Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications remains prohibitively expensive because of the necessity of resolving a broad range of spatiotemporal scales. In turn, practitioners often rely on coarse-grained approximations of the original PDEs, trading off accuracy for reduced computational resources. To mitigate the loss of detail inherent in such approximations, closure models are employed to represent unresolved spatiotemporal interactions. We present a framework for developing closure models for PDEs using synthetic data acquired through the method of manufactured solutions. These data are used in conjunction with reinforcement learning to provide closures for coarse-grained PDEs. We illustrate the efficacy of our method using the one-dimensional and two-dimensional Burgers' equations and the two-dimensional advection equation. Moreover, we demonstrate that closure models trained for inhomogeneous PDEs can be effectively generalized to homogeneous PDEs. The results demonstrate the potential for developing accurate and computationally efficient closure models for systems with scarce data.
View on arXiv@article{heimbach2025_2505.11308,
title={ Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data },
author={ Lothar Heimbach and Sebastian Kaltenbach and Petr Karnakov and Francis J. Alexander and Petros Koumoutsakos },
journal={arXiv preprint arXiv:2505.11308},
year={ 2025 }
}