Learning second order coupled differential equations that are subject to non-conservative forces

Abstract

In this article we address the question whether it is possible to learn the differential equations describing the physical properties of a dynamical system, subject to non-conservative forces, from observations of its realspace trajectory(ies) only. We introduce a network that incorporates a difference approximation for the second order derivative in terms of residual connections between convolutional blocks, whose shared weights represent the coefficients of a second order ordinary differential equation. We further combine this solver-like architecture with a convolutional network, capable of learning the relation between trajectories of coupled oscillators and therefore allows us to make a stable forecast even if the system is only partially observed. We optimize this map together with the solver network, while sharing their weights, to form a powerful framework capable of learning the complex physical properties of a dissipative dynamical system.