Stable Differentiable Modal Synthesis for Learning Nonlinear Dynamics

2026-01-15

AI Summary

This work proposes a differentiable modeling framework that integrates physical priors with data-driven learning to stably capture nonlinear dynamical systems, such as high-amplitude string vibrations. By combining the scalar auxiliary variable (SAV) method, neural ordinary differential equations (Neural ODEs), and physics-informed modal decomposition, the framework yields a differentiable model endowed with explicit numerical stability guarantees. This approach preserves the interpretability of physical parameters while ensuring long-term numerical stability. Experiments on synthetic nonlinear string vibration data demonstrate that the method accurately reconstructs the underlying system dynamics and generates high-fidelity audio, thereby validating its effectiveness and practicality for modeling complex nonlinear systems.

Technology Category

• Machine Learning: Multimodal Learning
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Mixed Discrete/Continuous Search

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• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• nonlinear dynamics
• differentiable modeling
• modal synthesis
• neural ODEs
• physical parameter interpretability

Innovation

• Scalar Auxiliary Variable
• Neural ODEs
• Modal Synthesis
• Stable Differentiable Modeling
• Nonlinear Dynamics