Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data

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.

Physics-Regularized Multi-Modal Image Assimilation for Brain Tumor Localization

Physical models in the form of partial differential equations represent an important prior for many under-constrained problems. One example is tumor treatment planning, which heavily depends on accurate estimates of the spatial distribution of tumor cells in a patient's anatomy. Medical imaging scans can identify the bulk of the tumor, but they cannot reveal its full spatial distribution. Tumor cells at low concentrations remain undetectable, for example, in the most frequent type of primary brain tumors, glioblastoma. Deep-learning-based approaches fail to estimate the complete tumor cell distribution due to a lack of reliable training data. Most existing works therefore rely on physics-based simulations to match observed tumors, providing anatomically and physiologically plausible estimations. However, these approaches struggle with complex and unknown initial conditions and are limited by overly rigid physical models. In this work, we present a novel method that balances data-driven and physics-based cost functions. In particular, we propose a unique discretization scheme that quantifies the adherence of our learned spatiotemporal tumor and brain tissue distributions to their corresponding growth and elasticity equations. This quantification, serving as a regularization term rather than a hard constraint, enables greater flexibility and proficiency in assimilating patient data than existing models. We demonstrate improved coverage of tumor recurrence areas compared to existing techniques on real-world data from a cohort of patients. The method holds the potential to enhance clinical adoption of model-driven treatment planning for glioblastoma.