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

TL;DR

This paper presents a neural network framework that learns second order coupled differential equations with non-conservative forces from real-space trajectory data, enabling stable predictions of complex dissipative systems.

Contribution

It introduces a novel network architecture combining a differential equation solver with a convolutional network to learn physical properties from trajectory data, including non-conservative forces.

Findings

Successfully models dissipative dynamical systems from trajectories

Enables stable forecasting with partial observations

Integrates differential equation learning with trajectory prediction

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.