Neural Harmonic Measure Operator

What it claims

NHMO learns the harmonic-measure density for a domain: the Poisson kernel dω_p/dσ on Ω×∂Ω. Prior learned Green's-function operators instead learn a singular volumetric kernel on Ω×Ω. Because the kernel depends only on geometry, it is fit once per shape distribution and reused for any boundary data. A separate learned lift handles Poisson sources.

On two complex 3D shapes (armadillo, bunny), with one shape-conditioned kernel covering both, the paper reports mean relative L2 error of 0.012 versus 0.142 for a Green's-function-style baseline.

How it works

• Kernel K_θ. A geometry encoder maps a discretization of Ω to a shape latent. A kernel head outputs a log-density g(p, ζ, ψ_Ω), with Fourier features of (p, ζ) and the distance ‖p−ζ‖ as extra inputs.
• At inference the kernel is normalized over the boundary quadrature weights. The Laplace term is therefore an exact convex combination of boundary values, so min h ≤ u_h ≤ max h holds by construction.
• A training-time penalty keeps the unnormalized sum near 1.
• Supervision. The kernel is trained from Walk-on-Spheres exit samples: M Brownian exit points per interior probe. Walks stop in an ε-shell and are projected to ∂Ω. Kernel supervision needs no FEM solutions or tetrahedral meshes.
• The closest baseline, NGF, by contrast trains on precomputed FEM fields.
• Lift v_φ. The lift is architected differently in each dimension:
• 2D: a U-Net on the ambient grid. Its inputs are the interior mask, h extended to the grid, f, and the kernel's own prediction u_h. Because it sees h, it also absorbs kernel fit error.
• 3D: a cross-attention head. It is queried by a Fourier embedding of p, with the shape latent and source-sample tokens (q_j, f(q_j)) as context. It does not see h. Its output is multiplied by max(0, −SDF(p)).
• Training is two-stage: kernel first, then lift.

flowchart LR
  G[Geometry Ω] --> E[Encoder ψ_Ω]
  E --> K[Kernel K_θ, normalized]
  H[Boundary data h] --> M[Boundary matvec u_h]
  K --> M
  F[Source f] --> L[Lift v_φ]
  E --> L
  M --> S((u = u_h + v_φ))
  L --> S

Evidence

• 3D shapes: armadillo and bunny, rel-L2 0.012 versus 0.142 against the GF-style baseline (Fig. 1).
• MCB-B 3D Poisson: the paper reports that NHMO beats four neural-operator baselines across all five categories. The fetched text cuts off before the numbers.
• 2D MNIST-shape testbed: NHMO is compared qualitatively against UPT, Transolver and NGF on OOD Laplace and Poisson examples, using finite-difference ground truth.
• Sanity check unique to this design: integrating the trained kernel against analytic harmonic functions on held-out shapes should return h(p).
• 2D test functions: x, xy, x²−y², eˣcos y.
• 3D test functions: low-degree solid spherical harmonics.
• End-to-end operators cannot run this check. Results are in §5.3, which is not in the fetched text.

How to use it

Inference cost structure:

1. Materialize K_eff = [w_j K_θ(p_i, ζ_j)] once per geometry.
2. For each new (h, f), the cost is one boundary matvec plus one lift forward pass.

Wall-clock numbers are in §5.4 and were not in the fetched text. No code, weights or licence are mentioned.

The same kernel can be reused for constant-drift Laplace by swapping the lift for a small drift-conditioned adapter. This is demonstrated on the bunny.

Limits

• OOD guarantee covers only the kernel. Linearity in h guarantees out-of-distribution behaviour under boundary-coefficient shifts for the kernel channel only. The authors state the lift carries no such guarantee.
• Kernel is per geometry distribution. It is trained per distribution of shapes, not universally. Generalization to shapes outside the training distribution is not addressed in the fetched text.
• WoS bias. WoS supervision carries an O(ε) bias from the stopping shell.
• Scope. Only d ∈ {2, 3} bounded Lipschitz domains with Dirichlet conditions are covered. Neumann and mixed conditions are not addressed.
• Truncated text. The fetched text ends in §4.5, so the quantitative tables, timings and the §6 analysis of the lift's h-dependence were not available to verify.