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

TL;DR

This work presents a physically-informed neural ODE framework for physical modelling synthesis, where modal decomposition yields a linear, finite-dimensional system and a neural network learns a dimensionless nonlinear modal coupling. The approach preserves explicit physical parameters, enabling extrapolation to unseen parameter settings, and leverages the Störmer–Verlet discretisation to maintain a second-order, physics-aligned integration scheme. Evaluations on a nonlinear oscillator and a 100-mode nonlinear transverse string demonstrate accurate capture of nonlinear dynamics with favorable computational efficiency compared to full nonlinear formulations, while preserving perceptual characteristics such as pitch glides. The method offers a pathway to data-driven yet physics-consistent distributed-systems modelling with potential for real-time and real-world identification applications in musical acoustics.

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.

Figures (6)

• Figure 1: Target and predicted nonlinear functions for the oscillator \ref{['eq: oscillator']}.
• Figure 2: Displacement trajectory for the selected test string. On the right, the relative absolute error between the target and predicted trajectories is shown, normalised by the maximum absolute value of target trajectory.
• Figure 3: Output for the selected test string at normalised position $\mathit{x_o}$ equal to 0.87.
• Figure 4: Displacements of the 1st, 20th, 40th mode for the selected test string at normalised position $\mathit{x_o}$ equal to 0.87 (initial 10 periods).
• Figure 5: MSE per mode for the initial 100 of the linear and predicted string trajectories for the test dataset.