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Neural Harmonic Measure Operator

Jinjin He, Sinan Wang, Yuchen Sun, Bo Zhu

arXiv:2609.35752v1cs.LGcs.NAmath.NA

TL;DR

Neural operators for elliptic PDEs retain volumetric computation, while existing boundary and stochastic methods require discretization or repeated per-query sampling. NHMO learns a geometry-only harmonic-measure boundary kernel and adds a learned source lift for Poisson problems. It outperforms four baselines across the five MCB-B 3D categories and is competitive with neural-operator baselines on the controlled 2D testbed.

  • Problem

    Existing neural operators retain volumetric computation, while boundary-integral methods require boundary discretization and dense solves and Walk-on-Spheres remains a per-query estimator.

    Source: §1 · p. 1

  • Method

    NHMO learns a geometry-only harmonic-measure density on the boundary from Walk-on-Spheres supervision and uses a learned lift for the Poisson source correction.

    Source: §1 · p. 2, §4.5 · p. 6

  • Results

    NHMO outperforms NGF, Transolver, LNO, and UPT across all five 3D MCB-B categories and is competitive with major neural-operator baselines on the controlled 2D testbed.

    Source: §5.2 · p. 7, §5.3 · p. 8

  • Takeaways & Limitations

    A fitted geometry-only kernel can be reused for arbitrary boundary data on the same shape without retraining, while the learned lift handles source corrections.

    Source: §1 · p. 2

    NHMO targets Dirichlet elliptic problems; Neumann or Robin conditions and other PDE types require generalized measures and remain future work.

    Source: §7 · p. 10

Abstract

from arXiv · show

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.

1 Introduction

Existing neural operators retain volumetric computation, while boundary and stochastic approaches either require dense boundary solves or repeated per-query sampling. NHMO learns a geometry-only boundary measure and combines it with a source lift for Poisson problems.

1 Introduction

  • Classical FEM and finite-difference solvers repeat discretization and solution when geometry, source, or boundary data changes, while neural operators still compute over the volumetric interior.Boundary-element methods require boundary discretization and dense solves, and Walk-on-Spheres remains a per-query estimator.Source: §1 · p. 1
  • Mean rel-L2 is 0.012 for NHMO versus 0.142 for the Green’s-function-style baseline across the two illustrated 3D shapes.Figure 1 compares reference fields, baseline predictions and errors, and NHMO predictions and errors.Source: Figure 1 · p. 2
  • NHMO learns a geometry-only density on Ω × ∂Ω, rather than a singular volumetric kernel on Ω × Ω, to solve elliptic PDEs on variable-shape domains.The boundary kernel can be reused for different boundary data on the same shape; a learned lift handles the Poisson source correction.Source: §1 · pp. 1–2
  • NHMO outperforms four neural-operator baselines on the MCB-B 3D Poisson benchmark and is validated on a controlled 2D testbed.The paper also demonstrates harmonic and drift adaptation experiments on complex 3D shapes.Source: §1 · p. 2

2 Related Work

Prior neural operators regress solutions or solution operators, while Green’s-function approaches learn volumetric kernels and WoS estimates solutions through repeated Brownian-exit sampling. NHMO instead models the boundary measure mediating Dirichlet solutions.

2 Related Work

  • Neural operators learn end-to-end maps from problem data to solutions; geometry-aware variants handle irregular meshes, but still regress solutions or solution operators.Examples include graph, Fourier, and branch–trunk architectures, as well as attention- and graph-based mesh methods.Source: §2 · pp. 2–3
  • Green’s-function methods learn volumetric kernels, while NGF—the closest prior work and principal baseline—trains on precomputed FEM solution fields.NGF integrates the source against the Green’s function in the volume and boundary data against its outward-normal derivative on the boundary.Source: §2 · p. 3
  • Walk-on-Spheres estimates elliptic solutions by Brownian-exit Monte Carlo, whereas NHMO uses WoS as supervision for its kernel rather than as a runtime estimator.Practical WoS walks stop near the boundary, introducing an O(ε) bias.Source: §2 · p. 3
  • NHMO parameterizes harmonic-measure density with a neural network and realizes the balayage decomposition with a learned source amortizer.The paper identifies this as the first such neural parameterization and learned source amortization.Source: §2 · p. 3

3 Background

Harmonic measure describes the boundary distribution of Brownian first exits and integrates boundary data to recover the Dirichlet Laplace solution. Balayage extends this representation to Poisson problems by separating a source-only zero-boundary term.

3.1 Harmonic measure

  • For a point p inside a domain, harmonic measure ωp is the probability distribution of where Brownian motion started at p first exits.Its family over interior points depends on domain geometry, not boundary data.Source: §3.1 · p. 3
  • Integrating continuous boundary data h against ωp gives the Dirichlet Laplace solution, and once ωp is known the representation works for any h.The probabilistic representation underlies Walk-on-Spheres Monte Carlo solvers.Source: §3.1 · p. 3
  • On a Lipschitz domain, harmonic measure has a surface-measure density, the Poisson kernel; NHMO learns this density rather than ωp itself.The figure depicts harmonic measure from Brownian exit locations, with red dots marking boundary patches E.Source: Figure 2 · p. 3, §3.1 · p. 4

3.2 Newtonian potential and balayage

  • Balayage sets w = u − Nf, making w harmonic with boundary values h − Nf|∂Ω before applying the harmonic-measure representation.The resulting source-only piece uf is independent of h, satisfies Δuf = f in Ω, and has zero boundary values; with f ≡ 0 the decomposition recovers the Laplace identity.Source: §3.2 · p. 4

4 Neural Harmonic Measures

NHMO represents elliptic PDE solutions as a boundary integral for boundary data plus a learned residual lift for the source contribution. Its geometry-only kernel is reused across boundary and source inputs, while the lift handles source effects and kernel approximation error.

4.2 Decomposition

  • The solution is factorized into an h-only boundary integral using Kθ and a learned residual field vφ approximating the zero-boundary Poisson particular solution.Setting Kθ to the harmonic-measure density and vφ to the particular solution recovers the decomposition identically.Source: §4.2 · p. 4
  • A fitted geometry-specific kernel handles every boundary/source pair on that shape, and its linear boundary channel changes proportionally under shifts in boundary coefficients.The lift carries no corresponding out-of-distribution guarantee.Source: §4.2 · p. 5

4.3 Boundary kernel Kθ

  • Kθ is a geometry-encoded log-density over interior and boundary points, normalized over boundary quadrature points at inference.The boundary integral is a convex combination of boundary values, and training regularizes its normalization.Source: §4.3 · p. 5

4.4 Field lift vφ

  • The lift uses the kernel’s boundary-integral prediction as a guide and learns a residual; for Poisson it carries the source-induced contribution that the boundary integral cannot represent.In 2D it is a U-Net receiving the domain mask, boundary data, source, and kernel prediction; in 3D it uses cross-attention over shape and source tokens and does not see h.Source: §4.4 · pp. 5–6

4.5 Training

  • Kernel training fits Walk-on-Spheres exit samples to a Gaussian KDE with negative log-likelihood, using probes from near-boundary, mid-interior, and deep-interior bands.Walks stop in an ε-shell, are projected to the boundary, and walks exceeding 128 steps are masked; normalization and mean-value-property regularizers are also applied.Source: §4.5 · p. 6
  • With Kθ frozen, the lift is trained by masked MSE against numerical reference solutions, recomputing the kernel prediction on the fly; no PDE-residual loss is used.Training uses one shape and one (h, f) instance per gradient step.Source: §4.5 · p. 6

5 Experiments

Experiments span intuitive demonstrations, controlled 2D MNIST tests, and the 3D MCB-B Poisson benchmark, followed by runtime and ablation analyses. NHMO performs strongly across these settings, with results also indicating where its remaining Poisson error and inference costs arise.

5.1 Intuitive demonstrations on complex shapes

  • On four graphics meshes, NHMO reaches mean rel-L2 error 0.012 versus 0.142 for the GF-style baseline; on a drifted 2D bunny slice, errors are 0.062 versus 0.323.The same fitted harmonic-measure density is reused across meshes; the drift case adds a small drift-conditioned adapter.Source: §5.1 · p. 7

5.2 2D MNIST: controlled cross-baseline benchmark

  • On 2D MNIST, NHMO has the lowest mean in-distribution error and lightest error tails on both splits; its median degradation is 1.25× under OOD shift versus 4×–8× for nonlinear baselines.The OOD split uses BC coefficients strictly outside the training range; the kernel-only variant also beats the nonlinear baselines on OOD.Source: §5.2 · p. 7, Table 1 · p. 7

5.3 3D MCB-B Poisson

  • NHMO outperforms NGF on all five MCB-B categories and beats Transolver, LNO, and UPT by wider margins; its macro-averaged OOD error rises from 0.193 to 0.263.The OOD comparison uses 40 shifted problems per category and no retraining.Source: Table 2 · p. 8, §5.3 · p. 8
  • Per-shape 3D errors have 95th percentiles below 0.50 in every category, while synthetic harmonic probes show errors ordered as expected for a harmonic-measure density.The smoothest probes have the lowest errors and second-order spherical harmonics the highest, including on the harder Motor and Fitting geometries.Source: §5.3 · p. 8, Table 3 · p. 9, Table 4 · p. 9

5.4 Runtime

  • For a Motor shape, sweeping 1,000 load cases takes NHMO about 18 seconds including geometry setup, versus about 77 seconds for NGF with a preloaded mesh.NHMO's geometry step runs once per shape and can be computed ahead of time for a shape library.Source: §5.4 · p. 9, Table 5 · p. 9

5.5 Ablations

  • In 2D ablations, changing the geometry representation moves median rel-L2 by only 0.27% on test and 0.09% on OOD, and increasing lift capacity gives no in-distribution gain.The reported robustness checks also find the median changes by less than 0.1% above 100 surface samples and WoS supervision converges above about 1,000 samples per query.Source: §5.5 · p. 9

6 Discussion

The discussion motivates a boundary-density kernel paired with an amortized source field, and examines how the lift handles residual dependence on boundary data.

Why a boundary density and an amortized source field

  • Unlike learned volumetric Green’s functions, NHMO trains its boundary density from WoS exit samples and avoids a new singular volume quadrature when the source changes.The discussion also notes that a learned volumetric integrand with a log singularity reached 6.7% / 5.2% mean / median error, and its boundary derivative was not a valid Poisson kernel.Source: §6 · p. 10

The boundary-data dependence of the lift

  • In 2D, the fitted kernel leaves a boundary-data-dependent residual, so the lift must see h or an explicit residual head must correct it.On 205 Laplace pairs, the lift correction correlates with that residual at median 0.97 and removes 71% of it.Source: §6 · p. 10
  • Separating source and boundary-residual roles gives 1.84% / 1.74% test mean / median and 2.56% / 2.39% OOD with a residual head, versus 2.1% / 2.0% and 2.6% / 2.5% for the two-term model.The 3D lift sees only geometry and source, as in the classical split; the 2D residual is linked mainly to KDE targets built on simplified polyline contours.Source: §6 · p. 10

7 Conclusion

NHMO builds a neural operator around harmonic measure for Dirichlet Laplace problems and extends it to Poisson sources with a learned lift. The authors suggest canonical objects from classical analysis may provide PDE-aligned inductive biases, while identifying current scope and workload boundaries.

7 Conclusion

  • NHMO models the boundary harmonic-measure density and pairs it with a learned balayage source lift for Poisson problems.The kernel can be tested independently with synthetic-harmonic probes.Source: §7 · p. 10
  • The current target is Dirichlet elliptic problems; Neumann/Robin conditions and other PDE types remain future work, and OOD sources may degrade the source lift.Per-shape precomputation also does not benefit workloads with only one problem per shape.Source: §7 · p. 10

B Harmonic measure: derivations and properties

For each interior point, harmonic measure represents the Dirichlet solution as an integral of boundary data against a probability measure. Under sufficient boundary regularity, its density is the negative outward-normal derivative of the Dirichlet Green’s function.

Harmonic measure

  • The Riesz representation theorem gives a unique boundary probability measure whose integral against h recovers u(p), and positivity yields the maximum principle.The stated existence result applies on bounded Lipschitz domains.Source: B Harmonic measure: derivations and… · p. 16

Green’s-function trace formula

  • For sufficiently regular boundaries, harmonic measure has a surface density equal almost everywhere to the negative nontangential outward-normal derivative of the Dirichlet Green’s function.The learned kernel approximates this density, while quadrature weights discretize surface measure.Source: B Harmonic measure: derivations and… · p. 17

Walk-on-Spheres as a sampler of ωp

  • Walk-on-Spheres chains uniform exits from the largest contained spheres, producing exit points distributed according to harmonic measure for the isotropic Laplacian.Averaging boundary values at these exits estimates u(p) directly; surface measure is needed when integrating the learned density, not to sample the exit law.Source: B Harmonic measure: derivations and… · p. 17
  • The implemented WoS sampler has ε-shell termination bias O(ε), masks walks exceeding 128 steps, and uses rasterized signed-distance fields to compute sphere radii.The uniform-sphere jump applies to the Euclidean isotropic Laplacian; drift or varying coefficients require transformed walks.Source: B Harmonic measure: derivations and… · p. 17

Further properties

C Architecture details

The appendix specifies NHMO’s architecture and training procedure, including a geometry-conditioned boundary kernel, a lift for field corrections, and staged losses and optimization schedules. The 3D configuration uses a distinct source-conditioned lift and kernel settings.

C Architecture details

  • The canonical 2D shape encoder maps boundary samples to 64 slice tokens and processes them with a 3-layer transformer; the kernel head uses cross-attention and quadrature-weighted softmax normalization.The 2D field lift is a symmetric U-Net; the canonical model totals approximately 11.2M parameters.Source: C Architecture details · pp. 17–18
  • The 3D MCB-B model uses a shape encoder, kernel head, and source-conditioned cross-attention lift whose output vanishes on the boundary.The lift attends to the frozen shape latent and source tokens, and receives neither boundary data h nor harmonic solution uh.Source: C Architecture details · p. 18

D.1 Kernel losses (Stage 1)

  • Stage 1 trains the kernel using Walk-on-Spheres exit-point likelihood plus mean-value, boundary-limit, and soft-mass losses; the 2D setup instead uses smoothed exit-point targets and KL plus L1 loss.In 3D, the mean-value loss compares normalized densities at an interior point and at sampled points on a sphere; the boundary-limit loss concentrates mass near a boundary anchor.Source: §D.1 · pp. 18–19

D.2 Loss-weight schedule

  • The kernel loss combines NLL, mean-value, boundary-limit, and mass-normalization terms with a step-dependent weight schedule.Warm-in suppresses the mean-value term early; the reported MCB-B results use 30,000 Stage 1 steps and do not use the post-80k branch.Source: §D.2 · p. 19

D.3 Stage 1 optimizer and LR schedule

  • The 3D kernel uses AdamW for 30,000 steps, with 1,000-step warmup, cosine decay, gradient clipping, and a log-density cap.Each step uses 2,000 surface samples and 512 interior anchors; the 2D kernel trains for 60,000 steps without a logit cap.Source: §D.3 · p. 19

D.4 Stage 2 optimizer and LR schedule, with warm-start

  • Stage 2 trains the lift on 512 source-probe samples per problem, using warmup-and-cosine optimization and typically 30,000–50,000 steps per category.Warm-start rounds reload the prior best lift weights; the 2D field lift trains for 10,000 steps, while the source-lift/residual variant uses 30,000 and 10,000 steps respectively.Source: §D.4 · p. 19

D.5 WoS supervision budget and variance

The appendix reports WoS supervision settings and variance, training costs, baseline implementation choices, and details of harmonic and drift demonstrations. The demonstrations include a shared-kernel harmonic export and a drift-conditioned adaptation that keeps the Laplace kernel frozen.

D.5 WoS supervision budget and variance

  • At the canonical 104 walks, supervision noise is well below kernel-fit error, and kernel quality is stable above approximately 1,000 walks per probe.Source: §D.5 · p. 20

D.6 Training cost

  • On one A100, a 2D kernel and field lift each take approximately one hour to train; a 3D kernel takes 8.8–24 hours per category.The stated 3D duration is for 30,000 steps; the Motor run used a shared GPU.Source: §D.6 · p. 20

E Baseline implementations

  • The benchmark uses official public implementations for Transolver, LNO, UPT, and BENO, with data loaders adapted to the study’s format.The 3D NGF comparison uses released data and the same test split; 2D NGF results come from a port because the authors did not release a 2D code path or evaluation data.Source: E Baseline implementations · p. 20

F Intuitive demonstrations: details

F.1 3D harmonic on common graphics meshes

  • The 3D harmonic demonstration tests eight cases across four graphics meshes and two boundary signals, fitting one shape-conditioned kernel across all four shapes.The visualization export adds a residual head trained against FEM references and distilled into a single forward pass.Source: §F.1 · p. 21, Table 9 · p. 21, Figure 6 · p. 21

F.2 Drift adaptation on a bunny slice

  • The bunny-slice drift demonstration compares a drift-conditioned adapter and residual head against a drift-conditioned Green-function-style baseline under a matched 4,000-step budget.Both methods warm-start from the same frozen Laplace kernel; the NHMO kernel itself is not retrained.Source: §F.2 · p. 22

G.1 Setup details

The appendix specifies the 2D MNIST setup and NGF port, then provides additional qualitative comparisons and an out-of-distribution 3D evaluation protocol. The 3D study shifts boundary-data and source coefficients beyond training ranges without retraining.

G.1 Setup details

  • The 2D benchmark uses MNIST digit masks upsampled to 256×256, samples two parametric boundary-condition families and four source families, and reports in-distribution and OOD splits.Digits 0, 6, 8, and 9 have multiply-connected domains.Source: §G.1 · pp. 22–23

G.2 NGF 2D port

  • The 2D NGF port preserves the official architecture and optimization settings while adapting geometry input and boundary handling to pixel grids.The aligned port differs from the initial port in several implementation choices, with the comparison reported in Table 11.Source: §G.2 · p. 24, Table 11 · p. 24

G.3 Additional qualitative comparisons

  • Figures 9 and 10 add 24 out-of-distribution shapes using the existing per-row layout and color-scale convention.Source: §G.3 · p. 24

H Additional MCB-B qualitative comparisons

I 3D coefficient-OOD study

  • The 3D coefficient-OOD study evaluates 10 shapes per category on four problems each, using boundary and source coefficients from U[1, 2] outside training ranges.It also includes a Laplace-only track with f ≡ 0 to isolate boundary extrapolation; the supplied passage identifies Table 12 as reporting results but gives no outcome values.Source: I 3D coefficient-OOD study · p. 24

J Inference speed: caching is implied by the factorization

NHMO’s geometry-only factorization allows the boundary kernel to be cached per shape and reused across boundary data and sources. The speed benefit is chiefly at deployment when many problems reuse the same geometries.

J Inference speed: caching is implied by the factorization

  • The effective kernel cache is a deterministic function of geometry and is fp32-bit-identical to recomputation across boundary data and sources.The reported cache uses about 8 MB per shape at 128×128 with 200 surface samples.Source: J Inference speed: caching is implied… · p. 24
  • Query folding computes shared context keys and values once, with identical fp32 predictions and unchanged relative L2 errors against FEM references.The faster-build path redraws samples, so its predictions are not bitwise identical to the original pipeline.Source: J Inference speed: caching is implied… · p. 25
  • The cache follows from NHMO’s geometry-only factorization; Transolver, UPT, LNO, and BENO mix geometry with problem inputs and admit no per-shape cache.NGF can in principle separate a geometry-only state from per-problem read-out.Source: J Inference speed: caching is implied… · pp. 25–26
  • Per-shape precompute costs O(nin nsurf dkernel), while per-problem cost includes O((nin + nsurf) d) and the lift U-Net term O(R^2 c D).The stated per-problem complexity class matches parametric baselines’ Galerkin-style forward.Source: J Inference speed: caching is implied… · p. 26
  • At K=1, NHMO pays the geometry step for each problem, so its deployment speed advantage is for repeated queries on the same geometries.For new mesh-based shapes, the geometry step is cheaper than meshing; grid-based baselines need no such step on grid inputs.Source: J Inference speed: caching is implied… · p. 26

K.1 Separating the lift’s roles: source lift and residual head

The ablations examine how the source lift and boundary residual correction complement the kernel, then test the model’s robustness across supervision, quadrature, corpus, and representation choices.

K.1 Separating the lift’s roles: source lift and residual head

  • 1.87 ± 0.05% test mean and 2.48 ± 0.05% OOD for the three-term model, versus 2.09 ± 0.03% and 2.60 ± 0.05% for the single lift.The three-term model’s source lift sees only (Ω, f); its residual head adds a boundary-dependent correction.Source: Table 13 · p. 27, §K.1 · p. 27
  • The single lift’s boundary correction correlates with the kernel-fit residual at median 0.97 and removes 71% of it; the source lift alone has 6.1% test mean versus 2.1% for the full model.The correction is small relative to ∥uh∥ (median 4.8%) and concentrates 82% of its spectral energy in the lowest tenth of radial frequencies.Source: §K.1 · p. 27

K.2 Lift removal (kernel-only vs. kernel + lift)

  • Kernel-only already beats every nonlinear end-to-end baseline on OOD; doubling lift parameters from 6.4M to 11.3M gives no in-distribution improvement and only marginal OOD gain.The ablation attributes the result to factorization rather than lift capacity.Source: §K.2 · p. 28, §K.3 · p. 28, Table 14 · p. 28

K.4 KDE bandwidth σ for Walk-on-Spheres supervision

  • The kernel is robust across a 5× KDE σ range (0.1%–0.5% of domain width) and above 1,000 walks per query; below 1,000, noisy KDE supervision degrades it.The canonical run uses 10,000 walks per query.Source: §K.4 · p. 28, §K.6 · p. 28, Table 15 · p. 28

K.8 Per-MNIST-class breakdown (single mixed-corpus uniformity)

  • Across 10 MNIST classes, one 5,000-shape mixed-corpus model has rel-L2 spread of 0.53% on test and 1.87% on OOD.Digit 8 is worst on OOD at 3.81%, yet beats every nonlinear end-to-end baseline’s overall mean.Source: §K.5 · p. 28, §K.8 · p. 29, Table 18 · p. 29

K.9 Representation invariance: SDF vs. point-cloud encoder

  • 0.27% absolute median on test and 0.09% on OOD are the costs of swapping the shape encoder from point cloud to SDF.The SDF encoder’s OOD/test gap is 1.14× versus 1.25× for the canonical encoder.Source: Table 19 · p. 29, §K.9 · p. 30

K.10 Learned volumetric integrand

  • The learned volumetric integrand reaches 6.7 / 5.2 test mean / median, no better than the source lift, while requiring O(NpNq) network evaluations per problem.Its boundary derivative integrates to ∼10^-3 and is negative over roughly half the boundary, meeting neither Poisson-kernel property reported here.Source: §K.10 · p. 30

K.11 Origin of the kernel-fit residual

  • Integrating the KDE targets gives 5.1% error despite 100× more walks, while walks on mask geometry give 0.07%; aligning KDE nodes to rasterized masks lowers kernel-only error from 5.9% to 3.2%.The mask-aligned kernel-plus-lift model reaches 1.8% / 1.7% test mean / median and 2.1% / 2.2% OOD, versus 2.1% / 2.0% and 2.6% / 2.5% for the Table 1 model.Source: §K.11 · p. 30

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