SO-PIFRNN: Self-optimization physics-informed Fourier-features randomized neural network for solving partial differential equations

Abstract

This study proposes a self-optimization physics-informed Fourier-features randomized neural network (SO-PIFRNN) framework, which significantly improves the numerical solving accuracy of PDEs through hyperparameter optimization mechanism. The framework employs a bi-level optimization architecture: the outer-level optimization utilizes a multi-strategy collaborated particle swarm optimization (MSC-PSO) algorithm to search for optimal hyperparameters of physics-informed Fourier-features randomized neural network, while the inner-level optimization determines the output layer weights of the neural network via the least squares method. The core innovation of this study is embodied in the following three aspects: First, the Fourier basis function activation mechanism is introduced in the hidden layer of neural network, which significantly enhances the ability of the network to capture multi-frequency components of the solution. Secondly, a novel derivative neural network method is proposed, which improves the calculation accuracy and efficiency of PIFRNN method. Finally, the MSC-PSO algorithm of the hybrid optimization strategy is designed to improve the global search ability and convergence accuracy through the synergistic effect of dynamic parameter adjustment, elitist and mutation strategies. Through a series of numerical experiments, including multiscale equations in complex regions, high-order equations, high-dimensional equations and nonlinear equations, the validity of SO-PIFRNN is verified. The experimental results affirm that SO-PIFRNN exhibits superior approximation accuracy and frequency capture capability.