arxiv: 2605.03497 · v1 · submitted 2026-05-05 · 💻 cs.LG

Recognition: unknown

GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains

James Rowbottom , Elizabeth L. Baker , Nick Huang , Ben Adcock , Carola-Bibiane Sch\"onlieb , Alexander Denker

Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:59 UTC · model grok-4.3

classification 💻 cs.LG

keywords diffusion modelsfunction spacesgraph convolutional networksfinite element methodirregular domainsresolution invariancescore-based modelsunstructured meshes

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The pith

Representing graph convolutional kernels as finite element functions allows diffusion models to generate functions on irregular domains while preserving resolution invariance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Score-based diffusion models in function spaces aim to model data as continuous functions rather than discrete vectors, which brings resolution invariance and flexibility with irregular data. Current implementations rely on Fourier neural operators that work best on regular grids and struggle with complex shapes or unstructured meshes. The paper introduces a backbone where generalized graph convolutional kernels are expressed as finite element functions. This change lets the model process non-convex, multiply-connected, and other non-trivial geometries directly. A reader would care because many scientific and engineering datasets live on real-world irregular domains where previous methods introduce bias or fail to generalize.

Core claim

The authors establish that a novel architecture representing generalised graph convolutional kernels as finite element functions serves as an effective backbone for score-based diffusion models in function spaces. This enables the models to handle unstructured meshes and complex domain topologies naturally, as shown through unconditional and conditional sampling experiments that maintain resolution invariance and achieve high fidelity on diverse geometries including non-convex and multiply-connected domains.

What carries the argument

Generalised graph convolutional kernels represented as finite element functions, which replace grid-biased operators to process irregular discretizations in the diffusion process.

If this is right

The proposed method maintains resolution invariance across varying discretizations of the same domain.
It achieves high fidelity when capturing functional distributions on non-trivial geometries.
Both unconditional and conditional sampling become feasible on complex domains.
The approach generalizes to non-convex and multiply-connected domains where grid-based methods fail.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This could facilitate the use of diffusion models in conjunction with traditional finite element simulations for generating physically consistent functions.
Similar representations might improve other types of operator learning beyond diffusion, such as in solving PDEs on irregular domains.
Testing on even more intricate topologies or higher-dimensional function spaces would further validate the resolution invariance property.

Load-bearing premise

That casting generalized graph convolutional kernels as finite element functions removes the grid biases of existing methods and enables effective modeling on unstructured meshes and complex geometries.

What would settle it

Training the model on a set of functions defined on one unstructured mesh of a complex domain and then checking whether samples generated on a much finer or differently connected mesh of the identical domain exhibit the same statistical properties and maintain high fidelity.

Figures

Figures reproduced from arXiv: 2605.03497 by Alexander Denker, Ben Adcock, Carola-Bibiane Sch\"onlieb, Elizabeth L. Baker, James Rowbottom, Nick Huang.

**Figure 1.** Figure 1: FEM convolution across increasing mesh resolutions. The reference patch (black square, w=0.15) and the learned filter (parameterised on a 5×5 grid) remain fixed in physical space, ensuring resolution-invariant aggregation over nodes (red dots). Graph Neural Operators Graph Neural Operators (GNOs) (Anandkumar et al., 2019; Wen et al., 2025) attempt to resolve the shrinking receptive field of standard MP-GN… view at source ↗

**Figure 2.** Figure 2: Q1 basis function for the FEM convolution for a reference patch of size w = 0.15 (black square). Every node i only activates four neighbouring Q1 basis functions (red, blue, orange, green). require a full forward pass per edge. Third, as the mesh is refined, the Monte-Carlo sum in (14) converges to the true integral R ϕp a dy at a rate determined solely by the properties of the signal a, since the basis ϕp… view at source ↗

**Figure 3.** Figure 3: Resolution generalisation for Gaussian blob sampling. Models are trained on a coarse triangular mesh with 2048 elements (top) and evaluated on a finer mesh with 8192 elements (bottom). We compare ground truth, CNN, FNO, GAOT, message passing GNN, and the proposed GRIFDIR. Resolution-dependent models degrade or produce artifacts at higher resolution, while resolution-invariant operators maintain spatial str… view at source ↗

**Figure 4.** Figure 4: Unconditional samples of the multi domain model. Ground Truth 10 Sensors 50 Sensors 100 Sensors FunDPS FunDAPS 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 view at source ↗

**Figure 5.** Figure 5: Sparse sensor reconstruction for Fun-DPS and FunDAPS with 10, 50 and 100 randomly placed sensors. We show a single sample per setting. Corresponding errors are given in view at source ↗

**Figure 6.** Figure 6: We use Fun-DPS to sample conditionally on the Poisson operator for the Gaussian blob dataset. The posterior mean and standard deviation are taken across 100 samples, with three such samples plotted on the right. Fun-DPS Fun-DAPS view at source ↗

**Figure 7.** Figure 7: Sparse sensor reconstruction with 75 randomly placed sensors with Fun-DPS and Fun-DAPS for the pinball dataset. For each method, we show the ground truth, posterior mean and standard deviation (computed with 100 samples), and three posterior samples view at source ↗

**Figure 8.** Figure 8: Two rollouts of the pinball dataset for different velocities µ from the training set. length scale = 0.0500 N = 1934 length scale = 0.1000 N = 557 length scale = 0.2000 N = 226 length scale = 0.4000 N = 140 view at source ↗

**Figure 9.** Figure 9: The pinball mesh for different length scales and number of elements N. provide the plots of the RMSE and energy scores. In view at source ↗

**Figure 10.** Figure 10: Unconditional samples of the pinball model. See view at source ↗

**Figure 11.** Figure 11: We compare the conditional methods Fun-DPS and Fun-DAPS for sampling from the conditional distribution arising from different numbers of sensors. Full quantitative results are in view at source ↗

**Figure 12.** Figure 12: The mean ℓ2-error of the conditional pinball model over the rollout time t. 22 view at source ↗

read the original abstract

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular discretisations. However, practical implementations have struggled to fully realise these benefits. Existing backbones like Fourier neural operators are often biased towards regular grids and fail to generalise to complex domain topologies. We propose a novel architecture for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, enabling the model to naturally handle unstructured meshes and complex geometries. We demonstrate the efficacy of our network architecture through a series of unconditional and conditional sampling experiments across diverse geometries, including non-convex and multiply-connected domains. Our results show that the proposed method maintains resolution invariance and achieves high fidelity in capturing functional distributions on non-trivial geometries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a workable graph-plus-FEM backbone for function-space diffusion on irregular meshes and shows sampling results on non-convex domains, but the resolution-invariance claim rests on experiments that may still carry mesh-dependent bias.

read the letter

The main takeaway is a new network architecture that represents generalized graph convolutional kernels as finite-element functions so score-based diffusion can run on unstructured meshes and complex topologies without defaulting to regular-grid assumptions like Fourier neural operators. They run unconditional and conditional sampling tests across several geometries, including non-convex and multiply-connected ones, and report that the outputs stay consistent when the mesh is refined while still matching the target functional distributions at reasonable fidelity. That combination of graph kernels with FEM is the concrete step forward; it directly addresses the limitation that existing function-space models hit when the domain is not a simple box or torus. The experiments are the part that lands: they actually try the method on the kinds of domains people care about in applications rather than staying with toy grids. The writing is straightforward about the motivation and the construction, and the abstract does not overclaim the theory side. The soft spot is exactly the one the stress-test note flags. Representing the kernels in an FEM basis does not automatically guarantee that the learned operator is unchanged under mesh refinement; the graph Laplacian discretization and the projection step can still introduce resolution-specific effects. The paper asserts invariance from the sampling results, but without reported ablations that fix the mesh sequence and vary only the refinement level, or a short argument showing the discretization commutes with the diffusion dynamics, the central claim is not yet strongly supported. Minor implementation details such as how boundary conditions are handled on multiply-connected domains are also left implicit. This paper is for people who need to generate or condition on function-valued data over real-world irregular geometries, such as in computational physics or imaging. A reader already working with graph or FEM discretizations will get the most out of the architecture description and the reported sampling behavior. It is worth sending to peer review because the practical gap it targets is real and the experiments are on the right class of domains, even though the invariance evidence will need tightening in revision.

Referee Report

2 major / 2 minor

Summary. The paper introduces GRIFDIR, a score-based diffusion model architecture for function spaces over irregular domains. It represents generalized graph convolutional kernels as finite element functions to achieve resolution invariance and handle unstructured meshes and complex (non-convex, multiply-connected) geometries, where prior methods like Fourier neural operators are biased toward regular grids. The approach is evaluated via unconditional and conditional sampling experiments demonstrating high-fidelity capture of functional distributions.

Significance. If the resolution-invariance claim holds without mesh-dependent artifacts, the work would meaningfully extend function-space diffusion models to scientific applications on realistic geometries, addressing a key practical limitation of existing backbones. The FEM-graph kernel construction is a targeted idea that could generalize beyond the reported experiments.

major comments (2)

[§4] §4 (Experiments) and the associated method derivation: the headline claim of resolution invariance rests on the assumption that the FEM projection of the graph convolutional kernel commutes with mesh refinement and leaves the learned score unchanged. No explicit verification (e.g., operator-norm bounds or convergence rates under h-refinement) is provided; the reported experiments on diverse geometries show visual fidelity but do not isolate whether discretization artifacts vary with mesh density, directly engaging the stress-test concern.
[§3.1] §3.1 (Architecture): the generalized graph convolutional kernel is cast as a finite-element function, yet the paper does not specify how the graph Laplacian discretization is chosen or whether it is independent of the underlying mesh density. If the effective operator changes with refinement, the diffusion dynamics become resolution-dependent, undermining the central invariance result.

minor comments (2)

The abstract and introduction would benefit from a concise statement of the precise function space (e.g., Sobolev or L2) in which the diffusion is defined.
Figure captions should explicitly state mesh resolutions and domain topologies for each panel to allow readers to assess the invariance claim visually.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which help clarify the presentation of our resolution-invariance results. We respond to each major comment below, providing additional explanation of the method's design while indicating revisions to strengthen the manuscript.

read point-by-point responses

Referee: [§4] §4 (Experiments) and the associated method derivation: the headline claim of resolution invariance rests on the assumption that the FEM projection of the graph convolutional kernel commutes with mesh refinement and leaves the learned score unchanged. No explicit verification (e.g., operator-norm bounds or convergence rates under h-refinement) is provided; the reported experiments on diverse geometries show visual fidelity but do not isolate whether discretization artifacts vary with mesh density, directly engaging the stress-test concern.

Authors: We agree that explicit operator-norm bounds or convergence rates under h-refinement are not derived in the original submission. The architectural choice to represent generalized graph convolutional kernels as finite-element functions is intended to ensure that the kernel lives in the underlying function space; by standard FEM approximation theory, the projection error vanishes with mesh refinement, so the learned score remains consistent in the continuous limit. We have revised the manuscript to add a short discussion of this consistency property (new paragraph in Section 3) together with a reference to classical FEM error estimates. In addition, we have inserted a brief analysis in Section 4 that reports sampling consistency (via distributional metrics) across successively refined meshes on the same underlying functions, thereby isolating discretization effects more explicitly than the original visual comparisons. A full operator-norm analysis lies outside the scope of the present work but could be pursued in follow-up research. revision: partial
Referee: [§3.1] §3.1 (Architecture): the generalized graph convolutional kernel is cast as a finite-element function, yet the paper does not specify how the graph Laplacian discretization is chosen or whether it is independent of the underlying mesh density. If the effective operator changes with refinement, the diffusion dynamics become resolution-dependent, undermining the central invariance result.

Authors: The graph is constructed from mesh connectivity, yet the convolutional kernel itself is parameterized directly as a function in the finite-element space rather than as a purely discrete matrix. The Laplacian discretization follows the standard weak-form FEM formulation (consistent mass and stiffness matrices), which converges to the continuous operator under refinement. Consequently, the effective diffusion dynamics are resolution-invariant in the function-space sense once the mesh adequately resolves the domain. We have updated Section 3.1 to state this discretization choice explicitly and to note its asymptotic independence from any particular mesh density, supported by the FEM convergence properties already implicit in the kernel representation. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation or claims.

full rationale

The paper proposes a new architecture representing generalized graph convolutional kernels as finite element functions for function-space diffusion models on irregular domains. It supports the resolution-invariance and fidelity claims through unconditional and conditional sampling experiments on diverse geometries including non-convex and multiply-connected domains. No equations or steps in the provided text reduce by construction to self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations; the central results rest on empirical demonstrations rather than tautological inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on specific free parameters, axioms, or new entities introduced.

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Cited by 1 Pith paper

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Related Work Graph DiffusionIt is important to distinguish the infinite-dimensional framework from recent works that apply GNNs within standard finite-dimensional diffusion models

11 GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains A. Related Work Graph DiffusionIt is important to distinguish the infinite-dimensional framework from recent works that apply GNNs within standard finite-dimensional diffusion models. For instance, Valencia et al. (2025) proposeGraph Diffusion Modelsto le...

2025
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FFM generalises flow matching to operate in infinite-dimensional function spaces

is a related framework to infinite- dimensional diffusion. FFM generalises flow matching to operate in infinite-dimensional function spaces. Rather than learning a vector field on a finite-dimensional latent, FFM defines a path of probability measures interpolating between a Gaussian measure (often N(0, C) ) and the data distribution, and learns a functio...

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Similar to the finite-dimensional setting, the conditional distribution A0|At =a t is generally intractable. Following (Zhang et al., 2025), we use a Gaussian approximation A0|At =a t ∼ N(ˆa0(at), r2 t C),(27) where C is the prior covariance operator and ˆa0(at) denotes a Tweedie estimate computed from the score model. Assume measurements are generated ac...

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The function-space (preconditioned) Langevin dynamics then take the form ˆa(j+1) 0 ←ˆa(j) 0 −η tC 1 r2 t C −1 ˆa(j) 0 −ˆa(0) 0 +∇Φ(ˆa(j) 0 ) + p 2ηt ϵj, ϵ j ∼ N(0, C),(29) with a decaying step sizeη t. 13 GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains Algorithm 1Function-Space Decoupled Annealing Posteri...

2024
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Coarse-mesh coordinates are encoded by a small MLP and added as positional embeddings before the first block

with AdaLN-Zero conditioning, capturing long-range dependencies that local message passing cannot resolve: h←h+γ 1 ⊙MHSA AdaLN(h;γ(t)) , h←h+γ 2 ⊙FFN AdaLN(h;γ(t)) , with output gates γ1,γ 2 zero-initialised. Coarse-mesh coordinates are encoded by a small MLP and added as positional embeddings before the first block. The decoder mirrors the encoder, mappi...

2022
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is then applied in image space, after which the predictions are interpolated back onto the original mesh. Default configuration: hidden dimension128, 4down/up-sampling stages with single convolutions per block, residual connections, attention at the bottleneck, max-pool downsampling and bilinear upsampling, no dropout. D.2. Evaluation Metrics For conditio...

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In Figure 11 we also 19 GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains Figure 8.Two rollouts of the pinball dataset for different velocitiesµfrom the training set. length scale = 0.0500 N = 1934 length scale = 0.1000 N = 557 length scale = 0.2000 N = 226 length scale = 0.4000 N = 140 Figure 9.The pinball...

1934