MFPC-PIML: Physics-informed machine learning based on multiscale Fourier feature phase compensation

The paradigm of physics-driven forward electromagnetic computation holds significance for enhancing the accuracy of network approximations while reducing the dependence on large-scale datasets. However, challenges arise during the training process when dealing with objective functions characterized by high-frequency and multi-scale features. These challenges frequently occur as difficulties in effectively minimizing loss or encountering conflicts among competing objectives. To overcome these obstacles, we carried out analysis leveraging the Neural Tangent Kernel (NTK) as our theoretical framework for analysis. Through this, we propose a novel architectural solution: a Multi-scale Fourier Feature Phase Compensation (MFPC) technology, according to Gaussian kernel mapping. This architecture leverages a Gaussian kernel to enhance the spectral representation of coordinate data, expanding the frequency domain coverage of Fourier feature mapping. Additionally, by compensating for phase loss inherent in conventional Fourier mapping, our approach effectively mitigates training difficulties, accelerates convergence, and significantly improves the model's accuracy in capturing high-frequency components. Our research comprises a range of challenging examples, including the high-frequency Poisson equation and the isotropic layered medium scattering model. Through these examples, we demonstrate the proficiency of our proposed method in accurately solving high-frequency, multi-scale Partial Differential Equation (PDE) equations. This highlights its potential not only in forward modeling but also in solving evolution and inverse problems.