Physically consistent model learning for reaction-diffusion systems

TL;DR

This paper develops a method for learning reaction-diffusion models from data that inherently satisfy physical laws like mass conservation and quasipositivity, ensuring well-posedness and interpretability.

Contribution

It introduces techniques to modify reaction terms for physical consistency and extends theoretical convergence results to these constrained RD systems.

Findings

Reaction terms can be systematically modified to satisfy physical constraints.

Theoretical convergence of the learning process to unique solutions is established.

Approximation results for quasipositive functions support physically consistent modeling.

Abstract

This paper addresses the problem of learning reaction-diffusion (RD) systems from data while ensuring physical consistency and well-posedness of the learned models. Building on a regularization-based framework for structured model learning, we focus on learning parameterized reaction terms and investigate how to incorporate key physical properties, such as mass conservation and quasipositivity, directly into the learning process. Our main contributions are twofold: First, we propose techniques to systematically modify a given class of parameterized reaction terms such that the resulting terms inherently satisfy mass conservation and quasipositivity, ensuring that the learned RD systems preserve non-negativity and adhere to physical principles. These modifications also guarantee well-posedness of the resulting PDEs under additional regularity and growth conditions. Second, we extend existing theoretical results on regularization-based model learning to RD systems using these physically consistent reaction terms. Specifically, we prove that solutions to the learning problem converge to a unique, regularization-minimizing solution of a limit system even when conservation laws and quasipositivity are enforced. In addition, we provide approximation results for quasipositive functions, essential for constructing physically consistent parameterizations. These results advance the development of interpretable and reliable data-driven models for RD systems that align with fundamental physical laws.