UEPI: Universal Energy-Behavior-Preserving Integrators for Energy Conservative/Dissipative Differential Equations

Abstract

Physical phenomena in the real world are often described by energy-based modeling theories, such as Hamiltonian mechanics or the Landau theory. It is known that physical phenomena based on these theories have an energy conservation law or a dissipation law. Therefore, in the simulations of such physical phenomena, numerical methods that preserve the energy-conservation or dissipation laws are desirable. However, because various energy-behavior-preserving numerical methods have been proposed, it is difficult to discover the best one. In this study, we propose a method for learning highly accurate energy-behavior-preserving integrators from data. Numerical results show that our approach certainly learns energy-behavior-preserving numerical methods that are more accurate than existing numerical methods for various differential equations, including chaotic Hamiltonian systems, dissipative systems, and a nonlinear partial differential equation. We also provide universal approximation theorems for the proposed approach.