Randomized Neural Networks for Partial Differential Equation on Static and Evolving Surfaces
This work addresses the problem of costly and inaccurate mesh-free solutions for surface PDEs in scientific and engineering applications, offering an incremental improvement over existing neural-network methods.
The authors tackled the challenge of solving partial differential equations on static and evolving surfaces by developing a randomized neural network method that fixes hidden-layer parameters and solves a least-squares problem for output coefficients, achieving favorable accuracy-efficiency performance in benchmarks.
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.