Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces

TL;DR

This paper introduces a randomized neural network method for efficiently solving surface PDEs on static and evolving geometries, avoiding costly mesh updates and nonconvex training issues.

Contribution

The work develops a novel RaNN approach with fixed random hidden layers and efficient least-squares output layer training for surface PDEs, including evolving surfaces.

Findings

Broad applicability demonstrated through numerical experiments

Favorable accuracy and efficiency on benchmark problems

Effective handling of static and evolving surface geometries

Abstract

Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the need for repeated mesh updates and geometry or solution transfer. While neural-network-based methods offer mesh-free discretizations, approaches based on nonconvex training can be costly and may fail to deliver high accuracy in practice. In this work, we develop a randomized neural network (RaNN) method for solving PDEs on both static and evolving surfaces: the hidden-layer parameters are randomly generated and kept fixed, and the output-layer coefficients are determined efficiently by solving a least-squares problem. For static surfaces, we present formulations for parametrized surfaces, implicit level-set surfaces, and point-cloud geometries, and provide a corresponding theoretical analysis for the parametrization-based formulation with interface compatibility. For evolving surfaces with topology preserved over time, we introduce a RaNN-based strategy that learns the surface evolution through a flow-map representation and then solves the surface PDE on a space--time collocation set, avoiding remeshing. Extensive numerical experiments demonstrate broad applicability and favorable accuracy--efficiency performance on representative benchmarks.