Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies

Xiang Li; Yue Song

cs.LGSep 27, 2026

Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies

Authors: Xiang Li, Yue Song

Organizations: Tsinghua University · Beijing University of Chemical Technology

Abstract

Learning PDE operators on complex domains requires capturing interactions among fields on vertices, edges, and faces, alongside global responses shaped by topology. Existing neural operators accommodate irregular geometries but often overlook these distinct field supports or their condition-dependent coupling. We introduce the Dynamic Kuramoto--Hodge Operator (DKHO), which combines topology-constrained interactions with learned coordination. DKHO encodes conditions on their native cochain supports, evolves Kuramoto-inspired relation states through the boundary and coboundary operators that compose the Dirac operator, and decodes non-harmonic and harmonic responses in orthogonal Hodge subspaces. Topology thus determines where information can flow, while learned dynamics adapts how it is exchanged to each PDE instance. Across porous-medium Darcy flow, torus transport--diffusion, and cavity magnetostatics, DKHO-large reduces prediction error by approximately 61% on average over leading baselines, while DKHO-small remains competitive using only 11.5--24.3% as many parameters. These results suggest that coupling topological structure with adaptive dynamics provides an effective inductive bias for accurate and parameter-efficient PDE operator learning on complex geometries and topologies.

Figures & tables

Figure 1: Schematic overview of the Dynamic Kuramoto–Hodge Operator. Dynamic Kuramoto–Hodge Operators operate on topological domains (a) , where PDE fields are represented as cochains at multiple ranks. A conditional Dynamic Simplicial Kuramoto module (b) coordinates adjacent ranks through boundary and coboundary interactions while dynamically modulating cross-rank exchange. Hodge-structured synthesis (c) decomposes the response into non-harmonic and harmonic components, which are recombined to produce the predicted PDE field.

Figure 2: Algorithm architecture overview of the Dynamic Kuramoto–Hodge Operator. DKHO separates operator learning into a local cochain branch and a topology-indexed harmonic branch. Conditional Dirac–Kuramoto dynamics modulate cross-rank exchange along boundary–coboundary routes, while Hodge synthesis combines the resulting non-harmonic and harmonic responses into the predicted field.

Figure 3: Qualitative comparison on porous-medium Darcy flow. For a representative test instance, we show the domain and input conditions alongside predictions of vertex potential, oriented edge flux, and face circulation. Insets display the local relative errors on each field’s native support.

Figure 4: Qualitative comparison on torus transport–diffusion. For a representative test instance, we show the initial condition and predictions of terminal concentration, oriented edge transport, and face-integrated mass. Insets display local errors on each field’s native support.

Method	Params (K)	MSE $\downarrow$	RelL2 $\downarrow$	Div Fid $\uparrow$	Curl MSE $\downarrow$	Vort Fid $\uparrow$	Enst Fid $\uparrow$	Energy Fid $\uparrow$	IoU $\uparrow$
GNO	231.5	2.30e-3	1.1030	0.5673	1.5409	0.4970	0.2387	0.3714	0.1400
FNO	227.4	4.59e-5	0.1189	0.9887	1.93e-2	0.9098	0.8721	0.9053	0.7242
MGN	246.6	1.66e-4	0.2532	0.9590	6.10e-2	0.8008	0.6970	0.8752	0.5555
DeepONet	238.9	1.07e-5	0.0814	0.9967	1.07e-2	0.9361	0.8356	0.9661	0.7984
Geo-FNO	252.7	4.23e-5	0.1152	0.9895	1.61e-2	0.9236	0.8870	0.9091	0.7234
HSD	265.4	7.20e-6	0.0547	0.9981	7.03e-3	0.9675	0.9513	0.9681	0.8268

Table 3: Magnetostatic response on a spherical shell. The flux-like

2

-form response is evaluated through a three-channel nodal-vector readout. Lower errors and higher fidelity and IoU scores indicate better performance. Bold and underline mark the best and second-best values.

Figure 5: Qualitative comparison on cavity magnetostatics. Predicted fields are shown around the inner cavity of the spherical shell. Insets display local errors on the nodal-vector readout.

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Geometry and cochains
Symbol	Definition
$(\mathcal{M},g)$ , $K$	Oriented Riemannian domain and its oriented simplicial discretization.
$K_{k}$ , $N_{k}$	Set of oriented $k$ -simplices and its cardinality.
$\Omega^{k}(\mathcal{M})$ , $C^{k}(K;\mathbb{R}^{c})$	Smooth $k$ -forms and discrete $k$ -cochains with $c$ channels.
$u_{k}$ , $a$	Physical output cochain and known PDE conditions.
Differential operators and topology

Appendix

Table 4: Notation for discrete fields and DKHO. The degree

k

specifies a geometric support; channel dimensions specify the quantities carried on that support.

System	Physical forms	Governing relations
Darcy flow	Potential $p\in\Omega^{0}$ ; line flux $q\in\Omega^{1}$	$q=-\kappa\,dp$ , $-\delta q=f$
Heat diffusion	Temperature $u\in\Omega^{0}$ ; heat flux $q\in\Omega^{1}$	$q=-\kappa\,du$ , $\partial_{t}u-\delta q=s$
Surface transport	Density $u\in\Omega^{0}$ ; transport $j\in\Omega^{1}$	$j=uv^{\flat}-\kappa\,du$ , $\partial_{t}u-\delta j=0$
Magnetostatics (3D)	Field $H\in\Omega^{1}$ ; flux $B\in\Omega^{2}$	$dB=0$ , $dH=J$ , $B=\mu\star H$
Maxwell system (3D)	$E,H\in\Omega^{1}$ ; $D,B,J\in\Omega^{2}$	$\partial_{t}B+dE=0$ , $\partial_{t}D-dH=-J$
Reaction–diffusion	Species $u\in\Omega^{0}$ ; diffusive flux $q\in\Omega^{1}$	$q=-\kappa\,du$ , $\partial_{t}u-\delta q=R(u)$

Appendix

Table 5: Representative PDEs in differential-form notation.

d

and

\delta

denote continuum derivatives here;

\delta=d^{*}

and

\Delta_{0}=\delta d

. Material coefficients are written in their simplest scalar form. The fields and operators are discretized on matching supports.

State	Value space	Role in a layer
Content $z_{k}$	$C^{k}(K;\mathbb{R}^{h})$	Condition-dependent representation from which the requested field is decoded.
Relation $\theta_{k}$	$C^{k}(K;\mathbb{R}^{c_{\theta}})$	Accumulated coordination state, coupled through incidence and read through sine/cosine maps.

Appendix

Table 6: Complementary latent states in DKHO. Both states are support-indexed; their channel dimensions and roles differ. Neither a relation coordinate nor its sine is a directly supervised physical field.

PDE family	Conditions, support assignment, and output observables
Poisson / Darcy	Source samples, boundary traces, and material coefficients; vertex values, oriented edge differences, and face-integrated source descriptors. Outputs: potential $C^{0}$ , line flux $C^{1}$ , and derived circulation $C^{2}$ .
Heat / diffusion	Initial temperature, diffusivity, and boundary data; vertex values with optional fixed diffusion filters of the initial state. Outputs: temperature $C^{0}$ and line-integrated heat flux $C^{1}$ .
Surface transport	Initial concentration, tangent velocity, and diffusivity; vertex concentration, oriented edge velocity, and initial mass summaries. Outputs: terminal concentration $C^{0}$ , edge transport $C^{1}$ , and face mass $C^{2}$ .
Incompressible flow	Body force, velocity traces, and pressure gauge; encode vector components or their oriented line integrals with boundary frames. Outputs: pressure $C^{0}$ and velocity one-cochains; normal flux uses degree $d-1$ .
Linear elasticity	Displacement, traction, and stiffness data; attach vector/tensor components to vertices or cells and integrate boundary loads. Outputs: vector displacement $C^{0}$ and vector-valued traction integrals on degree $d-1$ .
Electrostatics / magnetostatics	Sources, material data, and boundary indicators; input-derived source moments and support geometry. Outputs: scalar potential or, in 3D, surface-flux two-cochains, optionally reconstructed as node vectors.

Appendix

Table 7: Support-aware encoding of known PDE conditions. Examples connect physical inputs to numerical supports and output observables. Optional descriptors are computed from the inputs; the table is a construction library, not a list of evaluated tasks.

Operation	Work	Storage
Incidence and local geometry (offline)	$\mathcal{O}(E_{B}+N)$	$\mathcal{O}(E_{B}+N)$
Spectral bases (offline)	Eigensolver-dependent	Number of stored coefficients
Conditioning maps, one evaluation	$\mathcal{O}(Nhc_{\theta})$	$\mathcal{O}(Nc_{\theta})$
Relation feedback, one substep	$\mathcal{O}((E_{B}+N)c_{\theta})$	$\mathcal{O}(Nc_{\theta})$
Content update, one layer	$\mathcal{O}(Nh^{2}+Nhc_{\theta})$	$\mathcal{O}(Nh)$
Output projection and synthesis	$\mathcal{O}(N_{q}b_{q}r_{q})$	$\mathcal{O}(N_{q}b_{q})$ basis

Appendix

Table 8: Computational cost on a fixed complex. Online costs are per sample; preprocessing is reused across instances. The conditioning maps may be cached within each macro-layer.

b_{q}

denotes output-basis width.

Task and $(N_{0},N_{1},N_{2})$	Train	Val.	Test	$\beta$
Darcy: $(5477,16013,10535)$	4,000	500	500	$(1,2,0)$
Torus: $(2499,7497,4998)$	2,040	360	600	$(1,2,1)$
Cavity: $(3017,22344,38339)$	2,040	360	600	$(1,0,1)$

Appendix

Table 9: Benchmark meshes and nominal data partitions.

(N_{0},N_{1},N_{2})

counts vertices, edges, and faces;

\beta

gives the domain’s Betti numbers. The cavity additionally contains 19,010 tetrahedra. Partition usage is detailed in Appendix D.2 .

Target	S params	L params	Batch	Epochs (S/L)	Selected (S/L)
Darcy $C^{0}$	64,121	242,657	16	200/200	187/199
Darcy $C^{1}$	64,475	243,363	16	200/200	200/200
Darcy $C^{2}$	64,121	242,657	16	300/300	300/294
Torus $C^{0}$	63,546	241,506	16	300/300	292/289
Torus $C^{1}$	64,347	243,107	16	300/300	297/298
Torus $C^{2}$	64,314	243,042	16	300/300	299/294

Appendix

Table 10: DKHO capacities and training schedules. S/L denotes small/large. Epochs are the training budgets, and selected epochs minimize validation error. Parameters include the input and output heads. All rows use learning rate

10^{-3}

and seed 42.

Method	Darcy; torus $C^{1}/C^{2}$	Torus $C^{0}$	Cavity
GNO	Hidden 120, projection 68; 3 layers, radius 0.15	Hidden 120, projection 68; 3 layers, radius 0.15	Hidden 84, projection 96; 5 layers, radius 0.20
FNO	Width 21; 3 layers; $9\times 9$ modes	Width 20; 3 layers; $4^{3}$ modes	Width 21; 2 layers; $4^{3}$ modes
MGN	Hidden 72; 8 steps	Hidden 72; 8 steps	Hidden 64; 10 steps
DeepONet	Branch $(96,96,64)$ ; trunk $(68,68,64)$ ; 64 bases	Branch $(96,96,64)$ ; trunk $(68,68,64)$ ; 64 bases	Branch/trunk $(64,64,64)$ ; 74 bases
Geo-FNO	Width 18; 4 layers; $9\times 9$ modes	Width 8; 4 layers; $6^{3}$ modes	Width 12; 2 layers; $6^{3}$ modes
HSD	Residual width 12; 6 layers; $9\times 9$ modes	Residual width 12; 6 layers; $4^{3}$ modes	Residual width 14; 4 layers; $4^{3}$ modes

Appendix

Table 11: Baseline backbone configurations. Darcy and torus-form adapters share the first configuration. Spectral entries give width, depth, and retained modes per layer; HSD additionally uses 64 eigenmodes per degree and spectral hidden widths

(32,32)

Task	Decay	Batch	Epochs	Stopping rule
Darcy	$10^{-6}$	16	200	Full budget; select minimum validation error.
Torus $C^{0}$	$10^{-5}$	32	200	Full budget; select minimum validation error.
Torus $C^{1}/C^{2}$	$10^{-5}$	16	200	Full budget; select minimum validation error.
Cavity	$10^{-6}$	32	200	Full budget; select minimum validation error.

Appendix

Table 12: Training schedules for the native-input baseline comparisons. All use AdamW with initial learning rate

10^{-3}

and cosine decay. Epochs are maximum budgets.

Task	Native inputs	Added descriptors
Darcy	Coefficients, boundary traces, and support geometry	Degree-wise Laplacian bases (16 modes), heat descriptors, and global condition summaries
Torus	$u^{0}$ , tangent velocity, geometry, and fixed inter-degree transfers	Torus Fourier coordinates, low-order summaries of $u^{0}$ , and Laplacian, heat, mass, and energy descriptors
Cavity	Source/magnetization channels and shell geometry	Boundary/cavity indicators, Laplacian bases, source heat responses, and global source moments

Appendix

Table 13: Input descriptors for controlled comparisons. F augments the native inputs (N) with the descriptors in the final column; no target-field information is used.

Method	Native (N)			Matched (F)
Method	$C^{0}$	$C^{1}$	$C^{2}$	$C^{0}$	$C^{1}$	$C^{2}$
GNO	5.673e-4	9.834e-5	2.793e-5	1.625e-5	1.893e-5	2.938e-6
FNO	0.0063	0.0071	0.0022	0.0063	0.0071	0.0022
MGN	0.0076	4.724e-4	1.420e-4	5.984e-5	1.067e-5	3.029e-6
DeepONet	2.367e-4	0.0067	0.0023	1.406e-4	0.0069	0.0023
Geo-FNO	0.0154	0.0066	0.0021	0.0147	0.0069	0.0021

Appendix

Table 14: Darcy reconstruction error under input control. MSE (

\downarrow

) is reported for potential (

C^{0}

), flux (

C^{1}

), and circulation (

C^{2}

) under native (N) and matched (F) inputs.

Method	$C^{0}$			$C^{1}$			$C^{2}$
Method	RelL2	Grad	$S_{\beta_{0}}$	RelL2	CD	Curl	RelL2	CB	IoU
DKHO-small	0.0042	0.9985	0.9889	0.0264	0.9990	0.9986	0.0392	0.9996	0.9848
DKHO-large	0.0035	0.9989	0.9922	0.0155	0.9997	0.9996	0.0244	0.9998	0.9905
GNO	0.0058	0.9957	0.9916	0.0511	0.9968	0.9964	0.0371	0.9996	0.9861
FNO	0.1113	0.6837	0.8018	0.9736	0.6019	0.5066	0.9948	0.5460	0.6062
MGN	0.0112	0.9936	0.9823	0.0390	0.9984	0.9982	0.0379	0.9996	0.9835

Appendix

Table 15: Darcy accuracy and structure under matched inputs. Results on 500 test instances combine RelL2 (

\downarrow

) with differential and structural scores (

\uparrow

). Each block corresponds to the same output support as in Table 14 .

Method	Native (N)			Matched (F)
Method	$C^{0}$	$C^{1}$	$C^{2}$	$C^{0}$	$C^{1}$	$C^{2}$
GNO	0.0417	3.604e-4	1.255e-7	0.0153	3.995e-4	1.209e-7
FNO	0.0171	0.0013	4.796e-7	0.1170	0.0013	4.791e-7
MGN	0.0224	7.284e-5	4.842e-8	0.0120	7.779e-5	5.672e-8
DeepONet	0.0513	0.0013	5.160e-7	0.0811	0.0013	4.090e-7
Geo-FNO	0.0343	0.0014	5.669e-7	0.0172	0.0014	5.519e-7

Appendix

Table 16: Torus reconstruction error under input control. MSE (

\downarrow

) is reported for terminal concentration (

C^{0}

), edge transport (

C^{1}

), and face mass (

C^{2}

) under native (N) and matched (F) inputs.

Method	$C^{0}$			$C^{1}$			$C^{2}$
Method	RelL2	Grad	IoU	RelL2	CD	Curl	RelL2	CB	IoU
DKHO-small	0.1428	0.8440	0.8776	0.1310	0.9615	0.8026	0.1142	0.9666	0.9117
DKHO-large	0.1413	0.8472	0.8798	0.1262	0.9633	0.8187	0.1007	0.9737	0.9267
GNO	0.2075	0.8024	0.8097	0.3923	0.8111	0.5585	0.2122	0.9016	0.8258
FNO	0.5917	0.6326	0.5052	0.6687	0.6929	0.5116	0.4172	0.6174	0.5990
MGN	0.1790	0.8280	0.8423	0.1660	0.9338	0.7038	0.1252	0.9398	0.8872

Appendix

Table 17: Torus accuracy and structure under matched inputs. Results on 600 test instances combine RelL2 (

\downarrow

) with differential fidelity and high-response IoU (

\uparrow

), evaluated separately on each output support.

Method	Native (N)	Matched (F)
GNO	2.30e-3	1.374e-6
FNO	4.587e-5	8.932e-6
MGN	1.658e-4	1.331e-6
DeepONet	1.069e-5	5.792e-6
Geo-FNO	4.229e-5	9.891e-6
HSD	7.198e-6	1.522e-4

Appendix

Table 18: Cavity reconstruction error under input control. MSE (

\downarrow

) is evaluated on the three-channel nodal response using native (N) and matched (F) inputs.

Method	Native (N)			Matched (F)
Method	RelL2	Div	IoU	RelL2	Div	IoU
DKHO-small	–	–	–	0.0179	0.9996	0.9650
DKHO-large	–	–	–	0.0140	0.9997	0.9724
GNO	1.1030	0.5673	0.1400	0.0190	0.9996	0.9585
FNO	0.1189	0.9887	0.7242	0.0619	0.9979	0.7948
MGN	0.2532	0.9590	0.5555	0.0219	0.9997	0.9328

Appendix

Table 19: Cavity accuracy and structure under input control. RelL2 (

\downarrow

), divergence fidelity (Div,

\uparrow

), and IoU (

\uparrow

) are evaluated on the same 600 test instances in both protocols.

Target	Baseline	Params (K)		Ratio	RelL2 $\downarrow$
Target	Baseline	S	B	Ratio	S	B
Darcy $C^{0}$	DeepONet	64.1	559.1	11.5%	0.0042	0.0218
Darcy $C^{1}$	GNO	64.5	285.9	22.6%	0.0264	0.1129
Darcy $C^{2}$	GNO	64.1	285.9	22.4%	0.0392	0.1104
Torus $C^{0}$	FNO	63.5	309.1	20.5%	0.1428	0.2440
Torus $C^{1}$	MGN	64.3	303.2	21.2%	0.1310	0.1581

Appendix

Table 20: Parameter efficiency against the strongest native baseline. S denotes DKHO-small and B the selected baseline. The parameter ratio is

P_{\rm S}/P_{\rm B}

; counts are specific to each output model. Bold and underline mark the lower and higher RelL2 in each pair.

Target	Full	No Dirac	No phase	No harmonic
Darcy $C^{0}$	0.0042	0.0144	0.0062	—
Darcy $C^{1}$	0.0264	0.1124	0.0407	0.0310
Darcy $C^{2}$	0.0392	0.4698	0.0484	—
Torus $C^{0}$	0.1428	0.2877	0.1518	0.1465
Torus $C^{1}$	0.1310	0.3104	0.1469	0.1355
Torus $C^{2}$	0.1142	0.4072	0.1229	0.1688

Appendix

Table 21: Component ablations of DKHO-small. RelL2 (

\downarrow

) is evaluated under matched training settings. “No” denotes removal of one component. A dash marks an inapplicable harmonic ablation. Bold and underline indicate the best and second-best result in each row.

Metric	Darcy	Torus	Cavity
Instances	500	600	600
Support $q$	$24\%$	$20\%$	$18\%$
Spearman $\rho$	$0.500\pm 0.048$	$0.631\pm 0.123$	$-0.136\pm 0.050$
Dice	$0.505\pm 0.028$	$0.514\pm 0.102$	$0.582\pm 0.033$
Lift	$2.11$	$2.57$	$3.23$
Enrichment	$1.45\pm 0.02$	$1.64\pm 0.46$	$2.02\pm 0.21$

Appendix

Table 22: Population phase–PDE correspondence. Statistics are computed per instance and summarized as mean

\pm

standard deviation where shown. Enrichment and lift are dimensionless ratios; Perm. Dice is the mean after score permutation. Above chance is the fraction with Dice

>q

and enrichment

>1

. Columns use distinct task-specific diagnostics and are not ranked against one another.

Task	Case	Dice	Lift	Enr.	$\rho$	RelL2
Darcy	1	0.524	2.18	1.47	$0.541$	0.0197
	2	0.512	2.13	1.46	$0.510$	0.0195
	3	0.506	2.11	1.46	$0.498$	0.0164
Torus	1	0.700	3.50	2.98	$0.745$	0.1166
	2	0.663	3.32	1.98	$0.800$	0.0752
	3	0.653	3.26	2.04	$0.792$	0.0874

Appendix

Table 23: Phase–PDE diagnostics for illustrative held-out test instances. All three cases are visualized in Figure 6 . Enr. denotes intensity enrichment and RelL2 the prediction error of the same instance. Support fractions match Table 22 .

Figure 6: Spatial correspondence of DKHO phase coordination and rank-matched PDE structure across three held-out cases. Rows show Cases 1–3 from top to bottom; columns show Darcy faces, torus edges, and cavity nodes. All panels use the frozen DKHO-large checkpoint and the task-specific support fractions in Table 23 : $24\%$ (Darcy), $20\%$ (torus), and $18\%$ (cavity).

Explore similar work

Topology-Preserving Neural Operator Learning via Hodge Decomposition
May 13, 2026Dongzhe Zheng, Tao Zhong, Christine Allen-BlanchetteNeural Operators Graph Laplacians
Princeton University
Topological Neural Operators
Jun 8, 2026Lennart Bastian, Samuel Leventhal, Mustafa Hajij +1Neural Operators Topology
Imperial College London, UK · University of San Francisco, USA
The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics
Aug 10, 2026Petr Badolia, Leonid Obukhov, Dmitry Bylinkin +1Partial Differential Equations Oscillatory Dynamics
Basic Research of Artificial Intelligence Laboratory (BRAIn Lab) · Innopolis University

May 13, 2026, cs.LGJ/K move · Enter open · S save

Topology-Preserving Neural Operator Learning via Hodge Decomposition

Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette

Princeton University

Neural Operators Graph Laplacians Scalar Field Riemannian Manifolds

Open paper AskPDF arXiv

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Abstract

Explore similar work

Topology-Preserving Neural Operator Learning via Hodge Decomposition

Topological Neural Operators

The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics