Stable Differentiable Modal Synthesis for Learning Nonlinear Dynamics

TL;DR

This work examines how scalar auxiliary variable techniques can be combined with neural ordinary differential equations to yield a stable differentiable model capable of learning nonlinear dynamics and proposes an approach that leverages the analytical solution for linear vibration of system's modes to reproduce the nonlinear dynamics of the system.

Abstract

Modal methods are a long-standing approach to physical modelling synthesis. Extensions to nonlinear problems are possible, including the case of a high-amplitude vibration of a string. A modal decomposition leads to a densely coupled nonlinear system of ordinary differential equations. Recent work in scalar auxiliary variable techniques has enabled construction of explicit and stable numerical solvers for such classes of nonlinear systems. On the other hand, machine learning approaches (in particular neural ordinary differential equations) have been successful in modelling nonlinear systems automatically 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 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. 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.

Figures (5)

• Figure 1: The predicted displacement trajectory with the largest $\mathop{\mathrm{MSE}}\nolimits_{\mathrm{rel}}(\tilde{w}^n, w^n)$ (centre), the target displacement trajectory (left) and the relative absolute error between them (right).
• Figure 2: MSE per mode for the initial 100ms of the predicted and linear displacement trajectories compared to the target solution.
• Figure 3: The predicted audio output with the largest $\mathop{\mathrm{MSE}}\nolimits_{\mathrm{rel}}(\tilde{w}^n,w^n)$. Taken at normalised position $x_\mathrm{o} = 0.89$.
• Figure 4: Displacements of the 1st, 30th and 60th mode for the predicted trajectory with the largest $\mathop{\mathrm{MSE}}\nolimits_{\mathrm{rel}}(\tilde{w}^n,w^n)$. Taken at normalised position $x_\mathrm{o} = 0.89$. Initial 20 periods after the excitation are shown.
• Figure 5: Spectrogram of the predicted audio output with the strongest nonlinear effects. Taken at normalised position $x_\mathrm{o} = 0.89$.