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.

翻译：模态方法是物理建模合成领域长期采用的一种方法。将其扩展至非线性问题是可行的，从而得到耦合的非线性常微分方程组。近年来标量辅助变量技术的发展，使得为此类系统构建显式且稳定的数值求解器成为可能。另一方面，神经常微分方程在根据数据建模非线性系统方面取得了成功。在本工作中，我们研究了如何将标量辅助变量技术与神经常微分方程相结合，以产生一个能够学习非线性动力学的稳定可微模型。所提出的方法利用系统模态线性振动的解析解，使得系统的物理参数在训练后仍易于获取，而无需在模型架构中引入参数编码器。与我们先前使用多层感知机参数化非线性动力学的工作相比，我们采用了梯度网络，该网络允许对标量辅助变量技术所需的闭合形式且非负的势能进行解释。作为概念验证，我们生成了弦非线性横向振动的合成数据，并表明该模型可以通过训练重现系统的非线性动力学特性。文中提供了声音示例。