Neural Harmonic Measure Operator

AI summary

Overview

Research area: Machine learning for scientific computing, specifically neural operators that solve partial differential equations (PDEs) on variable-shape domains.

Technical level: Advanced. The paper assumes familiarity with elliptic PDEs, potential theory (harmonic measure, Green's functions, balayage), Monte Carlo solvers, and transformer architectures.

One-sentence scope: The paper introduces the Neural Harmonic Measure Operator (NHMO), a boundary-only neural operator that learns the geometry-dependent density of the harmonic measure so that one trained kernel solves Dirichlet and Poisson problems on a given shape for arbitrary boundary data and sources without retraining.

What This Paper Is About

Classical PDE solvers such as the finite element method (FEM) and finite differences discretize the volumetric interior of a domain and must re-run the entire pipeline whenever the geometry, source, or boundary data change. Existing neural operators amortize this cost but still operate on the bulk interior, so their compute scales with volumetric discretization rather than with the boundary. NHMO instead learns a probability density on the boundary alone — the harmonic measure — which depends only on geometry, and integrates it against whatever boundary data and source are supplied.

Key Contributions

1. A learnable boundary kernel for the harmonic-measure density. NHMO parameterizes the density dω_p/dσ as a transformer-based kernel K_θ(p, ζ; Ω) that depends on the geometry Ω only — not on boundary data h or source f — and is supervised by Walk-on-Spheres exit samples rather than by FEM solutions or tetrahedral meshes.

2. A Poisson extension via the balayage decomposition. The classical balayage split is realized with a zero-boundary-gauge residual lift v_φ that reuses K_θ to amortize the source-induced correction and avoids the singular volume quadrature that breaks direct evaluation of a learned volumetric kernel.

3. Reuse without retraining. Because the kernel is geometry-only, a single fitted K_θ handles every (h, f) pair on a shape, and the boundary term u_h is a linear functional of h, so coefficient shifts — including those outside the training range — change the prediction proportionally.

4. Empirical validation on two benchmarks. NHMO outperforms four prior baselines on the 3D MCB-B variable-shape Poisson benchmark across all five categories and is competitive with major neural-operator baselines on a controlled 2D MNIST testbed, plus two intuitive demos on complex 3D shapes (armadillo, bunny, fandisk, lucy) and a constant-drift Laplace experiment.

Main Findings

• 3D MCB-B Poisson benchmark: NHMO achieves relative L2 errors of 0.216 (Nut), 0.188 (Gear), 0.284 (Motor), 0.147 (Fitting), and 0.131 (Screws & Bolts), beating NGF (0.275, 0.243, 0.338, 0.160, 0.189) on all five categories and outperforming Transolver, LNO, and UPT by wider margins. The benchmark uses 20 unseen test shapes × 16 unseen (h, f) problems per category (320 test pairs), with 200 training shapes per category.

• Per-shape robustness: Median error sits below NGF's reported mean for all five categories, and the 95th-percentile error is below 0.50 on every category, so no single test shape fails catastrophically.

• Out-of-distribution behavior: When h and f are shifted outside their training ranges without retraining (40 problems per category), NGF's macro-averaged error rises from 0.241 to 0.615, while NHMO's rises from 0.193 to 0.263. On a Laplace-only track with the same boundary data, NHMO averages 0.099 against 0.60–0.64 for NGF, suggesting most of NHMO's remaining degradation comes from the source-conditioned lift.

• 2D MNIST controlled testbed: NHMO (kernel + lift) reaches in-distribution median/mean of 2.0/2.1 with p95 of 3.3, and OOD (BC coefficients drawn from U[+1,+2]) median/mean of 2.5/2.6 with p95 of 4.2. Under the OOD shift it degrades by 1.25× (median), while the nonlinear end-to-end baselines (Transolver, LNO, UPT, BENO) degrade by 4× to 8×.

• Kernel-only variant is already strong: The kernel-only variant (v_φ ≡ 0) beats all nonlinear end-to-end baselines on OOD (5.6/6.8 median/mean), confirming the lift is a small correction rather than the source of the gains.

• NGF's 2D port has heavy error tails: NGF's in-distribution Poisson errors reach p95 of 16.9% and max of 40.2%, against NHMO's 3.3% and 7.0%, consistent with its rank-limited bilinear source coupling.

• Seed stability: Across five training seeds of the lift, the 2D test mean is 2.09 ± 0.03% and the OOD mean 2.60 ± 0.05%.

• Harmonic-measure density check (3D): Integrating K_θ against analytically harmonic functions (x, xy, x²−y², Y_{2,0}, e^x cos y, e^x sin y) returns small errors ordered consistently with a true harmonic-measure density — smoothest probes lowest, second-order spherical harmonics highest — across all five categories.

• Intuitive 3D demos: With a single K_θ fitted once across four graphics meshes, NHMO reaches mean relative L2 of 0.012 versus 0.142 for a Green's-function-style baseline. On constant-drift Laplace over a 2D bunny slice, the same K_θ plus a small drift-conditioned adapter reaches 0.062 versus 0.323.

• Runtime: The geometry step (shape encoding and K_eff evaluation) takes 7.9 s (3D Nut) and 10.3 s (3D

Authors’ 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.