Neural Harmonic Measure Operator

Jinjin He; Sinan Wang; Yuchen Sun; Bo Zhu

cs.LGSep 28, 2026

Neural Harmonic Measure Operator

Authors: Jinjin He, Sinan Wang, Yuchen Sun, Bo Zhu

Organizations: Georgia Institute of Technology

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.

Figures & tables

Figure 1 : Intuitive demonstration on complex 3D shapes (top: armadillo; bottom: bunny), with a single shape-conditioned $K_{\theta}$ encoding both. Mean rel- $L_{2}$ is $0.012$ vs $0.142$ (Ours vs GF style; § 5.1 ). Columns: GT (reference solution), the Green’s-function-style (GF-style) baseline, its absolute error, our prediction, and our absolute error. Per row, fields share the left color bar and errors the right.

Figure 2 : Harmonic measure $(\omega_{p})$ from Brownian exit locations. Red dots denote small boundary patches (E).

Figure 3 : NHMO overview. Harmonic-measure density $K_{\theta}\approx d\omega_{p}/d\sigma$ with WoS paths and residual lift $v_{\varphi}$ , composed additively as in ( 6 ) (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: $K_{\theta}$ once fit for $\Omega$ solves any new boundary datum without retraining.

Figure 4 : 2D MNIST out-of-distribution (OOD) qualitative. Per row, a Laplace example (left) paired with a Poisson example (right); columns are GT (finite-difference reference) and the absolute error $|\text{pred}-\text{GT}|$ of UPT, Transolver, NGF, and Ours. Within each example (half-row), the four error panels share one color scale and GT has its own. Additional shapes in Appendix G.3 .

	test (in-dist)		test_ood
Method	median / mean	p95	median / mean	p95	OOD/test	per-problem
Transolver ( Wu et al., 2024 )	$4.4$ / $5.3$	$10.7$	$36.4$ / $36.2$	$55.2$	$8.3{\times}$	$12.5$ ms
LNO ( Wang and Wang, 2024 )	$5.2$ / $6.4$	—	$23.3$ / $33.2$	$81.6$	$4.5{\times}$	$13.2$ ms
UPT ( Alkin et al., 2024 )	$5.7$ / $6.3$	—	$26.5$ / $26.6$	—	$4.6{\times}$	$7.6$ ms
BENO ( Wang et al., 2024 )	$5.9$ / $7.0$	$16.1$	$44.1$ / $40.5$	$65.8$	$7.5{\times}$	—
NGF ( Yoo et al., 2025 ) (2D port)	$2.0$ / $3.9$	$16.9$	${3.9}$ / ${4.2}$	$6.0$	${1.93{\times}}$	$11.6$ ms

Table 1: 2D MNIST paramBC: relative

L_{2}

error (%) over the full interior, in-distribution and OOD (

U[+1,+2]

). p95: per-pair

95

th percentile. The OOD/test ratio (of medians) measures BC-coefficient extrapolation; lower is better. Per-problem inference is on a single A100 with the per-shape kernel matrix cached. NGF is released only in 3D; its row is our 2D port of the official code and training setup (Appendix G.2 ).

Method	Nut	Gear	Motor	Fitting	Screws & Bolts
Transolver ( Wu et al., 2024 )	0.320	0.281	0.407	0.180	0.221
LNO ( Wang and Wang, 2024 )	0.372	0.466	0.528	0.259	0.239
UPT ( Alkin et al., 2024 )	0.516	0.507	0.765	0.392	0.358
NGF ( Yoo et al., 2025 )	0.275	0.243	0.338	0.160	0.189
Ours (NHMO)	0.216	0.188	0.284	0.147	0.131

Table 2 : MCB-B Poisson benchmark: relative

L_{2}

error against the FEM reference, mean over

20

unseen test shapes

\times

16

unseen

(h,f)

problems per category. Lower is better. NGF, Transolver, LNO, UPT numbers are reported in NGF Table 2 under identical evaluation.

Figure 5 : Qualitative comparison on MCB-B Poisson. Per shape, five panels show GT (the FEM reference), NGF prediction, NGF error, our prediction, and our error; the cut face is colored by the field, the back half by a gray ghost surface. Within each row, GT and the predictions share one color scale and the four error panels share a second one; panels are stretched to a common aspect ratio. Two shapes per row across all five MCB-B categories. Additional shapes in Appendix H .

	Mean	Median	p95
Nut	0.216	0.215	0.329
Gear	0.188	0.144	0.378
Motor	0.284	0.265	0.450
Fitting	0.147	0.111	0.309
Screws	0.131	0.103	0.315

Table 3 : Per-shape distribution of relative

L_{2}

error (ours,

320

-pair test set).

	Geometry step (per shape)		Per problem
Setting	NHMO	baselines	NHMO	baselines
2D MNIST ( $128^{2}$ )	$6.6$ s	grid input	$4.0$ ms	$7.6$ – $13.2$ ms
3D Nut	$7.9$ s	$37$ s (tet meshing)	$7.5$ ms	$0.24$ s (NGF)
3D Motor	$10.3$ s	$48$ s (tet meshing)	$7.9$ ms	$0.25$ s / $77$ ms (NGF)

Table 5: Runtime on a single A100. The geometry step runs once per shape: shape encoding and

K_{\text{eff}}

for NHMO, tetrahedral meshing at the released resolution (fTetWild, CPU) for the MCB-B baselines.

Appendix figures & tables20 assets

Supplementary material from the paper’s appendix.

Appendix

Geometry
Symbol	Meaning
$\Omega\subset\mathbb{R}^{d}$	bounded Lipschitz domain in dimension $d\in\{2,3\}$
$\partial\Omega$	boundary of $\Omega$
$p,q\in\Omega$	interior points
$\zeta\in\partial\Omega$	boundary point
$\nu_{\zeta}$	outward unit normal to $\partial\Omega$ at $\zeta$

Appendix

Table 6: Notation used throughout the paper.

	2D MNIST	3D MCB-B
$\varepsilon$ (normalized domain)	$10^{-3}$	$10^{-3}$
step cap	$128$	$128$
walks per probe	$10^{4}$ (precomputed)	$4$ fresh exits per step (online)
probes	$32$ per shape	$8$ per gradient step
target	KDE, $\sigma=0.2\%$ of domain width, $512$ nodes	exit-point likelihood
masked	$0.0095\%$	$0.05$ – $0.40\%$ (by category)

Appendix

Table 7: WoS supervision settings of the reported kernels. Masked: fraction of walks that do not reach the

\varepsilon

-shell within the step cap.

	Nut	Gear	Motor	Fitting	Screws
walks per second	$7.6{\times}10^{8}$	$8.1{\times}10^{8}$	$7.5{\times}10^{8}$	$7.8{\times}10^{8}$	$7.8{\times}10^{8}$
mean steps per walk	$14.0$	$12.8$	$15.9$	$12.8$	$14.3$
masked fraction	$0.125\%$	$0.045\%$	$0.110\%$	$0.076\%$	$0.402\%$

Appendix

Table 8: Per-category WoS statistics in 3D (A100): throughput and mean number of steps per walk.

shape	$h$	Ours	GF style	ratio
armadillo	$\sin x$	0.011	0.117	$10.4{\times}$
armadillo	$\sin z$	0.011	0.103	$9.2{\times}$
bunny	$\sin x$	0.019	0.249	$13.2{\times}$
bunny	$\sin z$	0.016	0.214	$13.4{\times}$
fandisk	$\sin x$	0.011	0.142	$12.6{\times}$
fandisk	$\sin z$	0.012	0.138	$11.8{\times}$

Appendix

Table 9 : 3D harmonic on common graphics meshes: per-case rel-

L_{2}

against FEM reference.

Figure 6 : Additional intuitive 3D harmonic comparisons. Top: lucy. Bottom: fandisk. Same five-column layout as Figure 1 .

drift $\beta$	boundary $h$	Ours	GF style
$(2,0,0)$	$\sin x$	0.060	0.366
$(2,0,0)$	$\sin z$	0.062	0.280
$(-2,0,0)$	$\sin x$	0.062	0.378
$(-2,0,0)$	$\sin z$	0.065	0.326
$(0,0,2)$	$\sin x$	0.069	0.364
$(0,0,2)$	$\sin z$	0.061	0.293

Appendix

Table 10 : Drift adaptation: per-slice test rel-

L_{2}

on the bunny slice.

Figure 7 : Bunny drift-diffusion qualitative, $h(x,y,z)=\sin(6x)$ . Rows: four drift vectors $\beta=(1.5,1,0)$ , $(-2,0,0)$ , $(2,0,0)$ , $(0,0,2)$ . Columns 1–5: GT, GF style, GF style $-$ GT, Ours, Ours $-$ GT. Columns 6–8: drift-induced field $u_{\beta}$ minus the mean over the four drifts, highlighting the dipole structure aligned with each $\beta$ (Gaussian-blurred for clarity; metrics in Table 10 use unblurred fields).

Figure 8 : Bunny drift-diffusion qualitative, $h(x,y,z)=\sin(6z)$ . Same layout as Figure 7 .

	test mean / median	test p95 / max	OOD mean / median	OOD p95 / max
initial	$41.8$ / $22.1$	—	$24.5$ / $18.1$	—
aligned (40 epochs)	$3.87$ / $2.00$	$16.9$ / $40.2$	$4.20$ / $3.85$	$6.0$ / $10.1$

Appendix

Table 11 : NGF 2D port: initial port versus the port aligned with the official setup (relative

L_{2}

, %, mixed splits).

Figure 9 : 2D MNIST OOD qualitative (additional, set 1 of 2). $12$ shapes, random pick from remaining Laplace and Poisson cases; each GT tile is tagged with its problem type.

Figure 10 : 2D MNIST OOD qualitative (additional, set 2 of 2). $12$ shapes, random pick from remaining Laplace and Poisson cases; each GT tile is tagged with its problem type.

Figure 11 : Additional MCB-B Poisson qualitative comparisons. 14 shapes (7 rows $\times$ 2 shapes per row) in the same layout and color-scale convention as Figure 5 .

Track	Method	Nut	Gear	Motor	Fitting	Screws	Macro
Poisson-OOD	NGF (released)	0.678	0.605	0.616	0.627	0.548	0.615
	Ours	0.382	0.099	0.347	0.211	0.274	0.263
Laplace-OOD	NGF (released)	0.633	0.604	0.631	0.637	0.603	0.621
	Ours	0.127	0.037	0.162	0.070	0.101	0.099

Appendix

Table 12: 3D coefficient-OOD study: mean relative

L_{2}

error over

40

problems per category (identical problems for both methods).

Variant	head inputs	test	test_ood
kernel only	—	$7.7$ / $5.4$	$6.8$ / $5.6$
+ source lift $v_{\varphi}(\Omega,f)$	$\mathbf{1}_{\Omega}$ , SDF, $f$	$6.1$ / $4.5$	$6.3$ / $5.2$
+ residual head $r(\Omega,h,u_{h})$	$\mathbf{1}_{\Omega}$ , $h$ , $u_{h}$	$\mathbf{1.84}$ / $\mathbf{1.74}$	$\mathbf{2.56}$ / $\mathbf{2.39}$
main model: single lift $v_{\varphi}(\Omega,h,f)$	$\mathbf{1}_{\Omega}$ , $h$ , $f$ , $u_{h}$	$2.1$ / $2.0$	$2.6$ / $2.5$

Appendix

Table 13: From the kernel alone to the three-term variant (relative

L_{2}

, %, mean / median; Table 1 scale).

Variant	test (in-dist)	test_ood	OOD/test
$K$ -only (no $v_{\varphi}$ )	$5.4\%$ / $7.7\%$	$5.6\%$ / $6.8\%$	$\mathbf{1.0{\times}}$
$K+6.4$ M lift (canonical)	$\mathbf{2.0\%}$ / $\mathbf{2.1\%}$	$\mathbf{2.5\%}$ / $\mathbf{2.6\%}$	$1.25{\times}$
$K+11.3$ M lift (capacity scan, A2)	$2.0\%$ / $2.1\%$	$2.3\%$ / $2.5\%$	$1.15{\times}$

Appendix

Table 14 : Lift removal. Mixed (Laplace + Poisson) rel-

L_{2}

over interior pixels (median / mean).

$\sigma$ (fraction of domain width)	$K$ -only test median
$0.1\%$ (sharper)	${\sim}6.5\%$
$0.2\%$ (canonical)	$\mathbf{5.4\%}$
$0.5\%$ (smoother)	${\sim}6.0\%$

Appendix

Table 15 : KDE bandwidth scan.

K

-only test median.

Train shapes	$K$ -only test median	$K+$ lift test median
$200$	${\sim}7.0\%$	—
$500$	${\sim}6.0\%$	${\sim}3.5\%$
$1{,}000$	${\sim}5.7\%$	${\sim}2.5\%$
$5{,}000$ (canonical)	$\mathbf{5.4\%}$	$\mathbf{2.0\%}$

Appendix

Table 16 : Training-shape count.

K

-only test median rel-

L_{2}

as the corpus grows.

$n_{\text{surf}}$	median	mean	p $95$
$50$	$3.12\%$	$3.58\%$	$5.62\%$
$100$	$2.48\%$	$2.65\%$	$4.05\%$
$200$ (canonical)	$\mathbf{2.44\%}$	$\mathbf{2.58\%}$	$\mathbf{4.08\%}$
$400$	$2.43\%$	$2.56\%$	$3.98\%$

Appendix

Table 17 : Boundary-discretization scan at inference. Mixed rel-

L_{2}

on test_ood,

25

shapes

\times

{\sim}8

problems.

Digit class	$n_{\text{test}}$	test mean	test_ood mean
0	50	$2.14\%$	$2.89\%$
1	30	$2.19\%$	$2.95\%$
2	40	$1.86\%$	$2.60\%$
3	36	$1.94\%$	$2.26\%$
4	38	$2.15\%$	$2.48\%$
5	46	$2.02\%$	$2.48\%$

Appendix

Table 18 : Per-MNIST-class rel-

L_{2}

mean (mixed Laplace + Poisson, full-interior un-normalized).

Shape encoder	test (in-dist)	test_ood	OOD/test
Point-cloud (canonical)	$\mathbf{2.0\%}$ / $\mathbf{2.1\%}$	$\mathbf{2.5\%}$ / $\mathbf{2.6\%}$	$1.25{\times}$
$2$ D SDF-CNN ( $64\times 64$ )	$2.28\%$ / $2.43\%$	$2.59\%$ / $2.97\%$	$\mathbf{1.14{\times}}$

Appendix

Table 19 : Representation-invariance ablation: identical hyperparameters, swap shape encoder.

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Neural Shape Operator Surrogates -- Expression Rate Bounds

Helmut Harbrecht, Christoph Schwab

Neural Operators Partial Differential Equations Surrogates Expressivity

Open paper AskPDF arXiv

We prove error bounds for operator surrogates of solution operators for partial differential and boundary integral equations on families of domains which are diffeomorphic to one common reference (or latent) domain

D_{ref}

. The pullback of the PDE to

D_{ref}

via affine-parametric shape encoding produces a collection of holomorphic parametric PDEs on

D_{ref}

. Sufficient conditions for (uniformly with respect to the parameter) well-posedness are given, implying existence, uniqueness and stability of parametric solution families on

D_{ref}

. We illustrate the abstract hypotheses by reviewing recent holomorphy results for a suite of elliptic and parabolic PDEs. Quantified parametric holomorphy implies existence of finite-parametric, discrete approximations of the parametric solution families with convergence rates in terms of the number

N

of parameters. We obtain constructive proofs of existence of Neural and Spectral Operator surrogates for the shape-to-solution maps with error bounds and convergence rate guarantees uniform on the collection of admissible shapes. We admit principal-component shape encoders and frame decoders. Our results support in particular the (empirically reported) ability of neural operators to realize data-to-solution maps for elliptic and parabolic PDEs and BIEs that generalize across parametric families of shapes.

Abstract

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Neural Shape Operator Surrogates -- Expression Rate Bounds

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