Runge-Kutta Physics Informed Neural Networks: Formulation and Analysis

Abstract: In this paper we consider time-dependent PDEs discretized by a special class of Physics Informed Neural Networks whose design is based on the framework of Runge--Kutta and related time-Galerkin discretizations. The primary motivation for using such methods is that alternative time-discrete schemes not only enable higher-order approximations but also have a crucial impact on the qualitative behavior of the discrete solutions. The design of the methods follows a novel training approach based on two key principles: (a) the discrete loss is designed using a time-discrete framework, and (b) the final loss formulation incorporates Runge--Kutta or time-Galerkin discretization in a carefully structured manner. We then demonstrate that the resulting methods inherit the stability properties of the Runge--Kutta or time-Galerkin schemes, and furthermore, their computational behavior aligns with that of the original time discrete method used in their formulation. In our analysis, we focus on linear parabolic equations, demonstrating both the stability of the methods and the convergence of the discrete minimizers to solutions of the underlying evolution PDE. An important novel aspect of our work is the derivation of maximal regularity (MR) estimates for B-stable Runge--Kutta schemes and both continuous and discontinuous Galerkin time discretizations. This allows us to provide new energy-based proofs for maximal regularity estimates previously established by Kov\'acs, Li, and Lubich, now in the Hilbert space setting and with the flexibility of variable time steps.