将物理模态分析与神经微分方程结合，实现可学习的非线性动力学建模。
Stable Differentiable Modal Synthesis for Learning Nonlinear Dynamics
- 用标量辅助变量法构造稳定可微的非线性系统求解器
- 通过梯度网络建模非线性项，保持物理参数可解释性
- 适用于需物理可解释性的声学/振动系统建模任务
模态方法是物理建模合成的长期方法。对非线性问题的扩展可形成耦合的非线性常微分方程组。近期标量辅助变量技术已实现此类系统的显式且稳定的数值求解器。另一方面，神经常微分方程在从数据中建模非线性系统方面表现成功。本文研究如何将标量辅助变量技术与神经常微分方程结合，构建一个稳定可微的模型以学习非线性动力学。该方法利用系统模态线性振动的解析解，使训练后物理参数仍易于获取，无需在模型架构中加入参数编码器。相较于此前使用多层感知机参数化非线性动力学的方法，本工作采用梯度网络，使其符合标量辅助变量技术所需的闭式且非负势能要求。作为概念验证，我们生成了弦横向非线性振动的合成数据，并证明模型可训练以重现系统的非线性动力学行为。同时提供了声音示例。
原文摘要 · Abstract (English)
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.
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