Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators

Peiyao Chen; Zhouyuan Xu; Ran Ding; Jiansheng Fan; Chen Wang

cs.LGSep 5, 2026

Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators

Authors: Peiyao Chen, Zhouyuan Xu, Ran Ding, Jiansheng Fan, Chen Wang

Organizations: Tsinghua University

Abstract

Neural operators that accept arbitrary meshes are often treated as geometry-general, but unseen domain topology changes both the invariant and decaying subspaces of a PDE operator. We use Hodge heat flow as a controlled lens on this distinction and introduce TopoBox-3D, where tunnels and cavities vary Betti support while the exact Hodge decomposition separates the harmonic kernel from the positive spectrum. Across six architectures, models that infer topology implicitly suffer excess matched degradation in 37 of 45 model--task topology-OOD cells, yet cases that change harmonic dimension are not more strongly penalized on average. The dominant difficulty is instead spectral: the initial Rayleigh quotient is the most stable predictor of error, and spectral broadening adds information for edge and face cochains. Most strikingly, controlled probes show that explicit incidence and harmonic coordinates do not yield the best kernel-identity accuracy; nevertheless, TNO ranks first in mixed-input nonharmonic accuracy on all six tasks with nontrivial harmonic support. Together, these results establish topology as a distinct generalization axis beyond arbitrary-geometry compatibility and show that its influence extends across the Hodge spectrum rather than remaining confined to the harmonic kernel. More broadly, they suggest that global, low-frequency structural priors may help organize predictions in the faster-decaying complementary component, offering a new perspective on how neural operators may generalize across topology as well as geometry.

Figures & tables

Figure 1: TopoBox-3D geometry and Hodge diffusion. (a) Representative domain families. Family A and Family B share $(\beta_{1},\beta_{2})=(1,1)$ but differ geometrically, while tunnel-only, cavity-only, and mixed cases independently vary cycles and enclosed voids. (b) Schematic physical interpretations of Hodge diffusion for $k=0,1,2$ , corresponding to scalar, circulation, and flux fields.

Protocol	In-support topologies	Test-OOD topology	Generalization shift
A	$(\beta_{1},\beta_{2})=(1,1)$ , family A	$(1,1)$ , family B	fixed-topology geometry shift
B	$\beta_{1}\in\{0,1,2\},\ \beta_{2}=0$	$(3,0)$	unseen tunnel support ( $\beta_{1}$ -OOD)
C	$\beta_{1}=0,\ \beta_{2}\in\{0,1,2\}$	$(0,3)$	unseen cavity support ( $\beta_{2}$ -OOD)
D	$(\beta_{1},\beta_{2})\in\{0,1,2\}^{2}$	$(3,3)$	mixed topology-OOD

Table 1: TopoBox-3D protocols. In-support topologies apply to training, validation, and Test-IID.

Model	Processor	Structural access	Parameters
MGN 1	6-layer MPNN	same-degree simplex adjacency	1.162M
RIGNO	regional graph, 7 steps	coordinate-reconstructed graph	1.197–1.198M
Transolver	6-layer physics attention	coordinate tokens, local geometry	1.210–1.212M
GNOT	4-layer normalized attention	coordinate tokens, local geometry	1.135–1.136M
GAOT	MAGNO + transformer	coordinate latent graph	1.186M
TNO 2	4-layer cross-degree operator	$B_{1},B_{2},B_{3}$ + harmonic basis	1.168M

Table 2: Models and effective access to TopoBox structure. Parameter ranges cover

k=0

versus

k=1,2

	A0	A1	A2	B0	B1	B2	C0	C1	C2	D0	D1	D2
MGN	.0101	.2191	.3487	.0091	.2254	.2756	.0086	.1725	.3431	.0103	.2480	.3603
RIGNO	.0258	.2646	.3491	.0232	.2333	.3278	.0228	.1651	.3725	.0262	.2524	.3994
Transolver	.0354	.4188	.5034	.0347	.3837	.4399	.0317	.2755	.4917	.0351	.3655	.5730
GNOT	.0202	.3257	.4171	.0183	.2161	.4137	.0161	.0879	.4211	.0221	.2296	.4942
GAOT	.3190	.5408	.8479	.3199	.5413	.5166	.3253	.5174	.6656	.3178	.5476	.7098
TNO	.0155	.1029	.1760	.0144	.1216	.3848	.0131	.1622	.1840	.0161	.1362	.2478

Table 3: Median Test-IID mass-weighted relative

L^{2}

error over three seeds. Rank 1 , Rank 2 , and Rank 3 indicate the three lowest errors in each column.

	A0	A1	A2	B0	B1	B2	C0	C1	C2	D0	D1	D2
MGN	.0105	.2288	.4760	.0138	.2699	.3486	.0100	.1986	.4633	.0134	.2864	.4088
RIGNO	.0265	.2651	.4071	.0292	.2428	.3971	.0234	.1752	.4568	.0291	.2696	.4681
Transolver	.0379	.4160	.5933	.0493	.4230	.5789	.0336	.2823	.6522	.0406	.3863	.6555
GNOT	.0221	.3294	.4864	.0350	.3079	.5599	.0196	.1056	.5769	.0289	.2864	.6239
GAOT	.3205	.5297	.8581	.3155	.5359	.6856	.3236	.5216	.7928	.3140	.5413	.8253
TNO	.0161	.1086	.1770	.0216	.1058	.5210	.0160	.1881	.1428	.0196	.1169	.2218

Table 4: Median Test-OOD mass-weighted relative

L^{2}

error over three seeds. Rank 1 , Rank 2 , and Rank 3 indicate the three lowest errors in each column.

Figure 2: Matched topology penalty across all topology-OOD tasks. Each cell reports $Q=R_{p}/R_{A}$ for protocols B–D and degrees $k=0,1,2$ ; blue ( $Q<1$ ) indicates less degradation than the fixed-topology Protocol A baseline and red ( $Q>1$ ) indicates more. Up/down triangles mark 95% geometry-cluster intervals entirely above/below one; the right column gives the geometric mean across all nine tasks.

Figure 3: Drivers of relative error. (a) Distributions of univariate Spearman correlations over model–task cells. (b) Information added by spectral broadening $J_{T}$ after controlling for the Rayleigh mean: partial Spearman correlation and the increment in rank-regression $R^{2}$ , summarized across model–task cells at each degree.

Figure 4: Architecture-dependent capability asymmetry. (a) Controlled seed-0 kernel-identity error versus mixed-input nonharmonic error; bubble area is controlled bidirectional leakage. (b) Mean rank over the six tasks with nontrivial harmonic support; vertical segments span each model’s best-to-worst task rank. Geometry-driven implicit-topology processors tend to rank better on kernel identity, whereas TNO ranks first in mixed-input nonharmonic accuracy in all six tasks.

Appendix figures & tables16 assets

Supplementary material from the paper’s appendix.

Appendix

Split	A	B	C	D	Total
Train	800	800	800	800	3,200
Validation	120	120	120	120	480
Test-IID	200	200	200	200	800
Test-OOD	200	200	200	200	800
Total	1,320	1,320	1,320	1,320	5,280

Appendix

Table 5: Geometry counts per protocol. All fields and initial-condition regimes associated with one geometry remain in the same split.

Figure 5: Distributions computed from the formal 5,280-geometry manifest: mesh size, minimum tetrahedron quality, and primitive clearance. The mass at clearance 1 corresponds to geometries for which the relevant pairwise primitive-clearance constraint is vacuous.

Configuration	$\rho_{E}$	$\rho_{C}$	$\rho_{H}$
Non-harmonic	$1/2$	$1/2$	$0$
Weak harmonic	$4/9$	$4/9$	$1/9$
Balanced	$1/3$	$1/3$	$1/3$
Strong harmonic	$1/4$	$1/4$	$1/2$

Appendix

Table 6: Requested exact/coexact/harmonic energy fractions for

k=1,2

Model	Core configuration	$k=0$	$k=1,2$
MGN	width 160; 6 message-passing layers	1,162,241	1,162,881
RIGNO	node/edge width 128; 7 processor steps	1,197,059	1,198,083
Transolver	width 192; 6 layers; 8 heads; 32 slices	1,210,033	1,211,569
GNOT	width 96; 4 layers; 8 heads; 2 experts	1,134,921	1,135,689
GAOT	MAGNO width 48; transformer width 128 and 4 layers; $8\!\times\!4\!\times\!4$ latent grid	1,185,645	1,186,102
TNO	width 170; 4 cross-degree layers; harmonic branch active	1,167,563	1,167,563

Appendix

Table 7: Exact model configurations and trainable parameter counts. The

k=1

and

k=2

counts coincide for every model.

	A0	A1	A2	B0	B1	B2	C0	C1	C2	D0	D1	D2
MGN	1.037	1.048	1.385	1.507	1.197	1.265	1.179	1.150	1.362	1.292	1.147	1.129
RIGNO	1.027	1.002	1.136	1.259	1.041	1.211	1.027	1.061	1.220	1.108	1.068	1.169
Transolver	1.066	.993	1.179	1.407	1.073	1.322	1.059	1.019	1.293	1.131	1.057	1.167
GNOT	1.094	1.011	1.156	1.912	1.477	1.353	1.229	1.206	1.374	1.377	1.247	1.263
GAOT	1.005	.980	1.008	.986	.990	1.326	.994	1.008	1.187	.988	.988	1.161
TNO	1.039	1.051	1.005	1.496	.878	1.351	1.206	1.156	.776	1.218	.858	.894

Appendix

Table 8: OOD/IID degradation ratio

R

Figure 6: MGN on Protocol D, $k=0$ , under the balanced initial condition: target, prediction, and signed error for typical IID, typical topology-OOD, and challenging topology-OOD examples.

Figure 7: MGN on Protocol D, $k=1$ , under the balanced initial condition. Edge cochains are rendered through reconstructed vector-field streamlines using the same three-row selection convention described above.

Figure 8: TNO on Protocol D, $k=2$ , under the balanced initial condition. Face cochains are rendered through reconstructed vector-field streamlines using the same three-row selection convention described above.

Scope	Cells	$Q>1$	CI $>1$	CI $<1$	Geom. mean $Q$
Five implicit-topology models, all B/C/D degrees	45	37	29	5	1.110
Five implicit-topology models, harmonic dimension changes	20	17	13	1	1.083
TNO, all B/C/D degrees	9	5	5	4	1.034
TNO, harmonic dimension changes	4	0	0	4	0.827

Appendix

Table 9: Summary of matched topology penalties. “Harmonic dimension changes” comprises B-

k=1

, C-

k=2

, and D-

k=1,2

. Geometric means are computed by averaging

\log Q

Model	$k=0$	$k=1$	$k=2$
MGN	1.448 [1.325, 1.592] $\uparrow$	1.141 [1.099, 1.183] $\uparrow$	0.919 [0.874, 0.968] $\downarrow$
RIGNO	1.228 [1.111, 1.344] $\uparrow$	1.018 [0.997, 1.041]	1.075 [1.039, 1.110] $\uparrow$
Transolver	1.324 [1.194, 1.465] $\uparrow$	1.070 [1.049, 1.093] $\uparrow$	1.132 [1.086, 1.181] $\uparrow$
GNOT	1.795 [1.631, 1.989] $\uparrow$	1.532 [1.485, 1.588] $\uparrow$	1.169 [1.120, 1.220] $\uparrow$
GAOT	0.982 [0.971, 0.994] $\downarrow$	1.010 [0.997, 1.023]	1.316 [1.266, 1.363] $\uparrow$
TNO	1.438 [1.314, 1.562] $\uparrow$	0.833 [0.794, 0.874] $\downarrow$	1.344 [1.279, 1.408] $\uparrow$

Appendix

Table 10: Complete Protocol B matched penalties. Each cell is

\overline{Q}_{\mathrm{boot}}

[95% geometry-cluster interval].

Model	$k=0$	$k=1$	$k=2$
MGN	1.135 [1.043, 1.236] $\uparrow$	1.094 [1.044, 1.146] $\uparrow$	0.993 [0.944, 1.045]
RIGNO	1.003 [0.919, 1.094]	1.061 [1.031, 1.091] $\uparrow$	1.084 [1.048, 1.121] $\uparrow$
Transolver	0.992 [0.895, 1.100]	1.016 [0.992, 1.040]	1.105 [1.059, 1.150] $\uparrow$
GNOT	1.124 [1.024, 1.220] $\uparrow$	1.185 [1.149, 1.220] $\uparrow$	1.189 [1.140, 1.243] $\uparrow$
GAOT	0.989 [0.978, 1.000] $\downarrow$	1.029 [1.012, 1.045] $\uparrow$	1.178 [1.138, 1.222] $\uparrow$
TNO	1.161 [1.068, 1.265] $\uparrow$	1.099 [1.048, 1.154] $\uparrow$	0.771 [0.728, 0.820] $\downarrow$

Appendix

Table 11: Complete Protocol C matched penalties. Each cell is

\overline{Q}_{\mathrm{boot}}

[95% geometry-cluster interval].

Model	$k=0$	$k=1$	$k=2$
MGN	1.246 [1.137, 1.362] $\uparrow$	1.091 [1.055, 1.130] $\uparrow$	0.822 [0.781, 0.866] $\downarrow$
RIGNO	1.083 [0.987, 1.189]	1.046 [1.023, 1.070] $\uparrow$	1.022 [0.990, 1.057]
Transolver	1.058 [0.958, 1.178]	1.071 [1.051, 1.092] $\uparrow$	0.988 [0.949, 1.025]
GNOT	1.244 [1.127, 1.366] $\uparrow$	1.237 [1.200, 1.279] $\uparrow$	1.091 [1.047, 1.139] $\uparrow$
GAOT	0.983 [0.973, 0.994] $\downarrow$	1.006 [0.993, 1.018]	1.153 [1.118, 1.191] $\uparrow$
TNO	1.171 [1.077, 1.273] $\uparrow$	0.815 [0.771, 0.866] $\downarrow$	0.888 [0.823, 0.955] $\downarrow$

Appendix

Table 12: Complete Protocol D matched penalties. Each cell is

\overline{Q}_{\mathrm{boot}}

[95% geometry-cluster interval].

Figure 9: Cell-wise Spearman correlations for six principal descriptors. Rows are protocol–degree tasks and columns are models. Dashed cells indicate constant, unavailable predictors.

Figure 10: Marginal and conditional Jensen-gap correlations by model and task. The left panel reports $\rho_{s}(J_{T},E_{\rm rel})$ ; the right panel reports the partial Spearman correlation after controlling for $\mathcal{R}(w_{0})$ . Conditioning removes most of the global association but leaves task-dependent higher-order structure.

Figure 11: Extended topology audit. (a) The 12 tasks in the $(\Pi_{H},\Pi_{+})$ pressure plane, colored by the observed best model; dashed lines mark the pressure-group thresholds used for interpretation. (b) Pressure–error associations, read left-to-right and top-to-bottom: $\Pi_{+}$ versus task OOD error; $\Delta\Pi_{+}$ versus task OOD/IID ratio; $\Delta\Pi_{+}$ versus non-harmonic OOD/IID ratio; and component OOD/IID ratios for kernel-stable and kernel-shifted tasks. In the final panel, circles and triangles denote harmonic and non-harmonic components, respectively, and horizontal bars show group medians; all reported $\rho_{s}$ values are Spearman correlations. (c) Pure non-harmonic-to-harmonic leakage: total $L_{\perp\to H}$ on the left and the dimension-normalized quantity $L_{\perp\to H}/\sqrt{\beta_{k}}$ on the right. (d) Percentage of conventional mixed-input OOD error energy assigned to the harmonic component for each task–model pair.

Model	$I_{H}$	$L_{H\to\perp}$	$L_{\perp\to H}$	$E_{\perp}^{\mathrm{mix}}$
MGN	.187	.131	.025	.848
RIGNO	.165	.114	.016	.894
Transolver	.329	.171	.012	1.137
GNOT	.213	.146	.014	1.040
GAOT	.588	.049	.005	.709
TNO	.349	.290	.004	.256

Appendix

Table 13: Model-level topology audit. All errors are OOD medians; lower is better.

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