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Joint Spatiotemporal Spectral Neural Operators for Learning PDEs on Irregular Domains

Abdolmehdi Behroozi, Chaopeng Shen

arXiv:2608.29892v1cs.LG

TL;DR

PDE solution operators remain difficult to learn on irregular, geometry-dependent domains because standard spectral methods target regular domains and other approaches may require warping or geometric embeddings. GSNO combines graph-Laplacian spatial modes with temporal Fourier modes in a unified operator, achieving strong benchmark accuracy, efficient execution, and transfer across resolutions and related geometries.

  • Problem

    Learning accurate PDE solution operators on irregular and geometry-dependent domains remains challenging because regular-grid approximations fail to capture important geometric and boundary structure.

  • Method

    GSNO combines graph Laplacian eigenvectors for spatial representation with temporal Fourier transforms in a unified joint spectral kernel.

  • Results

    GSNO achieves strong accuracy across diverse PDE benchmarks, transfers zero-shot across mesh resolutions, attains the lowest errors on geometry-varying benchmarks, and reaches up to 2–3× faster per-epoch runtimes than strong baselines.

  • Takeaways & Limitations

    Compact graph–temporal spectral representations support accurate, scalable operator learning across varying resolutions and related geometries without model reconfiguration.

  • Takeaways & Limitations

    Experiments cover related geometry families and supervised benchmark PDEs, leaving substantially different topologies, real-world multiphysics systems, and sparse observational regimes for future evaluation.

Abstract

from arXiv · show

Learning solution operators for partial differential equations (PDEs) on irregular and geometry-dependent domains remains a central challenge in scientific machine learning. While spectral methods provide strong inductive biases for modeling global interactions, they are typically limited to regular domains, and existing neural approaches often require domain warping, interpolation, or costly geometric embeddings. We introduce the \textbf{Graph Spectral Neural Operator (GSNO)}, a neural operator that combines spatial graph spectral decompositions with temporal Fourier transforms through a unified space--time spectral kernel. This formulation enables globally coherent operator learning on non-Cartesian discretizations without domain warping or autoregressive rollouts. By replacing learned geometric embeddings with a graph Laplacian spectral basis, GSNO provides geometry-aware spectral learning with low parameter complexity. Across steady and unsteady PDE benchmarks on irregular and geometry-dependent domains, GSNO achieves strong accuracy with reduced runtime and parameter counts, while demonstrating robust zero-shot generalization across mesh resolutions and geometry families.

1 Introduction

PDE solution learning is computationally demanding, especially on irregular domains where standard grid-based neural operators miss geometric and boundary structure. GSNO addresses this gap with a unified graph–temporal spectral architecture designed for efficient, accurate, and transferable operator learning.

  • Motivation: Repeated high-resolution PDE solves can be computationally prohibitive in applications such as pollutant transport and fractured porous-media flow.Fine spatial and temporal resolutions are often required to capture multiscale dynamics.
  • Motivation: Traditional FDM, FVM, and FEM methods provide high precision but scale poorly with resolution, creating a trade-off between accuracy and computational cost.
  • Existing approaches: Neural operators learn mappings between function spaces and can generalize across mesh resolutions, with Fourier Neural Operators achieving strong results through efficient spectral representations.
  • Problem setting: Irregular geometries expose limitations of regular-grid approximations, which can miss critical boundary physics and fine-scale geometric details.Examples include propagating cracks, complex airfoils, and patient-specific anatomies.
  • Related work: Existing irregular-domain methods lack complementary capabilities, including explicit temporal modeling, robustness to complex topologies, or geometric spectral priors.
  • Proposed approach: GSNO combines graph Laplacian spatial modes with temporal FFTs to provide geometry-adaptive spectral learning, lower parameterization and training costs, strong benchmark performance, and zero-shot transfer.The claimed transfer scope includes unseen meshes, resolutions, and discretizations without retraining.

2 Methodology

GSNO learns PDE solution operators by combining graph spectral representations of irregular spatial domains with Fourier representations of time. Its layers perform joint spectral mixing, reconstruct physical fields, and retain local adaptivity through residual mappings.

  • Core architecture: GSNO combines graph Fourier transforms in space with classical Fourier transforms in time to learn global interactions on nonuniform, geometry-dependent discretizations.
  • Neural operator framework: A neural operator lifts inputs, applies iterative nonlinear layers containing pointwise maps and learnable global operators, then projects latent representations to outputs.
  • Joint spectral operator: GSNO instantiates its global operator as a joint graph–temporal spectral convolution that mixes nonlocal spatial structure and temporal dynamics in one coupled domain.
  • Joint spectral operator: Time-dependent GSNO operates on spatial nodes, time steps, and latent channels, while time-independent problems use the same formulation without the temporal Fourier transform.
  • Graph construction: The spatial graph uses mesh or triangulation-based connectivity, preserving geometric adjacency and mesh-induced structure for the graph Laplacian basis.
  • Graph spectral basis: GSNO retains truncated graph Laplacian eigenvectors as a graph Fourier basis, projecting spatial features into spectral coordinates and reconstructing them on the physical mesh.
  • Layer operation: Each layer follows graph projection, temporal transformation, learnable joint spectral mixing, inverse transformation, and graph reconstruction, alongside a local residual branch.
  • Geometry handling: For fixed geometries the Laplacian basis is reused, whereas varying geometries receive sample-specific bases while sharing the learnable GSNO parameters.The eigendecomposition is treated as preprocessing rather than a trainable model component.

3 Numerical Experiments

GSNO is evaluated across steady, geometry-varying, and time-dependent PDE benchmarks on irregular meshes, with comparisons covering accuracy, efficiency, resolution transfer, inverse modeling, and component ablations. Across these settings, the reported results show strong predictive accuracy, speed, and dependence on geometry-aware spectral components.

  • Benchmark scope: GSNO is compared with seven neural-operator baselines across nine steady-state and time-dependent PDE systems using matched datasets, meshes, splits, and optimization settings.The evaluation spans fixed and varying geometries, including Darcy flow, airfoil, pipe flow, hyper-elasticity, a 3D car, Navier–Stokes, shallow water, and 2D and 3D Burgers’ equations.
  • Steady-state benchmarks: 0.0083 relative L2 error is achieved for Darcy flow, improving over GNOT (0.0118) by 29.6%.GSNO also captures elevated hydraulic head near the notch and respects no-flow behavior along irregular boundaries.
  • Steady-state benchmarks: 0.0012, 0.0012, 0.0009, and 0.0008 relative L2 errors are reported for airfoil density, pressure, ux, and uy, respectively, with reductions of 42.9%, 40.0%, 35.7%, and 55.6% versus AMG.The model also attains the lowest per-epoch training time among the evaluated airfoil models.
  • Geometry-varying benchmarks: GSNO remains accurate across geometry-varying pipe, hyper-elasticity, and Shape-Net 3D car benchmarks, with predictions following reference flow or stress structures across distinct geometries.The Shape-Net car task uses approximately 32k unstructured points per sample, while pipe and hyper-elasticity vary geometry across samples.
  • Time-dependent benchmarks: 0.0152 error is obtained for long-horizon Navier–Stokes forecasting versus 0.0534 for Transolver, a 71% reduction, while training epochs are up to 5× faster.GSNO is reported as most accurate across temporal splits and shows errors up to an order of magnitude lower on finer meshes.
  • Time-dependent benchmarks: 0.0221 versus 0.0806 for Transolver is reported for the 1 →50 2D Burgers’ split, with more than 72% error reduction; the 5 →46 error is 0.0156.For 2D Burgers’, GSNO remains below 0.022 across splits, while the study also reports up to 10× lower error and 3.5× runtime speedup.
  • Generalization and efficiency: The Laplacian basis is recomputed on finer meshes while the trained spectral kernel is reused, enabling zero-shot transfer without retraining.The reported shallow-water transfer goes from Ns = 1832 during training to Ns = 3663 at evaluation.
  • Component ablations: Removing the learnable spectral kernel increases relative L2 error from 0.0083 to 0.0138 for Darcy flow and from 0.0156 to 0.0269 for Burgers’.The ablation retains the mesh-induced Laplacian basis, showing that spectral projection alone helps but learned mode mixing improves the operator’s expressivity.

4 Discussion, Limitations, and Conclusion

GSNO combines graph-Laplacian spatial modes with temporal Fourier modes to learn compact, geometry-aware operators that generalize across resolutions and varying domains. Its efficiency comes from low-rank spectral compression, while its current scope remains limited to related geometry families and supervised benchmark PDEs.

  • Method and spectral learning: GSNO jointly captures global spatial and temporal dependencies by combining graph-Laplacian eigenvectors with a temporal Fourier transform.The learned kernel operates in a unified space–time spectral representation while preserving geometric and physical structure.
  • Generalization: GSNO supports zero-shot transfer across mesh resolutions by recomputing the graph basis while keeping the learned spectral kernel fixed.This avoids retraining when the discretization changes.
  • Generalization: Across geometry-varying datasets, GSNO constructs sample-specific graph representations and achieves the lowest errors across the reported benchmarks.The shared model parameters operate with geometry-specific Laplacians and spectral bases.
  • Efficiency: Retaining only low-frequency spatial and temporal modes reduces memory and computation while preserving dominant physical modes needed for accuracy.The one-time eigendecomposition is reused during training and inference.
  • Efficiency: 2–3× faster per-epoch runtimes are reported than strong baselines while matching or exceeding their accuracy.The eigendecomposition cost is described as a small fraction of total training time.
  • Limitations and future scope: The experiments focus on related geometry families and supervised benchmark PDE systems, leaving substantially different topologies and sparse real-world multiphysics settings for future work.Broader geometry transfer may require additional adaptation or problem-specific alignment.

B.3 Spectral truncation error bounds via graph-Sobolev regularity

The appendix formalizes GSNO’s spectral operator class, truncation behavior, and permutation properties. Truncation error decreases with retained graph modes under graph-Sobolev regularity, while exact refinement consistency requires additional spectral convergence and stability assumptions.

  • Spectral truncation error: The graph-Sobolev truncation bound formalizes an accuracy–efficiency tradeoff: retaining more Laplacian modes decreases approximation error.The decay is faster when signal energy is concentrated in low graph frequencies.
  • Expressivity: GSNO realizes the joint graph–time spectral-multiplier operator class exactly when all spatial and temporal modes are retained.The result extends to networks composed of pointwise maps, joint multipliers, nonlinearities, and input/output lifts.
  • Spectral truncation error: With truncated spatial modes, the approximation error is bounded by the operator norm times the projection error, which decays with mode count for bounded graph-Sobolev signals.The proof uses unitarity of the joint spectral transform and pointwise action of the multiplier.
  • Expressivity: GSNO is universal within the class of neural operators whose global mixing is implemented by joint graph/time spectral multipliers.The universality statement is restricted to that operator class.
  • Permutation equivariance: Each GSNO layer is permutation equivariant under node relabeling when the spectral construction is interpreted through invariant subspaces or projectors.Individual eigenvectors may change sign or rotate within degenerate eigenspaces, so projector-level reasoning is required.
  • Refinement consistency: Mesh-refinement consistency is conditional on spectral graph convergence, regularity, and stability assumptions rather than established unconditionally by the appendix.The main paper validates refinement robustness empirically through zero-shot super-resolution experiments.

C.3 2D unsteady Burgers’ Equation

The benchmark suite evaluates GSNO on irregular, unstructured domains spanning fluid, wave, and solid-mechanics problems. The 2D Burgers experiment uses a six-petal domain with a central hole, while related cases include airfoil, Navier–Stokes, shallow-water, and hyper-elastic geometries.

  • 2D unsteady Burgers’ equation: The 2D Burgers benchmark uses a six-petal flower domain with a circular central hole and no-slip boundary conditions.The velocity field satisfies the two-dimensional viscous Burgers equation on an unstructured triangular mesh.
  • 2D unsteady Burgers’ equation: Burgers trajectories are generated from Gaussian-random-field initial conditions and integrated with fourth-order adaptive Runge–Kutta time stepping.The viscosity is set to ν = 0.2.
  • 2D unsteady Burgers’ equation: The Burgers dataset contains 1,000 samples split into 600 training, 200 validation, and 200 testing examples, with 51 temporal snapshots per trajectory.All models are trained for 1,000 epochs.
  • Benchmark coverage: The benchmark suite includes irregular domains such as an elliptical domain with a triangular hole, Lake Union, a tetrahedral square pyramid, and geometry-dependent hyper-elastic meshes.These cases cover unsteady fluid dynamics, shallow-water waves, three-dimensional transport, and solid mechanics.
  • Shallow-water benchmark: The shallow-water system uses conservative variables [h, hu, hv]⊤, finite-volume discretization, Rusanov flux, and reflective wall boundaries.The setup models wave propagation from a localized height perturbation.

D Amortized Cost of Graph Spectral Preprocessing

GSNO’s graph spectral preprocessing uses a sparse, truncated Laplacian basis and is incurred once before training. After amortization, this eigendecomposition contributes only a small fraction of per-epoch training cost.

  • Preprocessing design: The normalized graph Laplacian basis uses LOBPCG to compute the k_s smallest-eigenvalue eigenpairs, retaining k_s = 8 in the experiments.Sparse graph structure makes matrix–vector products scale with O(|E|) rather than dense-matrix cost.
  • Amortized cost: Across fixed-geometry benchmarks, one-time preprocessing takes 1.08–5.74 seconds and remains below 0.07% amortized overhead per epoch.Only one basis is required for the entire dataset in these benchmarks.
  • Overall implication: Across all benchmarks, preprocessing remains below 1.7% of per-epoch training cost, including datasets with hundreds or thousands of geometries.Measured preprocessing time depends on sparsity, connectivity, topology, and spectral separation, not only node count.
  • Benchmark context: The evaluation includes steady Darcy flow, airfoil flow, and Shape-Net 3D-car cases, with accuracy and efficiency reported across irregular meshes.The accompanying figures and tables compare predictions, resolution generalization, and training cost.

E.4 Additional results for 2D Burgers’ Equation

Additional 2D Burgers’ results show GSNO predicting both velocity components on an irregular flower-shaped domain while maintaining accuracy across resolutions. The accompanying analyses also examine temporal forecasting and efficiency.

  • Qualitative results: GSNO predictions cover horizontal and vertical velocity components over time on the irregular Burgers’ benchmark.The flower-shaped domain contains a circular hole, with 1,168 mesh nodes and 51 temporal snapshots.
  • Evaluation scope: The additional figures compare resolution-wise accuracy and training efficiency under varying temporal input-output settings.These analyses complement the qualitative predictions with generalization and runtime comparisons.
  • Forecast setup: The benchmark uses T_in = 5 input snapshots to forecast T_out = 46 future steps on the same-resolution mesh.The model is trained and evaluated with N_s = 1168 nodes.
  • Resolution generalization: GSNO consistently outperforms baselines across increasing spatial resolutions, achieving up to 10× lower error.The comparison reports relative L2 error and error ratios relative to GSNO.

E.5 Additional results for 3D unsteady Burgers’ equation

For 3D unsteady Burgers’ equation, GSNO combines strong temporal forecasting accuracy with the lowest reported training runtime across temporal configurations. The results emphasize efficiency for a high-dimensional time-dependent benchmark.

  • Training efficiency: GSNO requires the lowest runtime across all tested temporal configurations for 3D unsteady Burgers’ equation.The benchmark uses N_s = 1867, and runtime is measured as average seconds per epoch.
  • Forecast accuracy: GSNO achieves the best accuracy in every temporal configuration, with especially strong gains over competing methods at longer prediction horizons.The comparison uses relative L2, RMSE, and MAE.
  • Evaluation scope: The evaluation compares per-epoch runtime and accuracy across different input-output temporal settings for the 3D Burgers’ benchmark.Table E.5 provides the corresponding model comparison at N_s = 1867.

E.6 Additional results for 2D Navier–Stokes Equation

Additional 2D Navier–Stokes results examine zero-shot super-resolution, resolution generalization, and training efficiency. GSNO is evaluated from a coarse mesh to a finer mesh and compared with baselines across resolutions and temporal settings.

  • Evaluation scope: The additional results jointly assess qualitative super-resolution, resolution-wise accuracy, and temporal training efficiency for 2D Navier–Stokes.Table E.6 provides the corresponding model comparison at N_s = 1244.
  • Zero-shot super-resolution: GSNO performs zero-shot super-resolution by training on 972 nodes and evaluating on 1,903 nodes without retraining.The Navier–Stokes example uses viscosity ν = 10^-3 and compares ground truth with GSNO predictions.
  • Resolution generalization: GSNO achieves the lowest error across all tested Navier–Stokes mesh resolutions.The comparison reports relative L2 error and accuracy gaps relative to GSNO for T_in = 5 and T_out = 46.
  • Training efficiency: GSNO remains the most computationally efficient model across temporal input-output configurations.Runtime is measured per epoch at resolution N_s = 1244, with overhead reported relative to GSNO.

E.7 Additional results for Hyper-Elasticity Equation

GSNO recovers stress responses across hyper-elasticity geometries with varying internal voids. Quantitative comparisons report both the lowest errors and the fastest training among evaluated methods.

  • GSNO captures stress fields across solid configurations with different internal void patterns.
  • GSNO obtains the lowest relative L2, RMSE, and MAE values while maintaining the fastest per-epoch training time.

E.8 Additional results for Pipe Flow

The additional results cover GSNO predictions across curved pipe geometries and shallow-water meshes. GSNO combines accurate geometry and resolution generalization with efficient training across temporal settings.

  • Pipe Flow: GSNO captures velocity responses across curved pipe geometries with varying shapes.
  • Pipe Flow: GSNO achieves the lowest pipe-flow prediction errors and smallest per-epoch runtime among compared models.
  • Shallow Water Equations: GSNO transfers shallow-water predictions from a coarse mesh with 1832 nodes to a finer 3663-node mesh without retraining.
  • Shallow Water Equations: Across mesh resolutions, GSNO delivers the highest predictive accuracy, with improvements of up to 7× over other models.
  • Shallow Water Equations: GSNO achieves the best training efficiency across temporal input-output settings in the shallow-water benchmark.

F.1 GSNO Architecture, Training Setup, and Sensitivity to the Number of Spatial and Temporal Modes

GSNO’s spatial and temporal spectral modes control the balance among accuracy, computational cost, and generalization. Experiments identify mid-range settings, especially ks = 8 and kt = 8, as effective defaults.

  • Mode Sensitivity: Spatial modes ks select graph Laplacian eigenvectors, while temporal modes kt select Fourier modes for the spectral representation.
  • Spatial Modes: The spatial-mode sensitivity follows a U-shaped performance curve: very small ks underfit, whereas very large ks increase cost and slightly hurt generalization.
  • Spatial Modes: ks = 8 minimizes error with moderate runtime, while values between 6 and 10 generally work well.
  • Temporal Modes: The temporal-mode sensitivity shows that too few modes limit accuracy, whereas too many slow training and slightly degrade generalization.
  • Temporal Modes: kt = 8 provides the best balance between temporal accuracy, training cost, and generalization.
  • Default Configuration: The authors use ks = 8 and kt = 8 as a strong default configuration for 2D Navier–Stokes and related PDE benchmarks.

G Baseline Models Overview, Key Differences, and Hyperparameters

GSNO is compared with neural operators using dual networks, graph kernels, coordinate latents, domain warping, attention, and multi-graph constructions. Its distinguishing design combines fixed graph-Laplacian spatial spectra with explicit temporal Fourier spectra, alongside consistent retraining and computational evaluation.

  • Baseline Models: The comparison includes DeepONet, MGKN, CORAL, Geo-FNO, GNOT, Transolver, AMG, and SP2GNO across irregular-domain operator-learning settings.
  • Baseline Models: DeepONet uses branch and trunk networks for pointwise operator evaluation, whereas MGKN models graph-based spatial interactions without spectral temporal structure.
  • Baseline Models: CORAL uses coordinate-based latent representations, while Geo-FNO applies Fourier convolutions after learning a mapping to a latent uniform grid.
  • Baseline Models: GNOT and Transolver use attention-based architectures for irregular meshes, multiple inputs, diverse geometries, or boundary conditions.
  • Baseline Models: AMG constructs local, global, and physics graphs processed with dynamic graph attention to represent complementary spatial dependencies.
  • Key Differences: GSNO operates directly on physical meshes using Delaunay-based graphs and fixed Laplacian eigenvectors, avoiding learned domain warping.
  • Key Differences: GSNO combines graph-based spatial spectra and Fourier temporal spectra in a unified space–time kernel, while SP2GNO models time implicitly through stacked graph updates.
  • Evaluation Setup: All baseline models are retrained under identical data splits and training settings, with additional inference-time and memory-footprint comparisons reported for selected cases.
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