Neural Harmonic Measure Operator

2026-09-28

AI Summary

This study addresses the limited generalizability and repeated training requirements of solving elliptic partial differential equations on deforming domains. To overcome these limitations, this work proposes a Neural Harmonic Measure Operator that leverages geometry-dependent harmonic measures to decouple boundary data, parameterizing the boundary probability density via a Transformer kernel. By integrating Walk-on-Spheres sampling with Poisson decomposition and employing an auxiliary network to circumvent singular volume integrals, the proposed framework enables zero-shot inference for arbitrary boundary conditions and source terms after a single training phase. Comprehensive evaluations demonstrate that the method consistently outperforms four baseline models on the 3D MCB-B benchmark, while achieving 2D performance comparable to mainstream neural operators.

Abstract

We introduce Neural Harmonic Measure Operator (NHMO), a neural solver for elliptic PDE problems on variable-shape domains. The harmonic measure of a domain is the boundary probability distribution that, integrated against any boundary data, returns the Dirichlet Laplace solution. It depends only on the geometry, not on the boundary data. NHMO parameterizes the density of this measure as a transformer-based boundary kernel supervised by Walk-on-Spheres exit samples, so one trained kernel handles different boundary values on a shape with no retraining. We extend it to Poisson via a classical decomposition, with an auxiliary network amortizing the source-induced correction and avoiding the singular volume quadrature that breaks direct evaluation. At inference, new boundary values and new sources both yield PDE solutions by re-integration against the fitted kernel and lift, with no retraining. NHMO improves over four prior baselines on the MCB-B 3D variable-shape Poisson benchmark across all five categories, and is competitive with major neural-operator baselines on a controlled 2D testbed.

Problem

• Elliptic PDEs
• Variable-shape domains
• Harmonic measure
• Poisson equation
• Neural operator

Innovation

• Neural Harmonic Measure Operator
• Elliptic PDEs
• Transformer-based boundary kernel
• Walk-on-Spheres
• Variable-shape domains