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

Elena Celledoni, Brynjulf Owren, Chong Shen, Baige Xu, Takaharu Yaguchi

Keywords: energy-preserving integrator, energy-dissipating integrator, Hamiltonian system, numerical integrator, discrete gradient method

TL;DR: We propose a method for learning energy-preserving or -dissipating numerical schemes. Our approach admits the universal approximation property.

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.

Primary Area: Machine learning for sciences (e.g. climate, health, life sciences, physics, social sciences)

Submission Number: 12075