Stable Differentiable Modal Synthesis for Learning Nonlinear Dynamics

TL;DR

This paper introduces a stable, differentiable neural model combining scalar auxiliary variable techniques with neural ODEs to learn nonlinear dynamics in physical systems, enabling accessible physical parameters and improved interpretability.

Contribution

It proposes a novel integration of scalar auxiliary variable methods with neural ODEs for stable nonlinear dynamics modeling, using gradient networks for interpretability.

Findings

Successfully models nonlinear string vibrations

Maintains physical parameter accessibility after training

Demonstrates stability and interpretability of the approach

Abstract

Modal methods are a long-standing approach to physical modelling synthesis. Extensions to nonlinear problems are possible, leading to coupled nonlinear systems of ordinary differential equations. Recent work in scalar auxiliary variable techniques has enabled construction of explicit and stable numerical solvers for such systems. On the other hand, neural ordinary differential equations have been successful in modelling nonlinear systems from data. In this work, we examine how scalar auxiliary variable techniques can be combined with neural ordinary differential equations to yield a stable differentiable model capable of learning nonlinear dynamics. The proposed approach leverages the analytical solution for linear vibration of the system's modes so that physical parameters of a system remain easily accessible after the training without the need for a parameter encoder in the model architecture. Compared to our previous work that used multilayer perceptrons to parametrise nonlinear dynamics, we employ gradient networks that allow an interpretation in terms of a closed-form and non-negative potential required by scalar auxiliary variable techniques. As a proof of concept, we generate synthetic data for the nonlinear transverse vibration of a string and show that the model can be trained to reproduce the nonlinear dynamics of the system. Sound examples are presented.