieving accurate predictions and significant speedup compared to full-scale simulations.
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.