Physics-informed Fourier Basis Neural Network for Fluid Mechanics

Conventional machine learning approaches struggle to capture periodic patterns and solve quasi-periodic boundary problems in fluid mechanics. This study proposes a physics-informed Fourier basis neural network (PIFBNN) that integrates adaptive Fourier series with physical constraints to address canonical partial differential equations. The architecture preserves Fourier series' natural mathematical compatibility with periodic phenomena while incorporating trainable parameters (angular frequencies and weight coefficients) that enhance basis function flexibility and nonlinear learning capacity. We evaluate the framework on six fundamental fluid dynamics benchmarks: two-dimensional cylinder wake flow, lid-driven cavity flow, Kovasznay flow, Helmholtz equation, Burgers equation, and Allen-Cahn equation. Results demonstrate PIFBNN's consistent superiority over standard physics-informed neural network (PINN) in accuracy. Through sparse data reconstruction experiments and adjusting the activation functions of neural networks and comparing , we further validate the dual advantages of Fourier basis neural network (FBNN) over conventional artificial neural network (ANN): inherent periodicity handling and reduced sensitivity to activation function selection. The FBNN architecture maintains robust performance across different activation functions, as verified through systematic comparisons with ANN and PINN baselines. These findings position PIFBNN as a promising computational framework for complex fluid dynamics problems.