Learning PDEs from data on closed surfaces with sparse optimization

TL;DR

This paper proposes a novel approach, termed physical-informed sparse optimization (PIS), for learning surface PDEs, which incorporates both physical-informed model loss and regularization penalty terms in the loss function, enabling the identification of specific physical terms within the surface PDEs.

Abstract

The discovery of underlying surface partial differential equation (PDE) from observational data has significant implications across various fields, bridging the gap between theory and observation, enhancing our understanding of complex systems, and providing valuable tools and insights for applications. In this paper, we propose a novel approach, termed physical-informed sparse optimization (PIS), for learning surface PDEs. Our approach incorporates both $L_2$ physical-informed model loss and $L_1$ regularization penalty terms in the loss function, enabling the identification of specific physical terms within the surface PDEs. The unknown function and the differential operators on surfaces are approximated by some extrinsic meshless methods. We provide practical demonstrations of the algorithms including linear and nonlinear systems. The numerical experiments on spheres and various other surfaces demonstrate the effectiveness of the proposed approach in simultaneously achieving precise solution prediction and identification of unknown PDEs.

Figures (7)

• Figure 1: The relative $L_2$ errors of solutions for solving the identified model in Example 1 on the unit sphere for different nodes sizes $N$ (in log-log scale).
• Figure 2: Absolute errors of predicted solutions $u$ of the learned PDEs on the unit sphere in Example 1.
• Figure 3: Absolute errors of predicted solutions $u$ of the learned PDEs on the unit sphere in Example 2 at different time steps with $N=500$, and $0\%$ (first row) and $0.01\%$ (second row) noise level respectively.
• Figure 4: The absolute errors of predicted solutions from the learned PDE in Example 2 on general surfaces with samples of $0\%$ noise level. (top) Torus with $N=3968$. (middle) Cyclide with $N=3662$. (bottom) Bretzel2 with $N=9546$.
• Figure :