Learning Nonlinear Dynamics in Physical Modelling Synthesis using Neural Ordinary Differential Equations

TL;DR

This paper introduces a novel approach combining modal decomposition with neural ordinary differential equations to model nonlinear distributed musical systems, enabling data-driven synthesis of complex string vibrations with accessible physical parameters.

Contribution

The work demonstrates how neural ODEs can be integrated with modal analysis to model nonlinear string vibrations, maintaining physical interpretability without complex encoders.

Findings

Successfully modeled nonlinear string dynamics from synthetic data

The approach preserves physical parameters after training

Sound examples validate the model's effectiveness

Abstract

Modal synthesis methods are a long-standing approach for modelling distributed musical systems. In some cases extensions are possible in order to handle geometric nonlinearities. One such case is the high-amplitude vibration of a string, where geometric nonlinear effects lead to perceptually important effects including pitch glides and a dependence of brightness on striking amplitude. A modal decomposition leads to a coupled nonlinear system of ordinary differential equations. Recent work in applied machine learning approaches (in particular neural ordinary differential equations) has been used to model lumped dynamic systems such as electronic circuits automatically from data. In this work, we examine how modal decomposition can be combined with neural ordinary differential equations for modelling distributed musical systems. The proposed model leverages the analytical solution for linear vibration of system's modes and employs a neural network to account for nonlinear dynamic behaviour. Physical parameters of a system remain easily accessible after the training without the need for a parameter encoder in the network architecture. As an initial proof of concept, we generate synthetic data for a nonlinear transverse string and show that the model can be trained to reproduce the nonlinear dynamics of the system. Sound examples are presented.