PaperlayerView exact revision on arXiv

Source-linked AI summary

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

Rajat Sarkar, Venkataramana Runkana, Souvik Chakraborty

arXiv:2608.27883v1cs.LGphysics.comp-ph

TL;DR

Neural operators struggle with varying unstructured discretizations and pairwise graphs do not directly encode group-wise physical couplings. HALO addresses this with hypergraph spectral wavelets, Chebyshev filtering, and trainable tight-frame-regularized scales. Across eight benchmarks, it ranks first or among the strongest methods, while stable long-range forecasting and million-point industrial scaling remain open questions.

  • Problem

    Existing neural operators have difficulty with complex, variable unstructured geometries, while pairwise graph edges do not directly represent group-wise couplings.

  • Method

    HALO represents domains as hypergraphs and learns adaptive multi-scale spectral wavelet kernels with Chebyshev filters over a fixed sparse Laplacian.

  • Results

    Across all eight benchmarks, HALO ranks first or among the strongest methods from every operator family considered, with largest gains on unstructured problems and larger geometries.

  • Takeaways & Limitations

    HALO captures set-valued couplings directly while retaining resolution equivariance and a per-layer cost linear in the number of nodes.

  • Takeaways & Limitations

    Errors accumulate when inference extends substantially beyond training horizons, and experiments have not yet reached million-point industrial meshes or coupled multiphysics systems.

Abstract

from arXiv · show

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.

1 Introduction

Scientific operators must handle varying discretizations and higher-order physical couplings that regular-grid and pairwise graph methods do not directly represent. HALO addresses these challenges with hypergraph spectral wavelets and adaptive, efficient multi-scale filtering.

  • Operator learning seeks data-driven maps between function spaces that approximate PDE solution operators across problem instances.
  • Regular-grid spectral operators lose accuracy on complex unstructured geometries and cannot readily absorb samples with different meshes or node counts.
  • Pairwise graph edges are limiting because finite-element and finite-volume interactions often couple groups of nodes or conserved cells.
  • Hypergraphs represent arbitrary node groups directly, motivating their use for PDE operators that need higher-order, multi-scale, stable, and practical interactions.
  • HALO uses hypergraph spectral wavelets, Chebyshev filters, trainable scales, and tight-frame regularization for localized adaptive kernel integration without eigendecomposition.
  • HALO remains grid-agnostic and resolution-equivariant, matching the best regular-grid operators on structured domains and surpassing all baselines on unstructured ones.

2 Problem Formulation and Background

The paper formulates neural-operator learning as approximating PDE solution maps from finite paired observations on compatible discretizations. It develops hypergraph wavelet filtering to capture long-range and higher-order dependencies on unstructured domains.

  • 2 Problem Formulation and Background: The goal is to learn a solution operator N mapping input functions to output functions from finite paired observations.
  • 2 Problem Formulation and Background: The learned operator should approximate the continuous solution map and remain evaluable on compatible discretizations of the physical domain.
  • Kernel-Based Neural Operator: Kernel-based neural operators refine hidden representations through iterative update layers motivated by Green’s-function representations of PDEs.
  • Kernel-Based Neural Operator: Each layer combines a pointwise linear map, nonlinear activation, and kernel integration operator.
  • Kernel-Based Neural Operator: Kernel integration captures long-range function dependencies, while expressive computation over unstructured domains with higher-order interactions remains the central challenge.
  • 2.1 Hypergraph Wavelet Theory: A hypergraph allows each hyperedge to connect an arbitrary subset of vertices, with incidence and degree matrices defining its normalized Laplacian.
  • 2.1 Hypergraph Wavelet Theory: Spectral hypergraph filtering applies a band-pass kernel through the Laplacian eigendecomposition, and wavelet operators localize this filtering at chosen scales.
  • 2.1 Hypergraph Wavelet Theory: Chebyshev approximation avoids explicit eigendecomposition, costs O(Mnnz(Δ)), and uses M for spatial localization while s controls spectral selectivity.

3 Hypergraph Adaptive Wavelet Operator (HALO)

HALO lifts node data and geometry into a dataset-aware hypergraph, then applies adaptive Chebyshev wavelet blocks over a cached sparse Laplacian. Shared polynomial bases provide localized multi-scale filtering with linear scaling in node count.

  • Architecture: HALO lifts sampled fields, coordinates, and conditioning channels into latent node representations before decoding them to the target space.
  • Architecture: Coordinates enrich node features while defining the hypergraph geometry, whose sparse Laplacian is built once, cached, and reused across Adaptive Wavelet Blocks.
  • Architecture: The model supports steady-state inputs and autoregressive prediction from multiple recent snapshots.
  • Hypergraph Construction: Point clouds use Euclidean k-nearest-neighbor hyperedges, whereas meshes use connected k-ring patches that respect mesh topology.
  • Hypergraph Construction: Overlapping hyperedges encode local higher-order structure, with compact groups receiving greater weights than diffuse groups.
  • Adaptive Wavelet Block: An Adaptive Wavelet Block maps hidden node features through a fixed sparse hypergraph Laplacian using Chebyshev-realized multi-scale filters.
  • Adaptive Wavelet Block: The Meyer-like band-pass kernel vanishes at DC, peaks near x*=1.423, and decays at large spectral arguments.
  • Adaptive Wavelet Block: Wavelet scales start from a dyadic ladder and become positive trainable parameters through softplus reparameterization, adapting to target frequency content.

4 Experiments

HALO performs strongly across structured, unstructured, and large-scale aerodynamic PDE benchmarks, while its ablations show that connectivity, multiscale adaptation, and corrective paths contribute in task-dependent ways.

  • Overall benchmark results: HALO ranks in the top two on all eight benchmarks and is best or tied-best on seven, trailing only on Navier–Stokes.All results use identical train/test splits and resolutions; HALO values average three independent runs.
  • Unstructured meshes: 23.8% lower Cylinder error than the strongest adapted baseline and 68.6% lower than GNO demonstrate HALO’s advantage on variable-mesh unstructured data.HALO constructs its hypergraph per sample and remains permutation-equivariant without assuming a common discretization.
  • Large-geometry aerodynamics: On Aircraft, HALO reduces Transolver++ field error from 4.98 to 3.36 and surface-pressure error from 6.40 to 4.93.These correspond to improvements of 32.5% and 23.0%, respectively, on meshes reaching roughly 330k points.
  • Zero-shot transfer: HALO achieves the lowest target-grid error on three of four structured benchmarks in zero-shot transfer, with FNO ahead only on Navier–Stokes.Relative to the strongest zero-shot baseline, HALO lowers error by 52.9% on Allen–Cahn, 24.8% on Darcy, and 13.3% on Supernova; Navier–Stokes is the sole exception, where FNO is 15.6% better.
  • Effect of higher-order connectivity: Matched-support comparisons show higher-order grouping is task dependent: the pairwise control wins on Allen–Cahn, whereas the hypergraph wins on Darcy.Allen–Cahn error is 10.2% lower with pairwise support, while Darcy’s pairwise error is 21.5% higher.
  • Ablations: Darcy error decreases when hyperedge neighborhoods grow from k=16 to k=24, while fixed-scale and no-Delta ablations increase error by 2.8% and 62.3%.The fixed-scale comparison removes only 12 parameters; removing the Delta path also removes 311,296 trainable parameters, so its effect is not capacity-isolated.

5 Conclusion

HALO represents PDE domains as hypergraphs and learns adaptive spectral wavelet kernels, capturing set-valued couplings while avoiding eigendecomposition. Across eight benchmarks, it remains among the strongest operators, with gains clearest on unstructured problems and large geometries, but long-range forecasting remains unresolved.

  • HALO represents PDE domains as hypergraphs and learns kernels in the spectral wavelet domain to capture set-valued couplings directly.Trainable scales provide adaptive multi-scale filtering, while tight-frame regularization supports spectral coverage.
  • Chebyshev approximation avoids eigendecomposition and retains per-layer cost linear in the number of nodes.The construction preserves resolution equivariance across discretizations.
  • Across all eight benchmarks, HALO ranks first or among the strongest methods from every operator family considered.Its largest gains occur on unstructured problems and the largest geometries.
  • Errors accumulate when inference extends substantially beyond training horizons, leaving reliable long-range forecasting as an open problem.The reported accuracy applies to rollout horizons represented during training.

B.1 Supporting Approximation-Theoretic Results

The supporting results establish spectral bounds, approximation behavior, localization, and null-space properties for HALO’s wavelet construction. They also clarify that the piecewise-smooth kernel supports algebraic rather than exponential convergence guarantees.

  • The normalized hypergraph Laplacian has spectrum σ(Δ) ⊆ [0, 1], while on ordinary graphs it corresponds to a rescaled normalized graph Laplacian.The affinity factorization yields Δ = I − H̃H̃⊤ and establishes positive semidefiniteness.
  • The spectral surrogate is evaluated on 64 uniformly spaced points over [10^-3λmax, λmax], excluding the DC mode.The cutoff deliberately leaves near-null eigenvalues below the lower bound unconstrained.
  • The power method estimates λmax using sparse products with O(nnz(Δ)) work per iteration and O(n) memory.Symmetric positive semidefiniteness makes λmax the eigenvalue of largest modulus.
  • Chebyshev wavelet evaluation costs O(Mnnz(Δ)), and polynomial degree M confines the approximation to an M-hop neighborhood.The scale s controls spectral selectivity, whereas M controls spatial localization.
  • The Meyer-like kernel supports algebraic approximation rates, not exponential convergence, because it is only piecewise smooth across its breakpoints.The mapped quadrature integrand is therefore piecewise rather than globally polynomial.
  • When g(0) = 0, the wavelet analysis operator annihilates the null space of Δ, including the constant mode.This follows because every zero-eigenvalue component is multiplied by g(0).

C Benchmark Dataset Details

The evaluation covers eight PDE families across structured grids, unstructured meshes, 3D volumes, and aerodynamic geometries. The datasets include steady and time-dependent tasks, fixed and variable meshes, and resolutions ranging from small 2D meshes to 331,971-point aircraft surfaces.

  • The benchmark suite contains eight PDE benchmarks spanning regular grids, unstructured meshes, 3D volumes, and 3D aerodynamic geometries.The evaluation reports mean relative L2 error, with HALO results averaged over three independent runs.
  • Allen–Cahn models chaotic two-dimensional phase separation from an initial condition to a final phase field.The setting uses periodic boundary conditions and Gaussian-random-field initial conditions.
  • Darcy flow is a steady elliptic pressure problem in which permeability determines the pressure field.Samples vary through two-phase piecewise-constant permeability structures under homogeneous Dirichlet conditions.
  • Cylinder is an unstructured time-dependent flow benchmark whose mesh size varies across trajectories, from about 1,732 to 2,059 training nodes.The task predicts velocity and pressure 25 steps ahead.
  • Supernova is a 3D compressible-gas explosion benchmark generated with density-independent smoothed particle hydrodynamics.The simulation includes conservation laws, self-gravity, radiative heating and cooling, and an ideal-gas equation of state.
  • ShapeNet-Car uses one unstructured mesh of 32,186 points for steady aerodynamic fields and drag estimation across 889 car geometries.Each geometry is placed in a simulated wind tunnel at a constant inlet speed of 72 km/h.
  • Aircraft contains over 30 designs represented by 331,971 surface mesh points under five varying incoming-flow conditions.The benchmark targets high-fidelity large-scale aerodynamic prediction.

D Training Details

HALO training builds coordinate-derived hypergraphs, caches their sparse Laplacians, and uses Chebyshev wavelet blocks for steady prediction or full autoregressive rollout. Dataset-specific preprocessing selects neighborhood rules, while field-aware losses and balanced channel weighting support heterogeneous physical outputs.

  • Training and rollout: Time-dependent training feeds predictions back through the sliding window and accumulates loss across the complete rollout horizon.Inference similarly iterates the rollout noise-free, while steady tasks use a single operator application.
  • Preprocessing: HALO builds one hypergraph per discretization from k-nearest-neighbor or mesh-ring neighborhoods and computes a cached normalized Laplacian without eigendecomposition.The implementation estimates λmax by sparse power iteration and rescales the spectrum to [−1,1].
  • Adaptive Wavelet Blocks: The shared Chebyshev recurrence computes wavelet bases through sparse matrix products, with polynomial order controlling spatial localization and scale controlling spectral selectivity.The order-M approximation is supported within an M-hop neighborhood, with cost O(Mnnz(Δ)).
  • Hypergraph construction: Mesh-ring hyperedges follow connectivity, preventing neighborhoods from joining spatially close but geodesically distant nodes.The Airfoil construction combines cell hyperedges with two-ring neighborhoods, totaling 11,667 hyperedges.
  • Field-aware targets: Field-aware projections and losses separate heterogeneous physical channels, while per-channel normalization prevents high-energy channels from dominating auxiliary targets.On the reported variant, two channels improve from 0.222 and 0.183 to 0.106 and 0.117 under balanced weighting.

D.1 Sliced-ELLPACK Sparse Backend

The Sliced-ELLPACK backend accelerates the wide sparse products used by HALO’s Chebyshev recurrence while preserving the model and its numerical outputs. It dispatches between SELL-C-1 and CSR according to operand width and exploits near-uniform hypergraph row lengths.

  • Layout: SELL-C-1 partitions rows into locally padded slices, enabling coalesced GPU reads for wide dense operands.Slice height is selected from the operand width, and conversion is cached lazily.
  • Dispatch: The backend uses SELL on CUDA for K≥192 and CSR otherwise, while retaining mesh row order to preserve feature locality.Sorting Cylinder rows reduced padding but increased kernel time from 0.467 ms to 0.554 ms.
  • Performance: 0.171 ms: SELL-C-1 is faster than cuSPARSE CSR at 0.344 ms and COO at 1.799 ms on the Airfoil operator.The corresponding operator storage is 1.94 MiB for SELL-C-1, compared with 2.67 MiB for CSR and 4.39 MiB for COO.
  • Accuracy and runtime: 28%: SELL-C-1 reduces summed one-epoch suite time while preserving reported numerical accuracy.On a converged Cylinder checkpoint, raw-state relative L2 errors are 0.0250747, 0.0250765, and 0.0250790 for COO, CSR, and SELL.

D.2 Factored Hypergraph Laplacian

HALO can apply the hypergraph Laplacian either as an assembled sparse operator or through incidence factors. The factored form avoids densification and becomes preferable when many large per-sample operators would exceed memory, while learned wavelet scales adapt spectral coverage across PDEs.

  • Operator structure: Assembly replaces one-hop incidence connectivity with two-hop node coupling, densifying the Laplacian to 2.4×–4.4× the incidence nonzeros.A hyperedge with m members contributes m entries to H but m^2 entries to Δ.
  • Factored application: The factored apply evaluates the same Laplacian through two sparse incidence products and four elementwise scalings, without allocating the assembled operator.Both forms feed the same power iteration and differ only through floating-point summation order.
  • Arithmetic trade-off: 0.45×: the factored form uses this fraction of assembled multiply–accumulates on ShapeNet-Car, where assembly densification exceeds 2×.The factored path pays for a second sparse product and therefore can be arithmetically cheaper despite extra intermediate storage and launches.
  • Memory criterion: The form decision depends on resident operator count multiplied by individual operator size, not dimensionality alone.Cylinder’s 1,140 cached operators total 1.34 GiB and fit, whereas assembling ShapeNet-Car operators would raise resident memory from 10.4 to 33.9 GiB.
  • Boundary: Factored operators are not free: each apply writes and rereads an |E| × K intermediate and launches two sparse products.The device-memory saving is only 1.3× on a car mesh, versus 3.3× in host memory and on disk.
  • Learned wavelet scales: Learned scales specialize wavelet bands by PDE while tight-frame regularization maintains flat spectral coverage.For Navier–Stokes, mean s0 rises from 1.49 at ν=10^-3 to 1.68 at ν=10^-5; other benchmarks shift scales oppositely.

D.4 Zero-Shot Evaluation Details

The zero-shot evaluation trains models on a listed base grid and evaluates them on a target grid without fine-tuning. HALO performs strongly across structured benchmarks but shows measurable error growth when transferred to larger unstructured meshes.

  • Models are trained on the listed base grid and evaluated on the target grid without fine-tuning.
  • HALO has the lowest target-grid error on three of four structured benchmarks, while FNO remains ahead on Navier–Stokes.
  • On unstructured domains, evaluation uses 4.0× as many cells as training, with error increasing from 0.0113 to 0.0332 on Airfoil and 0.0664 to 0.0902 on Cylinder.

E.1 Structured-Grid Benchmarks

Held-out structured-grid diagnostics cover Allen–Cahn, Darcy, and Navier–Stokes examples. The figures compare inputs, references, HALO predictions, and absolute errors, including late-rollout behavior across viscosities.

  • Navier–Stokes: Lower-viscosity predictions make late-rollout difficulty visible as the dynamics become more chaotic.
  • Allen–Cahn and Darcy: Figure E.1 presents two unseen Allen–Cahn and Darcy inputs with corresponding ground truths, HALO predictions, and pointwise absolute errors.
  • Navier–Stokes: Figure E.2 compares Navier–Stokes forecasts at t=10 and 13 s for viscosities 10^-3, 10^-4, and 10^-5.

E.2 Unstructured-Mesh Flow Benchmarks

The Cylinder and Airfoil figures evaluate rollouts directly on native unstructured discretizations. Reference and prediction pairs track body-adjacent and downstream flow, while error maps identify difficult wake structures.

  • Figures E.3 and E.4 show Cylinder and Airfoil rollouts directly on their native unstructured discretizations.
  • Error maps locate harder wake structures behind the airfoil and cylinder, while reference and prediction pairs track body-adjacent and downstream flow.

E.3 3D Unstructured-Mesh Benchmarks

Three-dimensional held-out evaluations cover ShapeNet-Car and Aircraft on native unstructured meshes. HALO recovers major pressure and flow structures, with remaining errors concentrated near geometric features and boundaries.

  • ShapeNet-Car: ShapeNet-Car predictions on native 3D unstructured meshes recover surface-pressure regions, surrounding streamlines, and the wake momentum deficit.
  • Aircraft: Aircraft evaluations use native 3D unstructured meshes roughly ten times larger than ShapeNet-Car meshes.
  • Aircraft: Aircraft absolute-error maps localize discrepancies along sharp geometric features and surface boundaries.
Loading 2608.27883v1…