Physics-informed Neural Implicit Flow neural network for parametric PDEs

Abstract

The Physics-informed Neural Network (PINN) has been a popular method for solving partial differential equations (PDEs) due to its flexibility. However, PINN still faces challenges in characterizing spatio-temporal correlations when solving parametric PDEs due to network limitations. To address this issue, we propose a Physics-Informed Neural Implicit Flow (PINIF) framework, which enables a meshless low-rank representation of the parametric spatio-temporal field based on the expressiveness of the Neural Implicit Flow (NIF), enabling a meshless low-rank representation. In particular, the PINIF framework utilizes the Polynomial Chaos Expansion (PCE) method to quantify the uncertainty in the presence of noise, allowing for a more robust representation of the solution. In addition, PINIF introduces a novel transfer learning framework to speed up the inference of parametric PDEs significantly. The performance of PINIF and PINN is compared on various PDEs especially with variable coefficients and Kolmogorov flow. The comparative results indicate that PINIF outperforms PINN in terms of accuracy and efficiency.

Keywords: Kolmogorov flow; Neural Implicit Flow; Partial differential equations; Physics-informed Neural Network.