Upper Approximation Bounds for Neural Oscillators

Neural oscillators, derived from second-order ordinary differential equations (ODEs), have shown strong performance in stably learning causal mappings between long-term sequences or continuous temporal functions—such as mappings between loads and responses in complex nonlinear structural systems. However, theoretically quantifying the capacity of their network architectures remains a major challenge. This study investigates a neural oscillator model composed of a second-order ODE coupled with a multilayer perceptron (MLP). We establish upper approximation bounds for this architecture in two contexts: (1) approximating causal and uniformly continuous operators between continuous temporal function spaces, and (2) approximating uniformly asymptotically incrementally stable second-order dynamical systems. Theoretical analysis shows that in approximating second-order dynamical systems, the approximation error of the neural oscillator decays polynomially with the reciprocals of the widths of the utilized MLPs, thereby alleviating the curse of parametric complexity. The derived decay rates of the approximation errors are further validated through two numerical examples. These findings provide a solid theoretical foundation for the reliable application of neural oscillators in scientific and engineering domains.