Neural Harmonic Measure Operator

Jinjin He, Sinan Wang, Yuchen Sun, Bo Zhu
Georgia Institute of Technology
NeurIPS 2026
Paper OpenReview Code Checkpoints & Data
NHMO overview: harmonic-measure density with Walk-on-Spheres paths and the residual lift
Figure 1 — NHMO overview. Harmonic-measure density $K_\theta \approx d\omega_p/d\sigma$ with Walk-on-Spheres paths, and the residual lift $v_\varphi$, composed additively (bottom). Each WoS step lands on the largest circle inside $\Omega$; a walk stops in the $\varepsilon$-shell (drawn wider than in practice) and is projected to $\partial\Omega$. Right: once fitted for $\Omega$, $K_\theta$ solves any new boundary datum without retraining.

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.

Method

For a Poisson problem $\Delta u = f$ in $\Omega$ with $u = h$ on $\partial\Omega$, the classical balayage split writes the solution as a boundary integral of $h$ against the harmonic measure $\omega_p$ plus a source-only term that vanishes on the boundary. NHMO learns both pieces:

$$u(p) \;=\; \big\langle h,\, K_\theta(p,\cdot\,;\Omega) \big\rangle_{\partial\Omega} \;+\; v_\varphi(p;\Omega,h,f).$$

The kernel $K_\theta$ approximates the density $d\omega_p/d\sigma$ and depends on the geometry alone. It is trained from Walk-on-Spheres exit points, without solution data. Once it is evaluated for a shape, every new boundary condition is a quadrature against the same kernel matrix, and the lift $v_\varphi$ adds the source contribution.

Results

MethodNutGearMotorFittingScrews & Bolts
Transolver0.3200.2810.4070.1800.221
LNO0.3720.4660.5280.2590.239
UPT0.5160.5070.7650.3920.358
NGF0.2750.2430.3380.1600.189
NHMO (ours)0.2160.1880.2840.1470.131
MCB-B Poisson benchmark. MCB-B is a subset of MCB, with FEM solutions released by NGF. Relative $L_2$ error against the FEM reference, mean over 20 unseen test shapes $\times$ 16 unseen $(h, f)$ problems per category. Baseline numbers are from the NGF paper under identical evaluation.
Qualitative comparison with NGF on MCB-B Poisson
Figure 2 — Qualitative comparison on MCB-B Poisson. Per shape, five panels show the FEM reference, the NGF prediction and error, and our prediction and error; the cut face is colored by the field and the back half is drawn as a gray ghost surface. Within each row, the reference and the predictions share one color scale and the four error panels share a second one.
2D MNIST out-of-distribution examples
Figure 3 — 2D MNIST, out-of-distribution boundary coefficients. Per row, a Laplace example (left) and a Poisson example (right). Columns: finite-difference reference and the absolute error of UPT, Transolver, NGF, and NHMO. Within each example, the four error panels share one color scale.

BibTeX

@inproceedings{
he2026nhmo,
title={Neural Harmonic Measure Operator},
author={Jinjin He and Sinan Wang and Yuchen Sun and Bo Zhu},
booktitle={Advances in Neural Information Processing Systems},
year={2026},
url={https://openreview.net/forum?id=csUQ0IwX0R}
}