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Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators
Peiyao Chen, Zhouyuan Xu, Ran Ding, Jiansheng Fan, Chen Wang
TL;DR
Arbitrary-mesh compatibility does not establish whether neural operators generalize to domain topologies absent from training. The paper uses Hodge heat flow and the TopoBox-3D benchmark to test topology shifts across the harmonic kernel and positive spectrum, finding that degradation is not concentrated in the harmonic subspace. Its evidence is controlled but bounded by the benchmark’s linear, fixed-time dynamics and domain and boundary-condition choices.
Problem
Methods that accept irregular meshes or coordinates are not thereby shown to generalize to domain types outside training support.
Source: §1 · p. 1Method
Results
Takeaways & Limitations
The results suggest connecting global structural context with local and multiscale dynamics, without establishing that a harmonic basis or any single implementation is necessary.
Source: §6 · p. 9Evidence is limited to linear autonomous Hodge heat at fixed time on box-minus-void domains with homogeneous absolute boundary conditions, and protocol factors co-vary rather than isolating causal Betti effects.
Source: §5.2 · p. 6, §6 · p. 9
Abstract
from arXiv · showhide
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.
1 INTRODUCTION
The paper distinguishes compatibility with irregular geometries from extrapolation to unseen topology, then uses Hodge heat flow to examine where topology shifts affect learned operators. Its central finding is that degradation spans the Hodge spectrum, while explicit harmonic information does not make a model best at reproducing harmonic identity.
1 INTRODUCTION
- Accepting arbitrary meshes or coordinates does not establish generalization to domain types absent from training; diffeomorphic mappings cannot create tunnels or cavities.Standard evaluations of methods that permit topology variation rarely isolate whether test topology falls outside training support.Source: §1 · p. 1
- The paper defines topology-OOD by test Betti support outside training support, distinguishing it from geometry-OOD with extrapolative shape or pose but matching Betti support.This is a homological definition; Betti numbers do not fully classify domains.Source: §1 · p. 2
- Hodge heat separates a topology-dependent harmonic kernel from a geometry- and discretization-dependent positive spectrum, making topology generalization a test of kernel recognition and spectral filtering.For k-forms, dim ker Δk = βk(Ω), and the heat operator preserves the harmonic component while decaying the positive-spectrum component.Source: §1 · p. 2
- Topology-OOD usually adds degradation for implicitly informed models, but the strongest error signal comes from positive-spectrum content rather than the invariant subspace.The introduction also reports that TNO is not best at harmonic identity, yet has the lowest mixed-input nonharmonic error on tasks with nontrivial harmonic support.Source: §1 · p. 2
2 RELATED WORK AND BACKGROUND
Prior work broadens neural operators’ input compatibility and develops methods for transfer across geometries, but the paper distinguishes those advances from controlled tests of extrapolation beyond observed Betti support.
Neural operators and irregular-geometry representations
- Irregular-domain operators use coordinate maps, indicators, nonuniform Fourier transforms, graphs, attention, and neural fields, broadening the geometries and discretizations they can ingest.Input compatibility alone does not specify extrapolation to geometry families absent from training.Source: §2 · p. 2
Operators for cross-geometry generalization
- Cross-geometry methods align domains through transformations or assemble local operators, while other studies address changing cracks and topology-varying surface PDEs.The paper frames its contribution as isolating extrapolation beyond observed Betti support, not claiming topology-changing PDE learning is unprecedented.Source: §2 · pp. 2–3
- Higher-order networks encode oriented simplices and incidence, and Hodge-structured operators incorporate diffusion or harmonic structure, but the paper contributes a controlled comparison rather than a new Hodge architecture.It also audits whether learned maps preserve kernel identities.Source: §2 · p. 3
3 TOPOBOX-3D BENCHMARK
TopoBox-3D separates geometry extrapolation from extrapolation across homological support by varying tunnels and cavities in box-minus-void domains. It pairs these domain families with Hodge heat tasks across scalar, circulation, and flux cochains.
3 TOPOBOX-3D BENCHMARK
- TopoBox-3D constructs connected box-minus-void domains with through-tunnels and closed cavities, measuring topology after tetrahedralization rather than relying only on generator labels.The benchmark checks chain-complex identities and recomputes ranks and Euler characteristic on the numerical domain.Source: §3 · p. 3
- The benchmark varies tunnel and cavity support independently, while geometry-OOD uses tilted tunnels and rotated ellipsoidal cavities instead of standard aligned tunnels and spherical cavities.The enforced clearances yield β(Ω) = (1, g, c, 0).Source: §3 · p. 3
- Hodge heat tasks cover k = 0 scalar conduction, k = 1 edge-based circulation, and k = 2 face-based flux, with four exact/coexact/harmonic energy mixtures.The k = 2 operator retains its B3 contribution; k = 0 also includes smooth mean-zero fields as a topology-insensitive control.Source: §3 · p. 3
- Four protocols hold Betti support fixed for geometry-OOD or extrapolate beyond observed tunnel, cavity, or combined support; each uses 800 training and 200 Test-OOD geometries.Across protocols, the benchmark contains 5,280 domains and 63,360 PDE instances, with geometry ID as the split unit and Test-OOD excluded from model selection.Source: §3 · p. 4, Table 1 · p. 4
4 EXPERIMENTAL SETUP
The experiments compare six approximately capacity-matched architectures with different access to mesh and topological structure. Training and evaluation use matched IID/OOD protocols, in-support validation, and geometry-clustered uncertainty estimates.
4 EXPERIMENTAL SETUP
- The six approximately capacity-matched models span four structural-access levels, but differ in processors, tokenization, and native inputs, so ranks are not treated as causal effects of one feature.The analysis emphasizes within-model IID-to-OOD degradation and access strata.Source: §4 · p. 4, Table 2 · p. 4
- TNO receives cross-degree incidence maps and a harmonic basis, whereas five implicit-topology models are not given Betti numbers or harmonic bases.MGN preserves same-degree simplex adjacency; RIGNO and GAOT construct regional or latent graphs, while Transolver and GNOT use coordinate-indexed attention tokens.Source: §4 · p. 4
- Models train for 300 epochs with AdamW and three seeds, using mass-weighted relative MSE; evaluation reports the corresponding mass-weighted relative L2 error.Validation is restricted to training topology support, and the selected checkpoint is evaluated once on Test-IID and Test-OOD.Source: §4 · p. 4
- MGN is a degree-specific MeshGraphNets-style adaptation, while TNO implements the published cross-degree incidence and harmonic mechanism because official code was unavailable.These are implementation qualifications, not additional architecture comparisons.Source: §4 · p. 4
- Ratios are formed within seed before taking the three-seed median, and uncertainty resamples IID and OOD geometry IDs while retaining each selected geometry’s four initial conditions.Ranks are averaged across the 12 protocol–degree tasks, and geometry-cluster bootstrap intervals use 1,000 resamples.Source: §4 · p. 5
5 RESULTS
Across the benchmark, predictive strengths vary by cochain degree, topology-OOD degradation is widespread, and spectral properties help explain error. Explicit topology information aligns with stronger mixed-input nonharmonic accuracy, although it does not yield the best kernel-identity accuracy.
5.1 PREDICTIVE PERFORMANCE
- TNO has mean IID/OOD ranks of 1.58/1.67 and ranks in the top three on all 12 tasks, while MGN wins 5/12 tasks in each split and leads every k = 0 protocol and seed.Together they account for 11/12 task wins in both splits; TNO is stronger on most k = 1, 2 tasks but is not uniformly dominant, with B-k = 2 OOD error 0.5210 versus MGN’s 0.3486.Source: §5.1 · p. 5
5.2 GEOMETRY VERSUS TOPOLOGY GENERALIZATION
- 37 of 45 implicit-model topology-OOD comparisons have Q > 1 (82.2%), with geometric mean Q = 1.110, and harmonic-dimension changes do not amplify the penalty.Among 20 implicit-model cells with harmonic-dimension changes, 17 have Q > 1 and mean Q is 1.083; the matched ratio is a stress-test contrast, not a causal estimate of a Betti-number effect.Source: §5.2 · p. 6
5.3 ERROR DRIVERS AND SPECTRAL BROADENING
- The initial Rayleigh quotient is the most stable error predictor, with median Spearman correlation 0.784 across 72 model–task cells and the largest single-variable R2 in 43/72 cells.Its dominance is associated with mean positive-spectrum frequency; larger values place more energy in rapidly decaying modes and leave a weaker final-time signal to reconstruct.Source: §5.3 · p. 6
- After controlling for the Rayleigh mean, spectral broadening adds little overall but becomes a modest predictor for edge and face cochains.Incremental rank-R2 is 0.0003, 0.038, and 0.086 for k = 0, 1, and 2, respectively; descriptors such as token count and topological complexity are also more informative at k = 2, though weak globally.Source: §5.3 · pp. 6–7
5.4 TOPOLOGY PRIORS EXTEND BEYOND THE KERNEL
- TNO ranks first in mixed-input nonharmonic accuracy on all six tasks with nontrivial harmonic support, while geometry-driven processors tend to rank better on kernel identity.The mixed-input metric does not isolate a pure positive-spectrum filter, and the architecture comparisons do not identify which ingredient causes the gain; the results motivate a factorized kernel and complementary-spectrum design target.Source: §5.4 · pp. 7–8, Figure 4 · p. 8
6 BEYOND ARBITRARY GEOMETRY: DISCUSSION, LIMITATIONS, AND OUTLOOK
Topology generalization extends beyond geometry compatibility: topology changes the harmonic kernel while geometry and discretization reshape the positive spectrum. The discussion interprets low-frequency structural information as possible context for decaying components, while bounding the evidence to a controlled benchmark.
6 BEYOND ARBITRARY GEOMETRY: DISCUSSION, LIMITATIONS, AND OUTLOOK
- Topology-OOD difficulty is not confined to the invariant harmonic subspace: decaying spectral content dominates the error signal, and TNO is strongest on that content despite not leading harmonic-identity accuracy.The authors suggest low-frequency structural information may provide context for more rapidly decaying components.Source: §6 · p. 8
- The evidence is bounded by linear, autonomous, fixed-time Hodge heat on box-minus-void domains with homogeneous absolute boundary conditions; it is not a surrogate for general physical PDE performance.The authors also note that Betti numbers omit knotting, torsion, and other distinctions.Source: §6 · p. 9
- The next test is whether global low-frequency domain information improves learning of local, high-frequency, or rapidly decaying behavior in realistic fields, varied boundary conditions, nonlinear evolution, and time-dependent or multiphysics problems.The authors do not prescribe a Hodge-specific architecture or evaluation protocol.Source: §6 · p. 9
A.1 DISCRETE HODGE DECOMPOSITION AND HEAT SEMIGROUP
- An exact Hodge heat operator preserves harmonic components as identity and has no cross terms with the orthogonal complement.The harmonic space has dimension βk under homogeneous absolute boundary conditions; the complementary operator acts on the positive spectrum.Source: §A.1 · p. 13
- The controlled audit measures mass-normalized harmonic, nonharmonic, and cross-component probe errors, while mixed-input nonharmonic error is not a pure nonharmonic-probe response.All four quantities are medians over finite probe or test records, not operator norms or suprema over the full cochain space.Source: §A.1 · p. 13
A.2 REFERENCE-DOMAIN INTERPRETATION AND THE MATCHED RATIO
- Within a diffeomorphic family, pullbacks preserve cohomology and harmonic-kernel dimension; domains with different Betti vectors cannot be identified by such a pullback.The matched ratio compares topology-OOD degradation against fixed-topology geometry extrapolation, without implying that equal Betti vectors guarantee diffeomorphism.Source: §A.2 · pp. 13–14
A.3 TIME DISCRETIZATION AND JENSEN SPECTRAL BROADENING
Crank–Nicolson time discretization preserves harmonic modes and contracts mass, while a Jensen-gap statistic captures spectral broadening. The benchmark’s absolute boundary-condition convention gives the harmonic spaces their stated cohomological meaning.
A.3 TIME DISCRETIZATION AND JENSEN SPECTRAL BROADENING
- Crank–Nicolson preserves harmonic modes exactly and is mass-contractive; for resolved modes, its update gives a second-order temporal approximation to the exact heat semigroup.The semidiscrete evolution is Mk ẇ(t) + κKkw(t) = 0, advanced with N steps of size Δt = T/N.Source: §A.3 · p. 14
- The reported Jensen gap is nonnegative and vanishes only when spectral energy lies at a single eigenvalue; its leading nonzero term measures variance across the full spectrum, including harmonic mass.The reported statistic substitutes 100-step Crank–Nicolson retention for exact heat retention; the minimum observed value was 4.33 × 10−4.Source: §A.3 · p. 14
A.4 BOUNDARY CONDITIONS
- Homogeneous absolute boundary conditions yield harmonic spaces representing absolute de Rham cohomology, with dim Hk = βk; the reported rankings and model audit have not been validated for alternative boundary complexes.For k = 0, the conditions reduce to homogeneous Neumann conditions and leave one constant harmonic mode per connected component.Source: §A.4 · p. 14
B DATASET DETAILS AND VISUALIZATIONS
TopoBox-3D varies tunnel and cavity configurations, verifies topology on the meshed domains, and pairs geometry splits with controlled cochain fields and initial conditions. The benchmark also documents its geometry and field-generation conventions.
B.1 GENERATION, SPLITTING, AND QUALITY CONTROL
- The formal audit found zero errors across 5,280 geometries and 63,360 PDE instances, checking Betti numbers, Euler characteristic, nullities, chain identities, fields, energy fractions, and heat contractivity.Accepted domains use oriented tetrahedral meshes; within each training, validation, and Test-IID split, admissible Betti pairs are allocated as evenly as the split permits.Source: §B.1 · p. 15
B.2 FIELDS AND INITIAL CONDITIONS
- Initial conditions combine smooth ambient fields and, when available, an independently generated harmonic component to target requested exact, coexact, and harmonic energy fractions.When harmonic dimension is zero, unavailable harmonic energy is redistributed; for k = 0, the constant harmonic mode is suppressed.Source: §B.2 · p. 16
- For k = 2, adapters rescale face-cochain inputs and restore the same scale to predictions before loss and error calculations, leaving physical units and relative-error definitions unchanged.Table 6 gives the requested exact/coexact/harmonic fractions for k = 1, 2.Source: §B.2 · p. 16, Table 6 · p. 16
C IMPLEMENTATION AND COMPLETE MODEL RESULTS
Across models without full chain-complex and harmonic-basis inputs, matched topology-OOD degradation is usually greater than the fixed-topology comparison, without a larger average penalty when harmonic dimension changes. The appendix also documents model implementations, training, ratio construction, and illustrative predictions.
C.1 ADAPTERS, OPTIMIZATION, AND COMPLETENESS
- The implementations include distinct graph, attention, regional, latent-grid, and incidence-based processors; only TNO receives the full chain complex and active harmonic basis.All 216 runs reached 300 epochs, with best validation checkpoints selected before Test-IID or Test-OOD evaluation.Source: §C.1 · pp. 16–17, Table 7 · p. 17
C.3 REPRESENTATIVE PREDICTIONS
- Representative predictions are selected by pooled error quantiles on the balanced initial condition and illustrate target, prediction, and signed error; edge and face cochains are rendered as reconstructed vector-field streamlines.The reconstructions are for visualization only: reported errors use the original oriented cochains and mass matrix.Source: §C.3 · pp. 17–18, Figure 6 · p. 18, Figure 7 · p. 19
C.2 COMPLETE DEGRADATION RATIOS
- The matched penalty Q compares each topology-OOD degradation ratio against the Protocol-A fixed-topology ratio; Q above one indicates greater relative IID-to-OOD degradation in the topology-OOD protocol.The tabulated Qboot points use 1,000 geometry-cluster resamples; interval arrows mark 95% intervals entirely above or below one.Source: §C.2 · p. 17, §D.1 · p. 20, Table 10 · p. 20
D.1 COMPLETE MATCHED TOPOLOGY PENALTIES
- 37/45 matched estimates exceed one across the five models without the full chain complex and harmonic basis, with geometric mean 1.110; harmonic-dimension-changing cells have geometric mean 1.083.TNO is heterogeneous across the nine B/C/D tasks, with 5/9 estimates above one and geometric mean 1.034; its four harmonic-dimension-changing comparisons are all below one, with geometric mean 0.827.Source: Table 9 · p. 20, §D.1 · p. 20
D.2 FULL DRIVER ANALYSIS
The initial Rayleigh quotient is the most consistent error predictor across the analyzed model–task cells, while spectral broadening contributes additional information chiefly for edge and face cochains. Structural descriptors are weaker overall, though several become more informative for face cochains.
D.2 FULL DRIVER ANALYSIS
- R(w0) has median Spearman correlation 0.784 and median single-variable R2 0.585, and gives the largest R2 in 43/72 cells.Target retention is also strongly predictive, but is partly coupled to the relative-error denominator; harmonic energy fraction is protective in all 36 estimable higher-order cells.Source: §D.2 · pp. 21–22
- For harmonic energy fraction, median ρs = −0.521 and median R2 = 0.257 across 36 higher-order cells, with all 36 geometry-cluster tests at p < 0.05.Target retention qT has median ρs = −0.693 and median R2 = 0.455, but is not an independent mechanism because target norm enters the relative-error denominator.Source: §D.2 · p. 22
- At k = 2, median R2 rises to 0.195 for log nk, 0.189 for βk, 0.252 for β1 + β2, and 0.143 for log λ+1.These descriptors are weaker globally, but discretization size, topological complexity, and the low end of the positive spectrum become more informative for face cochains.Source: Figure 9 · p. 21, §D.2 · p. 22
- JT has median marginal ρs = 0.503, but its median partial correlation falls to 0.009 and incremental rank-R2 to 0.010 after controlling for R(w0).The conditional contribution is degree-dependent: incremental rank-R2 is 0.0003, 0.038, and 0.086 for k = 0, 1, and 2, respectively.Source: §D.2 · p. 22, Figure 10 · p. 22
D.3 EXTENDED TOPOLOGY-PRESSURE DIAGNOSTICS
The extended audit separates task-side spectral pressure from model-side subspace behavior. Positive-spectrum pressure tracks error and degradation more consistently than harmonic-dimension shifts do, while leakage and error allocation vary substantially by architecture.
D.3 EXTENDED TOPOLOGY-PRESSURE DIAGNOSTICS
- The audit keeps task-side pressure separate from model-side capability because a single difficulty score would obscure kernel changes, OOD spectral difficulty, and shifts away from training support.Π+ aggregates OOD medians of R(w0) and JT; the positive-spectrum shift instead measures movement of OOD spectral/discretization distributions from IID support.Source: §D.3 · pp. 22–23
- ΠH is ≃0.8 for the four kernel-changing tasks and ≃0 for the others, making it a near-binary kernel-shift indicator in this benchmark rather than a graded topology-complexity measure.The task positions in the (ΠH, Π+) plane are colored by observed best model.Source: §D.3 · p. 23, Figure 11 · p. 24
- Across 12 task medians, Π+ tracks absolute OOD error (ρs = 0.909, p = 4.2 × 10−5), while ∆Π+ tracks task-level OOD/IID ratio (ρs = 0.608, p = 0.036).Across 36 model–task non-harmonic cells, ∆Π+ is associated with positive-branch degradation (ρs = 0.800, p = 4.8 × 10−9); ΠH has weak, nonsignificant association with harmonic degradation (ρs = 0.148, p = 0.389), but stronger association with non-harmonic degradation (ρs = 0.504, p = 0.0017).Source: §D.3 · p. 23, Figure 11 · p. 24
- Total non-harmonic-to-harmonic leakage generally grows with βk, while dividing by √βk flattens the curves, consistent with accumulation over more harmonic directions rather than uniformly worse leakage per direction.The probe uses pure non-harmonic inputs and measures leakage into the harmonic output.Source: §D.3 · p. 23
- GAOT assigns 50.6–95.9% of OOD error energy to the harmonic branch across six tasks, whereas TNO assigns 0.2–1.2%.This is an allocation of existing error energy, not a total-accuracy score.Source: §D.3 · pp. 23–24
- TNO’s mixed-input nonharmonic error is 0.256, but its pure-harmonic identity error is 0.349 and reverse leakage is 0.290; RIGNO has the lowest empirical identity error at 0.165.GAOT has the smallest reverse leakage (0.049) but the largest identity error (0.588), showing that low cross-subspace leakage does not ensure harmonic-subspace preservation.Source: §D.3 · p. 24
E ADDITIONAL LIMITATIONS
The paper bounds its conclusions to a synthetic homological benchmark and a specific Hodge heat setting. Its matched comparisons, diagnostics, and architecture rankings also carry attribution and statistical limitations.
E ADDITIONAL LIMITATIONS
- TopoBox-3D varies numerically verified Betti vectors within connected box-minus-void domains built from disjoint primitive tunnels and cavities, so conclusions do not cover arbitrary topology in CAD, anatomical, porous, or multiscale domains.Betti numbers do not classify fundamental groups, knotting, torsion, or other embedding-dependent structure.Source: E ADDITIONAL LIMITATIONS · p. 25
- The benchmark tests linear autonomous Hodge heat with κ = 1, T = 0.1, and homogeneous absolute boundary conditions, not relative, mixed, interface, or physically imposed boundaries or broader operator classes.The evidence does not establish the same behavior for forced, nonlinear, coupled, history-dependent, time-varying, or multiphysics operators.Source: E ADDITIONAL LIMITATIONS · p. 25
- Protocols B–D co-vary Betti support with boundary count, mesh size, discretization, and positive spectrum, so matched ratios are stress-test contrasts rather than causal Betti effects.Architecture comparisons are not single-factor ablations; TNO is a reimplementation, and three seeds do not support fine distinctions between neighboring models.Source: E ADDITIONAL LIMITATIONS · p. 25
- Error-driver results are associative, pressure summaries use only 12 task regimes, and controlled probes use 286 seed-0 geometry records rather than the multi-seed conventional evaluation.ΠH is nearly binary in this benchmark, and Emix⊥ is reported separately from full mixed-input evaluation rather than as pure-probe operator error.Source: E ADDITIONAL LIMITATIONS · p. 25