Stable Differentiable Modal Synthesis for Learning Nonlinear Dynamics

2026/01/31 by V. M. Zheleznov, Victor Zheleznov, Stefan Bilbao +2
Computer Science · Engineering · Physics and Astronomy · #Model Reduction and Neural Networks #Neural Networks and Applications #Neural Networks and Reservoir Computing #cs.LG #cs.SD #eess.AS #physics.comp-ph

paper · pdf · doi:10.17743/jaes.2026.0269

published as J. Audio Eng. Soc., vol. 74, no. 7/8, pp. 513-523 (2026 Jul./Aug.) · Accepted for publication in Journal of the Audio Engineering Society (special issue on New Frontiers in Digital Audio Effects)

arxiv created 2026/03/15 · openalex publication_date 2026/07/20 · openalex created_date 2026/07/21 · openalex updated_date 2026/07/21 · arxiv updated 2026/08/03

Abstract

Modal methods are a long-standing approach to physical modeling 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 modeling nonlinear systems from data. 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. 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 the authors’ previous work that used multilayer perceptrons to parametrize nonlinear dynamics, gradient networks are employed to allow an interpretation in terms of a closed-form and nonnegative potential required by scalar auxiliary variable techniques. As a proof of concept, the authors 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.

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