Physics-informed Fourier Basis Neural Network for Fluid Mechanics

Abstract

Solving partial differential equations (PDEs) is an important yet challenging task in fluid mechanics. In this study, we embed an improved Fourier series into neural networks and propose a physics-informed Fourier basis neural network (FBNN) by incorporating physical information to solve canonical PDEs in fluid mechanics. The results demonstrated that the proposed framework exhibits a strong nonlinear fitting capability and exceptional periodic modeling performance. In particular, our model shows significant advantages for the Burgers equation with discontinuous solutions and Helmholtz equation with strong periodicity. By directly introducing sparse distributed data to reconstruct the entire flow field, we further intuitively validated the direct superiority of FBNN over conventional artificial neural networks (ANN) as well as the benefits of incorporating physical information into the network. By adjusting the activation functions of networks and comparing with an ANN and conventional physics-informed neural network, we proved that performance of the proposed FBNN architecture is not highly sensitive to the choice of activation functions. The nonlinear fitting capability of FBNN avoids excessive reliance on activation functions, thereby mitigating the risk of suboptimal outcomes or training failures stemming from unsuitable activation function choices.hese results highlightthe potential of PIFBNN as a powerful tool in computational fluid dynamics.