Neural Harmonic Measure Operator

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.

2. Document Verification & Archival Data

• Contributing Researchers: Jinjin He, Sinan Wang, Yuchen Sun, Bo Zhu
• Submission Date: September 28, 2026
• Full Preprint Document: Download Official PDF
• Permanent Archive Record: arXiv:2609.35752v1

3. Academic Citation Reference

Standard Reference (APA Format):

Jinjin He, et al. (2026). Neural Harmonic Measure Operator. arXiv:2609.35752v1. https://arxiv.org/abs/2609.35752v1

Academic Field: Machine Learning | Document Identifier: arXiv:2609.35752v1

BibTeX Entry:

@article{arxiv_2609.35752v1,
  author    = {Jinjin He and Sinan Wang and Yuchen Sun and Bo Zhu},
  title     = {{Neural Harmonic Measure Operator}},
  journal   = {arXiv preprint arXiv:2609.35752v1},
  year      = {2026},
  url       = {https://arxiv.org/abs/2609.35752v1}
}