MS-IUFFNO: Multi-scale implicit U-net enhanced factorized fourier neural operator for solving geometric PDEs

Abstract

Geometric partial differential equations (geometric PDEs) are defined on manifolds in Riemannian space, specifically tailored for modeling the temporal evolution of surfaces in natural sciences and engineering. For varying initial surfaces (initial conditions), traditional numerical methods require re-solving the equation even for the same geometric PDE, which significantly hinders the efficiency of simulations. The efficient predictive capabilities of neural networks (NNs) makes them a powerful tool for solving differential equations. The solution of geometric PDEs governs the continuous evolution of the surface over time, making it challenging for most NNs to handle solution prediction for geometric PDEs with varying initial surfaces. We propose a novel neural operator-based framework for solving geometric PDEs. Once trained, our model can predict the solution of the same geometric PDE under arbitrary initial conditions (initial surfaces). To the best of our knowledge, this is the first attempt to solve geometric PDEs using the neural operator. Firstly, we employ a learned continuous Signed Distance Function (SDF) representation method (DeepSDF) to convert the initial mesh surface into an implicit level-set representation, thereby avoiding the difficulties associated with solving explicit geometric PDEs. Subsequently, by integrating the multi-scale module, we design a Multi-Scale Implicit U-Net enhanced Factorized Fourier Neural Operator (MS-IUFFNO) for solving implicit geometric PDEs. The innovative structure of the neural operator substantially improves the prediction accuracy and long-term stability for solving geometric PDEs with reduced computational complexity. In addition, we construct datasets to train neural operators to solve the mean curvature flow and Willmore flow, which are representative of geometric PDEs. Finally, a numerical benchmark is conducted to compare MS-IUFFNO to several classical neural operator models for solving the mean curvature flow and Willmore flow, where results show that our model exhibits superior performance in terms of prediction accuracy, extrapolation capability, and stability.