Beyond Pairwise Graphs in Science: Hypergraph Adaptive Wavelet Operators for Parametric PDEs

Rajat Sarkar; Venkataramana Runkana; Souvik Chakraborty

cs.LGAug 28, 2026

Beyond Pairwise Graphs in Science: Hypergraph Adaptive Wavelet Operators for Parametric PDEs

Authors: Rajat Sarkar, Venkataramana Runkana, Souvik Chakraborty

Organizations: Department of Applied Mechanics Indian Institute of Technology Delhi New Delhi, 110016, India · TCS Research Tata Consultancy Services, India · Department of Applied Mechanics Yardi School of Artificial Intelligence (ScAI) Indian Institute of Technology Delhi New Delhi, 110016, India

Abstract

Physical systems are often modeled by solution operators that map input fields, parameters, geometries, or past states to steady or future physical states. Learning these maps is difficult, especially for time-dependent systems that must assimilate history and remain stable under autoregressive rollout. Many neural operators work best on regular, structured grids, while realistic simulations often require unstructured meshes or point clouds to resolve complex geometries; in such settings, grid-centric representations can lose accuracy. Graph neural operators handle these domains through message passing or spectral graph filtering, but pairwise edges do not directly capture group-wise couplings among mesh cells, local neighborhoods, or conservation volumes. We introduce the Hypergraph Adaptive waveLet Operator (HALO), which lifts the domain to a hypergraph and learns in its spectral wavelet domain. HALO avoids explicit hypergraph-Laplacian eigendecomposition through Chebyshev polynomial wavelet filters, giving localized spectral kernels at linear sparse-matrix cost. Its trainable dyadic wavelet scales are regularized toward tight-frame coverage, allowing the frequency response to adapt to each PDE while encouraging stable multi-scale spectral coverage. Across 2D and 3D benchmarks on structured and unstructured discretizations, HALO achieves best or near-best accuracy among frequency-, transformer-, DeepONet-, state-space-, and graph-based baselines and sustains stable multi-step rollouts. The same model scales to industrial aerodynamic geometries: on meshes of a few hundred thousand points it is on par with, or better than, the strongest fixed-discretization transformers, while remaining resolution-equivariant.

Figures & tables

Figure 1: HALO architecture: coordinate-aware input uplift, a stack of $L$ Adaptive Wavelet Blocks operating on the cached sparse hypergraph Laplacian, output projection, and a global residual connection.

Panel of Figure 2 — Figure 2: Dataset-aware hypergraph construction. Point clouds use Euclidean $k$ -nearest-neighbor groups, whereas meshes use connected $k$ -ring patches that respect the supplied topology.

Figure 3: Adaptive Wavelet Block. A shared Chebyshev basis supports every learned wavelet scale; analytic Base and low-rank Delta responses are mixed across scales and combined with the skip path before normalization and activation.

Model Specification			2D Regular Grid			2D Unstructured Grid		3D Volume
Family	Operator	Res.-Eq.	Darcy	Navier–Stokes	Allen–Cahn	Airfoil	Cylinder	Supernova
Problem regime			Steady-state	Time-dep.	Steady-state	Time-dep.	Time-dep.	Time-dep.
Spectral	FNO [ 7 ]	✓	1.08	13.67	2.39	–	–	55.76
	Geo-FNO [ 9 ]	✓	1.08	13.67	2.39	2.80	59.06	55.76
	WNO [ 8 ]	✓	0.84	38.49	6.43	–	–	55.31
	LSM [ 29 ]	✗	0.65	22.25	4.43	1.38	59.55 ∗	52.64

Table 1: Mean relative

L_{2}

error across the six field-prediction benchmarks (

\times 10^{-2}

); lower is better. For HALO, the second line gives the standard deviation over three independent training runs. Baseline ranks are computed independently for each dataset.

Model Specification		ShapeNet-Car				Aircraft
Model	Res.-Eq.	Vol. $\downarrow$	Surf. $\downarrow$	$C_{D}$ $\downarrow$	$\rho_{D}$ $\uparrow$	Field $\downarrow$	$C_{l}$ $\downarrow$	$R_{l}^{2}$ $\uparrow$	Surf. $\downarrow$
PointNet [ 32 ]	✗	4.94	11.04	2.98	95.83	15.20	9.50	98.20	16.90
Graph U-Net [ 33 ]	✗	4.71	11.02	2.26	97.25	–	6.30	95.30	16.10
MGN [ 13 ]	✗	3.54	7.81	1.68	98.40	10.20	3.80	99.30	11.30
GNO [ 11 ]	✓	3.83	8.15	1.72	98.34	11.40	3.10	99.10	12.90
Geo-FNO [ 9 ]	✓	16.70	23.78	6.64	82.80	–	–	–	–

Table 2: Aerodynamic prediction on two large 3D unstructured geometries (

\times 10^{-2}

). HALO entries are mean

\pm

standard deviation over three independent runs. Cell colors follow the baseline-rank and HALO legend in Table 1 ; bold marks the best overall result, including ties. “–” denotes an unsupported setting. MGN: MeshGraphNet.

Figure 4: Large-geometry prediction. Columns show the reference and absolute errors of Transolver++ and HALO. ShapeNet-Car includes surface pressure and volume velocity; Aircraft shows surface pressure. Labels report mean relative- $L_{2}$ error, and $\times$ / $\checkmark$ marks resolution equivariance.

Figure 5: Zero-shot super-resolution on structured grids and unstructured meshes. The field montage contrasts the coarse training discretization, fine reference, HALO prediction, and absolute error for representative Allen–Cahn and Airfoil tests. The benchmark ledger reports all six base- and target-resolution errors. Reference and prediction share color limits; error maps use logarithmic scales.

Graph type	Allen–Cahn	Darcy
Hypergraph	$0.4614\%{\pm}0.0058\%$	$0.4125\%{\pm}0.0115\%$
Pairwise	$0.4142\%{\pm}0.0179\%$	$0.5011\%{\pm}0.0170\%$

Table 3: Relative-

L_{2}

error of the hypergraph operator versus a pairwise graph with matched local support, on Allen–Cahn (

k{=}8

) and Darcy (

k{=}24

). Entries are mean

\pm

standard deviation in percent.

Neighborhood size $k$	Relative $L_{2}$ error
$k{=}24$	$0.4125\%{\pm}0.0115\%$
$k{=}16$	$0.4786\%{\pm}0.0109\%$

Table 4: Effect of hyperedge neighborhood size on Darcy at fixed capacity (

1{,}797{,}023

parameters), so only connectivity changes. Entries are mean

\pm

standard deviation.

Scale setting	Parameters	Relative $L_{2}$ error
Trainable scales	$575{,}629$	$0.4614\%{\pm}0.0058\%$
Fixed scales	$575{,}617$	$0.4741\%{\pm}0.0195\%$
Single scale	$280{,}709$	$0.5258\%{\pm}0.0023\%$

Table 5: Learned multiscale wavelet filtering on Allen–Cahn, comparing the trainable-scale control with frozen and single-scale banks. Entries are mean

\pm

standard deviation.

Model	Parameters	Relative $L_{2}$ error
w/ Delta path	$575{,}629$	$0.4614\%{\pm}0.0058\%$
w/o Delta path	$264{,}333$	$0.7488\%{\pm}0.0028\%$

Table 6: Delta-path ablation on Allen–Cahn over two seeds; entries are mean

\pm

standard deviation.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Meaning	Symbol	Meaning
$\mathcal{N}:\mathcal{A}\!\to\!\mathcal{U}$	Solution operator to be learned	$T_{m},\,\mathbf{T}_{m}$	Chebyshev polynomial and basis tensor
$a^{(i)},\,u^{(i)}$	Input / output functions of sample $i$	$c_{s,m}$	Chebyshev coefficients of $g(s\cdot)$
$D\subset\mathbb{R}^{d}$	Bounded physical domain, dimension $d$	$Q,\,\theta_{q}$	Quadrature-node count and angles
$N$	Number of training pairs	$x^{*}\!\approx\!1.423$	Peak location of the kernel $g$
$\mathcal{K}_{\phi}^{(\ell)},\,\kappa_{\phi}$	Kernel-integration operator and kernel	$s_{j}^{\mathrm{init}},\,\lambda_{j}^{*}$	Dyadic-ladder initial scale, peak eigenvalue
$v_{\ell},\,W_{\ell},\,\sigma$	Hidden state, pointwise map, nonlinearity	$\rho_{j}$	Softplus pre-activation for scale $s_{j}$

Appendix

Table A.1: Notation used throughout the paper and this appendix.

		Regular Grid			Unstructured Mesh
	Configuration	Allen–Cahn	Darcy	Navier–Stokes	Airfoil	Cylinder
Physics	Task	Final state	Pressure	Flow forecast	Flow forecast	Flow forecast
	Input	Initial field	Permeability	Past states	Past fields	Past fields
	Output	Final field	Pressure	Future states	Future fields	Future fields
Data	Geometry	Periodic grid	Regular grid	Periodic grid	Shared mesh	Variable mesh
	Dimension	2D	2D	2D + time	2D + time	2D + time

Appendix

Table C.1: Benchmark dataset details and operator-learning tasks for the five 2D structured-grid and unstructured-mesh settings.

		Structured Grid	Unstructured Grid
	Configuration	Supernova	ShapeNet-Car	Aircraft
Phys.	Task	3D forecast	Steady flow	Steady flow
	Input	Past fields	Geometry	Geometry
	Output	Future fields	$u,v,w,p$	$C_{p},\rho,u,v,w,p$
Data	Geometry	Cubic simulation domain	Car + wind-tunnel domain	Aircraft exterior
	Prediction support	Volume	Volume + surface	Surface

Appendix

Table C.2: Dataset details for the three 3D benchmarks, grouped by structured and unstructured discretization. Prediction support states whether the target fields occupy the volume, the surface, or both.

		Regular Grid					3D Volume
	Configuration	Allen–Cahn	Darcy	Navier–Stokes			Supernova
				$\nu{=}10^{-3}$	$\nu{=}10^{-4}$	$\nu{=}10^{-5}$
Training	Normalization	Pointwise Gaussian					$\log(1{+}x)$
	Loss	$L_{2}{+}.1\nabla{+}.02\mathrm{TF}$	$L_{2}{+}.1\nabla{+}.1\mathrm{TF}$	$L_{2}{+}.1\mathrm{TF}$	$L_{2}{+}.1(\nabla{+}\mathrm{TF})$	$L_{2}{+}.1(\nabla{+}\mathrm{TF})$	$L_{2}{+}.05\nabla{+}.1\mathrm{TF}$
	Epochs	2000	500	600	700	500	300
	LR / WD	$10^{-3}/10^{-6}$	$5{\times}10^{-4}/10^{-5}$	$5{\times}10^{-4}/10^{-4}$	$7{\times}10^{-4}/10^{-4}$	$10^{-3}/10^{-4}$	$10^{-3}/10^{-6}$

Appendix

Table C.3: HALO configurations for 2D and 3D regular-grid benchmarks. “–” denotes no temporal rollout;

w_{\mathrm{end}}

is the endpoint of the linear per-step ramp of Eq. ( 34 ), so

w_{\mathrm{end}}{=}1

means uniform weighting; Supernova is normalized by a

\log(1{+}x)

transform followed by a channel-wise

z

-score computed over

128

trajectories.

	Configuration	Airfoil	Cylinder	ShapeNet-Car	Aircraft
Training	Normalization	Nondim. + ch. $z$ -score		ch. $z$ -score
	Loss	$L_{2}{+}.1\mathrm{TF}$	$L_{\rm data}{+}\nabla{+}.1\mathrm{TF}$	$L_{\mathrm{MSE}}{+}.1\mathrm{TF}$	$L_{\rm bal}{+}.1\mathrm{TF}$
	Epochs	500	600	500	200
	LR / WD	$10^{-3}/10^{-5}$	$10^{-3}/10^{-6}$	$10^{-3}/10^{-5}$	$10^{-3}/10^{-5}$
	Scale LR/WD	$10^{-4}/0$	$10^{-4}/0$	$10^{-3}/0$	$10^{-3}/0$
	Optimizer	AdamW	AdamW	AdamW	AdamW

Appendix

Table C.4: HALO training and architecture configurations for the unstructured 2D and 3D mesh benchmarks. “–” denotes no temporal rollout;

w_{\mathrm{end}}

is the endpoint of the linear per-step ramp of Eq. ( 34 ), so

w_{\mathrm{end}}{=}1

means uniform weighting.

L_{\rm data}=L_{v}+1.25L_{p}

weights Cylinder’s velocity and pressure channels directly, in place of a single relative-

L_{2}

term.

L_{\rm bal}

is the channel-balanced relative

L_{2}

of Eq. ( D.3 ), which normalizes each Aircraft field separately.

Figure C.1: The two large 3D unstructured aerodynamic benchmarks, each immersed in an incoming wind: (a) ShapeNet-Car (drag) and (b) Aircraft (lift).

Dataset	COO (s)	SELL-C-1 (s)	Speedup
Allen–Cahn ( $64^{2}$ )	15.11	14.36	$1.05\times$
Darcy ( $85^{2}$ )	135.71	112.86	$1.20\times$
Navier–Stokes ( $\nu=10^{-5}$ , $64^{2}$ )	459.12	434.88	$1.06\times$
Airfoil	444.78	243.08	$\mathbf{1.83\times}$
Cylinder	485.55	247.05	$\mathbf{1.97\times}$
Aircraft	222.60	192.13	$1.16\times$

Appendix

Table D.1: Mean epoch time for the coordinate and SELL-C-1 backends. Each entry is a two-epoch A/B measurement on an NVIDIA A40 at the dataset’s best-model configuration, with identical seed, batch order, and learning-rate schedule; the sparse format is the only change. The gain is largest on the denser unstructured-mesh operators.

Benchmark	Ops	$n$	$\mathrm{nnz}(H)$	$\mathrm{nnz}(\Delta)$	Density	MACs	Asm./run	Form
Shared topology: one operator serves the whole run
Allen–Cahn, NS $\,{\times}3$	$1$	$4{,}096$	$36{,}864$	$102{,}400$	$2.78\times$	$0.72\times$	$1.20$ MiB	assembled
Darcy	$1$	$7{,}225$	$180{,}625$	$568{,}139$	$3.15\times$	$0.64\times$	$6.56$ MiB	assembled
Airfoil	$1$	$3{,}982$	$96{,}321$	$230{,}300$	$2.39\times$	$0.84\times$	$2.67$ MiB	assembled
Supernova	$1$	$32{,}768$	$294{,}912$	$1{,}200{,}286$	$\mathbf{4.07\times}$	$0.49\times$	$13.99$ MiB	assembled
Per-sample topology: one operator per mesh, all cached

Appendix

Table D.2: The factored-versus-assembled decision, measured on the operators each benchmark actually builds. Ops is the number of distinct operators a run holds resident; Density is

\mathrm{nnz}(\Delta)/\mathrm{nnz}(H)

, how much fatter assembly makes the operator; MACs is

2\,\mathrm{nnz}(H)/\mathrm{nnz}(\Delta)

, the multiply–accumulates of one factored apply relative to one assembled apply, the factor of two being the second sparse product the factored form pays. MACs counts arithmetic only, and so excludes the intermediate the factored path writes and the second kernel launch. Assembly densifies on every benchmark, so the factored form is arithmetically cheaper on every benchmark, yet neither column tracks the form used. Green marks the two benchmarks kept factored; gray marks the two cells a reader would expect to force that form and which do not—Supernova’s densification, the second highest in the suite, and Cylinder’s operator count, the largest. What tracks the form is Asm./run , the operator count times the assembled size of one. Cylinder’s meshes vary in size; its row reports the representative operator and scales it by the cached count. Sizes are binary.

	Cylinder	ShapeNet-Car	Aircraft
Distinct operators	$1{,}140$	$889$	$30$
Factored, per operator	$0.69$ MiB	$11.94$ MiB	$37.99$ MiB
Assembled, per operator	$1.20$ MiB	$39.09$ MiB	$75.20$ MiB
Factored, whole run	$0.77$ GiB	$\mathbf{10.37}$ GiB	$\mathbf{1.11}$ GiB
Assembled, whole run	$\mathbf{1.34}$ GiB	$\mathbf{33.94}$ GiB	$\mathbf{2.20}$ GiB
Form used	assembled	factored	factored

Appendix

Table D.3: Why Cylinder assembles and the two geometry benchmarks do not, with the factored benchmarks shaded. All three carry a per-sample mesh and memoize one operator per sample, so all three hold a resident set that scales with the dataset; they differ by up to

63\times

in the size of a single operator, and it is the product that decides. Factored size is the cached blob—integer incidence pairs and the two scaling vectors; assembled size is the corresponding CSR operator.

Dataset	$\lambda_{\max}$	$s_{0}$	$s_{1}$	$s_{2}$	$s_{3}$	$s_{4}$
Dyadic initialization ( $\lambda_{\max}{=}1$ )	–	1.423	2.845	5.691	11.381	22.762
Allen–Cahn ( $J{=}3$ )	0.997	1.240	2.858	6.337	–	–
Darcy	0.998	1.149	2.504	5.661	12.645	28.198
Navier–Stokes $\nu{=}10^{-3}$	0.997	1.492	2.926	6.241	11.257	22.636
Navier–Stokes $\nu{=}10^{-4}$	0.997	1.584	3.023	6.788	11.092	22.713
Navier–Stokes $\nu{=}10^{-5}$	0.997	1.677	3.225	6.268	11.397	23.090

Appendix

Table D.4: Final learned wavelet scales of the best HALO model on each benchmark, averaged over the

L

Adaptive Wavelet Blocks (per-block values vary by at most a few percent).

Structured-Grid Zero-Shot Super-Resolution
Dataset	Grid	FNO	WNO	GNO	HALO
Allen–Cahn	Base $64^{2}$	0.0239	0.0643	0.0215	0.0046
Allen–Cahn	ZS $128^{2}$	0.0244	0.2801	0.1870	0.0115
Darcy	Base $85^{2}$	0.0108	0.0084	0.0346	0.0041
Darcy	ZS $421^{2}$	0.0121	0.0109	0.0469	0.0082
NS $\nu{=}10^{-4}$	Base $64^{2}$	0.0408	0.1275	0.4046	0.0442

Appendix

Table D.5: Mean relative

L_{2}

error; lower is better. Bold marks the best result in each row; ZS denotes evaluation on the zero-shot target grid.

Panel of Figure 24 — Figure E.1: Representative held-out Allen–Cahn and Darcy predictions. Each dataset contains two unseen inputs; rows show the input, ground truth, HALO prediction, and pointwise absolute error.

Panel of Figure 25 — Figure E.2: Held-out Navier–Stokes autoregressive rollouts at three viscosities. For each viscosity, columns show the $t=10$ and $13\,\mathrm{s}$ forecasts and rows show the input, ground truth, HALO prediction, and absolute error.

Panel of Figure 26 — Figure E.3: Held-out Cylinder rollout on the variable native mesh.

Panel of Figure 27 — Figure E.4: Held-out Airfoil rollout for the same test trajectory.

Panel of Figure 28 — Figure E.5: Held-out ShapeNet-Car predictions on the native 3D unstructured mesh. (a) surface pressure. (b) volume velocity magnitude with streamlines. Columns show the ground-truth field, the HALO prediction, and the pointwise absolute error.

Panel of Figure 29 — Figure E.6: Held-out Aircraft surface-pressure predictions on native 3D unstructured meshes at Mach $7.0$ , angle of attack $0^{\circ}$ , and sideslip angle $2^{\circ}$ : (a) aircraft 92, with $4.42\%$ relative- $L_{2}$ error; and (b) aircraft 93, with $5.83\%$ relative- $L_{2}$ error. Within each panel, columns show the ground truth, HALO prediction, and pointwise absolute error.

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