Neural oscillators that originate from the second-order ordinary differential equations (ODEs) have shown competitive performance in learning mappings between dynamic loads and responses of complex nonlinear structural systems. Despite this empirical success, theoretically quantifying the generalization capacities of their neural network architectures remains undeveloped. In this study, the neural oscillator consisting of a second-order ODE followed by a multilayer perceptron (MLP) is considered. Its upper probably approximately correct (PAC) generalization bound for approximating causal and uniformly continuous operators between continuous temporal function spaces and that for approximating the uniformly asymptotically incrementally stable second-order dynamical systems are derived by leveraging the Rademacher complexity framework. The theoretical results show that the estimation errors grow polynomially with respect to both the MLP size and the time length, thereby avoiding the curse of parametric complexity. Furthermore, the derived error bounds demonstrate that constraining the Lipschitz constants of the MLPs via loss function regularization can improve the generalization ability of the neural oscillator. A numerical study considering a Bouc-Wen nonlinear system under stochastic seismic excitation validates the theoretically predicted power laws of the estimation errors with respect to the sample size and time length, and confirms the effectiveness of constraining MLPs' matrix and vector norms in enhancing the performance of the neural oscillator under limited training data.

翻译：源于二阶常微分方程的神经振荡器在学习复杂非线性结构系统的动态载荷与响应之间的映射关系方面已展现出具有竞争力的性能。尽管取得了这一经验性成功，理论上量化其神经网络架构的泛化能力仍有待发展。本研究考虑了一种由二阶常微分方程后接多层感知机构成的神经振荡器。通过利用Rademacher复杂性框架，推导了其在逼近连续时间函数空间之间的因果且一致连续算子时的概率近似正确泛化上界，以及逼近一致渐近增量稳定的二阶动力系统时的泛化上界。理论结果表明，估计误差随MLP规模和时间长度呈多项式增长，从而避免了参数复杂性的维数灾难。此外，推导出的误差界表明，通过损失函数正则化约束MLP的Lipschitz常数能够提升神经振荡器的泛化能力。针对随机地震激励下Bouc-Wen非线性系统的数值研究，验证了估计误差关于样本量和时间长度的幂律关系与理论预测一致，并证实了在有限训练数据下约束MLP矩阵与向量范数对提升神经振荡器性能的有效性。