Neural Harmonic Measure Operator
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.
Related
- Conditioned random walks on linear groups I: construction of the target harmonic measure
- Singularity of harmonic measure for finitely supported random walks
- Singularity of harmonic measure on Poisson--Voronoi tessellations in hyperbolic spaces
- Failure of dimension drop for harmonic measure on Ahlfors regular flat boundaries
- Failure of dimension drop for harmonic measure on Ahlfors regular boundaries of dimension close to $1$
BibTeX
@misc{he2026neural,
title = {{Neural Harmonic Measure Operator}},
author = {Jinjin He and Sinan Wang and Yuchen Sun and Bo Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35752},
eprint = {2609.35752},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35752},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Neural Harmonic Measure Operator
AU - Jinjin He
AU - Sinan Wang
AU - Yuchen Sun
AU - Bo Zhu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35752
UR - https://arxiv.org/abs/2609.35752
AB - 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.
ER -