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

2026-03-02

AI Summary

This work proposes a mesh-free method based on randomized neural networks (RaNN) to address the challenges of geometric complexity and frequent remeshing in the numerical solution of partial differential equations on both static and evolving surfaces. By fixing the parameters of the random hidden layer and solving for the output coefficients via least squares, the approach leverages a flow map to model surface evolution, thereby eliminating the need for traditional remeshing. The framework uniformly accommodates diverse surface representations—including parametric, implicit, and point-cloud forms—and integrates spatiotemporal collocation with interface compatibility theory for rigorous analysis. Numerical experiments on benchmark problems demonstrate high accuracy and efficiency on both topologically preserved evolving surfaces and static geometries, confirming the method’s versatility and robustness.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Computational social science

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.

Problem

• surface partial differential equations
• static surfaces
• evolving surfaces
• mesh updates
• geometry transfer

Innovation

• Randomized Neural Networks
• Surface PDEs
• Mesh-free Methods
• Flow-map Representation
• Least-squares Training