FRACTAL MULTI-SCALE PHYSICS-INFORMED NEURAL NETWORKS FOR FRACTAL-MODIFIED NONLINEAR OSCILLATORS

This paper proposes fractal multi-scale Physics-informed neural networks (FMPINNs) for solving fractal-modified nonlinear oscillators. The present scheme converts the numerical solution of differential equation into the optimization problem involving governing equation and boundary conditions. Similar to meshless methods, the developed PINN does not require mesh generation or numerical integration. By embedding appropriate Fourier feature mappings into corresponding parallel sub-networks, the fitting ability of the proposed method for high-frequency and multi-scale objective functions is enhanced. Some numerical experiments demonstrate that compared to classical PINNs, our method is more effective especially in handling such high-frequency problems, which provides innovative ideas for solving fractal nonlinear oscillators.