Neural Harmonic Measure Operator
NHMO learns a geometry-only kernel that solves variable-shape elliptic PDEs without retraining.
Neural Harmonic Measure Operator (NHMO) parameterizes a domain's harmonic-measure density as a transformer-based boundary kernel supervised by Walk-on-Spheres exit samples. Because the measure depends only on geometry, different Dirichlet boundary values on the same shape are solved by re-integration without retraining. Poisson problems use a classical decomposition plus an auxiliary network that amortizes the source correction and avoids singular volume quadrature. NHMO improves on four prior baselines across all five categories of the MCB-B 3D variable-shape Poisson benchmark and is competitive with major neural-operator baselines on a controlled 2D testbed.
- Harmonic measure depends on geometry, not on boundary data.
- A transformer kernel is trained with Walk-on-Spheres exit samples.
- New boundary values and sources need re-integration, not retraining.
- NHMO beats four baselines on every MCB-B 3D Poisson category.
Full article160 words · extracted from arxiv.org · click to collapse
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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.35752