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.

翻译：模态方法是一种历史悠久的物理建模合成方法。该方法可扩展至非线性问题，包括弦的高振幅振动情形。模态分解会导出一个密集耦合的非线性常微分方程组。近期标量辅助变量技术的研究成果，为此类非线性系统构建了显式且稳定的数值求解器。另一方面，机器学习方法（特别是神经常微分方程）已成功实现从数据中自动建模非线性系统。本研究探讨了如何将标量辅助变量技术与神经常微分方程相结合，构建能够学习非线性动力学的稳定可微分模型。该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，无需在模型架构中设置参数编码器。作为概念验证，我们生成了弦非线性横向振动的合成数据，并证明该模型可通过训练复现系统的非线性动力学特性。文中同时提供了声音示例。