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

Jingbo Sun, Fei Wang

TL;DR

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.

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.

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

TL;DR

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.

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.
Paper Structure (13 sections, 13 theorems, 194 equations, 7 figures, 9 tables)

This paper contains 13 sections, 13 theorems, 194 equations, 7 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Randomized Neural Networks
  3. RaNN methods for PDEs on static surfaces
  4. Stationary partial differential equations
  5. Parametrization-based approaches
  6. Theoretical analysis of the parametrization-based approach
  7. Level-set-based approaches and point-cloud-based RaNN approaches.
  8. Time-dependent partial differential equations
  9. RaNN methods for PDEs on evolving surfaces
  10. RaNN methods for capturing surface evolution
  11. RaNN Methods for solving PDEs
  12. Numerical Experiments
  13. Summary

Key Result

Lemma 3.4

For each $i$ there exist $0<\lambda_i\le \Lambda_i<\infty$ such that and $g_i^{\alpha\beta},\sqrt{g_i}\in C^{k-1}(\overline{D_i})$.

Figures (7)

  • Figure 1: Architecture of Randomized Neural Networks
  • Figure 2: Pointwise absolute errors of the two point sets.
  • Figure 3: Predicted temperature distribution on the cup-shaped surface.
  • Figure 4: Predicted heat distribution on the bunny surface.
  • Figure 5: Evolving ellipsoid and corresponding mean curvature distribution.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Remark 3.1: Why value and normal derivative are sufficient
  • Remark 3.2
  • Remark 3.3: Fixing the additive constant on closed surfaces
  • Lemma 3.4: Uniform ellipticity on each chart
  • Theorem 3.5: Elliptic regularity on a closed connected surface (graph norm)
  • Lemma 3.6: Gluing: zero mismatches imply a global $H^2(\Gamma)$ function
  • Lemma 3.7: Lifting of interface mismatch traces
  • Theorem 3.8: Graph-norm control with interface mismatch traces
  • Lemma 3.9: Upper bound of the training graph norm
  • Corollary 3.11: Graph-norm control in terms of the computable $Z$-norm
  • ...and 16 more