On the Performance Boundary of Fourier-Enhanced Physics-Informed Neural Networks for High-Order Nonlinear Oscillators: A MEMS-Motivated Numerical Case Study

Numerical prediction of high-order nonlinear oscillators using conventional physics-informed neural networks (PINNs) is severely hampered by spectral bias, which prioritizes low-frequency components and leads to significant errors in capturing multi-harmonic vibration responses. This work is a numerical-method-oriented investigation rather than an experimental MEMS validation study. To mitigate spectral-bias issues, we employ a Fourier multiscale physics-informed neural network (FMPINN), which integrates random Fourier feature mapping with a parallel multi-sub-network architecture. We note that the individual building blocks of FMPINN are existing components in scientific-machine-learning literature, and our core contribution is not architectural innovation. While this enhancement yields substantial accuracy improvements—reducing relative L2 errors by one to two orders of magnitude under weak-to-moderate nonlinearity—the central contribution of this work lies in the quantitative characterization of its applicability boundary. Through six systematically designed numerical cases of a fifth-order Duffing oscillator reduced-order model motivated by MEMS resonator dynamics, we demonstrate that the performance of FMPINN saturates under strong nonlinear excitation. Contrary to intuition, simply increasing the Fourier-feature dimension cannot resolve this saturation. Spectral analysis reveals that this bottleneck stems from the initialization-determined finite effective bandwidth of the fixed, frozen frequency mapping, which induces spectral-folding-dominated artifacts in the presence of dense high-order harmonics. This provides a plausible mechanistic interpretation supported by our numerical experiments, rather than a mathematically proven fundamental bound. Our findings establish a practically useful numerical benchmark for the nonlinear stiffness threshold of Fourier-augmented PINN frameworks and underscore the need for adaptive, rather than fixed-scale, frequency embedding strategies. All evaluations are conducted against synthetic noise-free Runge–Kutta 45 benchmarks; experimental MEMS measurement data are not used in the present work.