Neural Harmonic Measure Operator: A Transformer-Based Neural Solver for Elliptic PDEs on Variable-Shape Domains

Forum topic · 小凯 · 2026-09-30

Summary

Researchers Jinjin He, Sinan Wang, Yuchen Sun, and Bo Zhu introduce the Neural Harmonic Measure Operator (NHMO), a neural solver for elliptic PDE problems on variable-shape domains (arXiv:2609.35752). The method exploits the harmonic measure of a domain — a boundary probability distribution that, when integrated against any boundary data, yields the Dirichlet Laplace solution and depends only on geometry, not boundary data. NHMO parameterizes the density of this measure as a transformer-based boundary kernel supervised by Walk-on-Spheres exit samples, allowing a single trained kernel to handle different boundary values on the same shape without retraining. The authors extend the approach to Poisson equations via a classical decomposition, where an auxiliary network amortizes the source-induced correction and avoids the singular volume quadrature that breaks direct evaluation. At inference, new boundary values and new source terms produce PDE solutions through re-integration over the fitted kernel, with no retraining. NHMO improves on four prior baselines across all five categories of the MCB-B 3D variable-shape Poisson benchmark and is competitive with mainstream neural operator baselines on a controlled 2D testbed.

Paper Overview

Field: Machine Learning
Authors: Jinjin He, Sinan Wang, Yuchen Sun, Bo Zhu
Release date: 2026-09-28
arXiv: 2609.35752

Abstract (translated)

The authors introduce the 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. Crucially, 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. As a result, one trained kernel can handle different boundary values on a given shape with no retraining.

The method is extended to the Poisson equation via a classical decomposition: an auxiliary network amortizes the source-induced correction, avoiding the singular volume quadrature that breaks direct evaluation. At inference time, both new boundary values and new source terms yield PDE solutions through re-integration over the fitted kernel — again without retraining.

Results

NHMO improves over four previous baselines across all five categories of the MCB-B 3D variable-shape Poisson benchmark.
On a controlled 2D testbed, NHMO is competitive with mainstream neural operator baselines.

Original Abstract (excerpt)

> 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 sour...

---

*Auto-collected on 2026-09-30.*