Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators
Authors: Albert Matveev
Sanmitra Ghosh
Aamal Hussain
James-Michael Leahy
Michalis Michaelides
Venue: NeurIPS 2025 spotlight ()
Keywords: neural operator
Fourier neural operator
diffusion multipler
bayesian inference
variational inference
uncertainty quantification
Primary area: machine_learning_for_sciences

Abstract: Operator learning is a powerful paradigm for solving partial differential equations, with Fourier Neural Operators serving as a widely adopted foundation. However, FNOs face significant scalability challenges due to overparameterization and offer no native uncertainty quantification -- a key requirement for reliable scientific and engineering applications. Instead, neural operators rely on post hoc UQ methods that ignore geometric inductive biases. In this work, we introduce DINOZAUR: a diffusion-based neural operator parametrization with uncertainty quantification. Inspired by the structure of the heat kernel, DINOZAUR replaces the dense tensor multiplier in FNOs with a dimensionality-independent diffusion multiplier that has a single learnable time parameter per channel, drastically reducing parameter count and memory footprint without compromising predictive performance. By defining priors over those time parameters, we cast DINOZAUR as a Bayesian neural operator to yield spatially correlated outputs and calibrated uncertainty estimates. Our method achieves competitive or superior performance across several PDE benchmarks while providing efficient uncertainty quantification.

Light-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators
Albert Matveev
Sanmitra Ghosh
Aamal Hussain
James-Michael Leahy PhysicsX London, UK
Michalis Michaelides
albert www.teeeee@yhsicc...eee ai
Abstract
Operator learning a powerful paradigm for solving partial differential equations, with Fourier Neural Operators serving as widely adopted foundation. However, FNOs face significant scalability challenges due overparameterization and offer no native uncertainty quantification key requirement for reliable scientific and engineering applications. Instead, neural operators rely on post hoc UQ methods that ignore geometric inductive biases. In this work, we introduce DINOZAUR: a diffusion-based neural operator parametrization with uncertainty quantification. Inspired by the structure of the heat kernel, DINOZAUR replaces the dense tensor diffusion multiplier that
multiplier in FNOs with a weeeeniionniitinndeeeeeee has a single learnable time parameter per channel, drastically reducing parameter count and memory footprint without compromising predictive performance. By defining priors over those time parameters, we cast DINOZAUR as a Bayesian neural operator to yield spatially correlated outputs and calibrated uncertainty estimates. Our method achieves competitive superior performance across several PDE benchmarks while providing efficient uncertainty quantification. The code is available htttt::///// :/g::::///iiit thub. com cowm/Plyssiicaaaa///NAAAAA
Introduction
Partial differential equations (PDEs) fundamental mathematical tool that describe wide range of physical phenomena observed in the real world. Numerous problems in science and engineering require formulating and solving PDEs, often presenting significant computational and methodological challenges to researchers and practitioners. A common strategy for solving PDEs via conventional numerical methods, such finite difference, finite element, finite volume method []]. While accurate, they become prohibitively expensive in complex engineering tasks, especially where rapid iteration or real-time response is critical, such design optimization or control (2).
In recent years, there has been a growing trend toward replacing computationally intensive numerical solvers with neural sedwork-based surrogate models trained to approximate PDE solutions. Since discretized PDE domains are commonly represented using point clouds, meshes, or graphs, geometric deep learning (3] has emerged promising paradigm for this class of problems. In particular, message passing layers, initially developed for geometric and relational data, have proven effective due their structural resemblance physical convolution operators. These layers especially suitable for capturing the spatial dependencies inherent in PDE systems, making them well-suited for learning dynamics over irregular non-Huclideaa domains 4 (5)
However, geometric deep learning methods often lack key properties required for solving PDEs, motivating alternative approaches. A PDE its parameters, equation structure, and boundary conditions defines an operator that maps input functions to solution functions. Neural operators
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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I
a
P
1/-
F
ARTI
FFN
1(41
2 M+1
Q -
MM 100000
g o I
i2
Nop
Figure I: Overview of DINOZAUR. NO block No..r. is revised by updating the integral transform I, with diffusion multiplier exp(-Xx7, and including gradient features Go, add anisotropy. Feed-forward network (FFN) is applied to mix two sets of features.
(NOs) 6 have recently emerged as a leading framework for approximating such solution operators and, more generally, for tackling the operalor leurning problem implicitly present in PDEs. These models incorporate inductive biases that improve performance and offer discretization invariance and universal approximation 77,, making them strong alternative to geometric deep learning. A prominent example is the Fourier Neural Operator (FNO) [9]. which learns transformations in the spectral domain via the Fast Fourier Transform. While FNOs restrictive in requiring input data on a regular grid, they remain a widely used backbone neural operator architectures [10]. Despite their success problems such Darey flow and Burgers' equation, FNOs are known be overparameterized [1] [2]. Their design leads to exponential growth with domain dimensionality, causing overfitting and significant memory overhead.
Moreover, PDE solvers often inform downstream decision-making, so it is imperative for any substitute method to quantify the uncertainty its output. Uncertainty quantification (UQ) is common strategy to improve model robustness by estimating confidence in its predictions. It enhances model outputs by accounting for different sources of error, helping guide more informed and reliable downstream use. Uncertainty modeling has long relied on Bayesian inference [3], ensembles [14,, and conformal prediction [15] to produce confidence estimates. While some of these methods have been adapted to deep learning with varying success [8) UQ for GNNs and neural operators [9] remains underdeveloped, often relying post hoc methods ill-suited to their structure.
To address the scalability issues and limited UQ methods in FNOs, we present DINOZAUR - diffusion-based neural operator parametrization with uncertainty quantification. Our diffusion multi- plier inspired by the heat kernel in the Fourier basis and requires only one learnable parameter per channel, resulting in superior scalability without sacrificing predictive performance. This physically motivated choice of multiplier allows us to place meaningful probabilistic priors on the learnable time parameters, recover the posterior via inexpensive variational inference (VI) [20] of these parameters, and efficiently sample spatially correlated functions. We summarize our contributions as follows:
We introduce dimensionalitt-inneeeedd spectral multiplier parametrization, reducing the parameter count by several orders of magnitude compared to equivalent FNO-based architectures.
We define a Bayesian formulation of the heat kernel and build an interpretable uncertainty quantification method sample spatially correlated outputs.
We demonstrate that our method is scalable, maintains strong predictive performance across multiple PDE benchmarks, and achieves superior UQ estimates compared to conventional probabilistic methods.
2
Related work
The area designing learnable approximation for PDE solutions vast and rapidly growing 21
Given the increasing availability of data, the need to incorporate inductive biases, and the inherently geometric structure of many physical problems, two classes deep learning models have come to dominate the field: geometric deep learning models and neural operators.

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offer flexible framework for learning on non- non-Euclidean domains. Many GNNs implement spatial convolution by performing
Geometric deep learning Graph neural networks (GNNs) 123
learnable aggregation of local neighhorhoods 24 25
Neural PDE solvers increasingly rely on spatial
GNN layers to model and predict physical phenomena across various domains, including control (26].
deformable material simulation (27). and spatiotemporal mesh evolution 4. Specific architectures
target building robust autoregressive solvers [5] and encoding the geometric boundary conditions [28).
However, their performance often suffers from sensitivity discretization, limiting generalization across mesh resolutions. Spectral GNNs define convolutions using the graph Laplacian's ' eigen-
decomposition [29] [30,, often with polynomial approximations of spectral filters [3 32 These have seen success in 3D shape analysis, particularly with functional maps [33]. DiffusionNet [34] exemplifies this approach by using the spectral form of the heat kernel for a simple and interpretable kernel parametrization. It exhibits strong performance and generalization; we note that such a kernel is applicable in arbitrary domains.
Neural operators Neural operators are powerful class of models for learning mappings between infinite-dimensional spaces in a resolution-invariant manner [7]. offering an alternative to GNNs. Lu al. 6 introduced DeepONet, inspired by operator learning and universal approximation theory, using a branch-trunk architecture to encode inputs and evaluate the solution at arbitrary locations. A parallel line of work introduced the Graph Neural Operator (GNO) architecture [35 36], transferring the message passing intuition from GNNs to NOs. A family of neural operators decomposes the input function into a basis defined by one of the popular transforms: Fourier (9. Laplace (37] Wavelet [38]
In particular, FNOs are popular for their performance and interpretability despite needing a regular grid sampling, and are commonly used other networks, such as Geo-FNO (39, U-FNO
40 and
Geometry-Informe Neural Operator (GINO) [10], to name few. However, FNOs face scalability and generalization issues due to exponential parameter growth with domain dimensionality [T. Efforts to improve this, such as FFT factorization [MЛ, low-rank tensor decomposition [12] and MLPs in the spectral domain [4I] have reduced growth but still rely on spatial dimensions.
Functional uncertainty quantification UQ in deep learning has been widely studied, with pop- ular methods including MC Dropout [16). Laplace approximation [42] ensembles [43]. and non- parametric conformal prediction [7), which typically require extra computations or careful cali- bration. In the context of functional UQ crucial for neural operators the landscape is further limited. Recent approaches include applying a post hoc Laplace approximation to the final layer of the NO [44), using conformal methods after training [45) learning an uncertainty measure as an auxiliary output [46]. Another family of approaches leverages diffusion-based generative models for uncertainty quantification in physics surrogates 47-50) While these methods use diffusion models to capture statistics of high high-dimensional dynamical systems and generate realistic samples, they require solving a differential equation for reverse diffusion numerically through multiple denoising steps. which is computationally intensive compared to deterministic forward passes in neural operators. Despite these various approaches, none of them model uncertainty within the NO itself; instead, they adopt generic methods for neural networks and neglect the inductive biases of neural operators.
Diffusion multiplier parametrization for neural operators
3.1 Problem setting
We consider the task of learning a parametric surrogate map the initial and boundary conditions of PDE to its solution. Let S2 C RRd be a bounded open set representing a PDE domain in d-dimensional Fuclidean space. Let also A = A(Q; (Rds) be a Banach space of real-valued functions defined on Ω and U = U(2; Rdd.) ) be space of solution functions. We associate a e A with a function that parametrizes the partial differential operator L. and trace operator B,. Then, we form the PDE:
(a(u)(x) = f(x) Vxr E Ω [a[u[(x) = g(z) WI E an,
where [xx is fixed forcing function and (zx is boundary condition. This formulation is general enough to include varying geometric boundaries by including the signed distance function of a geometry in functions from A and defining solution extension to the whole domain [51 10]
3

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We assume the PDE induces continuous solution map N: A + L4 that maps inputs to solutions u, and we are seeking find its parametrized upproximation No with p parameters. Given the discretized observations, (An, Mn } n==1 we pose problem of the empirical risk minimization:
mim PER
1u -on No
(2)
N
We construct the neural operator in the usual form sequence of lifting layer P, M NO blocks No. and projection layer 2, where the NO block No: given by:
Ui 1 (1) No,luu)(<)) = W us (x)) + by + Kim (rr(x)))) i M,
i=
where Ky is a kernel, matrix Waskip E Rdd xd. b, E iR is a bias and de is the number of channels. FNO restricts domain to be periodic torus Ω = T kernels to be stationary and represents the integral transform I in the following way:
[x) = J, my (2 y) = (RA
(4)
where F is Fourier transform, and (k) acts as unique complex-valuct matrix if k < kmax zd and as zero otherwise.
Given the block's width de and maximal truncation modes kmax (hinaxx-.... dd max E 2 tensor R2 - [R_,, [A)]sss contains O 11 J=1 kg k learnable parameters, yielding quadratic depen- dency on the number of channels and exponential growth of parameters the dimensionality d of the
domain Q. In this paper, we target the form of kernel Key and propose an alternative parametrization.
3.2 DINOZAUR architecture
Diffusion multiplier Motivated by the success of DiffusionNet 34 and its resemblance to FNO, we choose to restrict the form of Kiy to that the heat kernel for our architecture. Consider the heat equation on the torus = Td;
d,u(T, = Au(T,x)) v(0,z) = vx(x),
(5)
with a solution [52 given by:
.) 11 (xxp)
@ (1)])
(6)
where @ is the Hadamard product and A = - 11:12 We refer to R2 = lexp(- - du ))))kmmmm as
diffusion multiplier. Figure illlstrates the updated block architecture.
When integrated into the FNO block, R2 R- x x is tensor with the only learned Rd. Each is an independent scalar defining diffusion time
parameters being T = (T' (....... pd)) within channel с of the NO block. Times 7 control the adjustable receptive field of the convolution kernel and are not related to the time variable that may be present in the original PDE ( We note that our multiplier parametrization does not depend on dimensionality d of the domain 52 or on the truncation modes kmax since DS is precomputed and fixed.
Gradient features The diffusion multiplier propagates information across the domain, enabling efficient function convolution. However, the heat kernel is isotropic by construction, which may limit its applicability. We found that including gradient features mitigates that issue and yields better experimental results.
Since alrcady map Fourier basis, it's natural to calculate gradients in the spectral domain. Raw gradients can be ill-defined due to non-perine non-periodic boundaries or uneven sampling. Following Sharp apply linear transformation Wgrad
al. 34 improve robustness and add some flexibility, Idd. x with a hyperbolic tangent non- con-eineyr to normalize the output in the physical domain. We denote the gradient features operator Go and define its output at channel follows:
Vo(x)= F-1 [ [2
(x(v)((x) = tanh Σ
(x), Waradd Wordd Vo)( c= I...,,do. de.

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p(r(D) No.
plu.(()aa.;))
{r]
ART
(a)
(b)
Figure 2: (a) Probabilistic graph for DINOZAUR, showing how the latent diffusion-times, т, relate to the observations and non-mechanistic model parameters, 0.0. (b) We define priors for and sample multiple time parameters from posterior in each block at inference time, which translates into spatially correlated uncertainty at the output.
Since v E Re, its Jacobi matrix is Ve E Rd By taking the inner product, we reduce the spatial dimensionality d and end up with compatible shape. We report the results with and without the gradient features in Section 4.1
Gradient features are applied to the output of the integral transform I, and stacked with it inside the DINOZAUR block, doubling the number of channels. We add matrix Wmix E Rdxx2de to map all features back to the width of the block. The final block architecture is:
[ I,, (x:](x) (go. o IT)=(((())
0i+1 (I)) = Tiship vz(z) + b + Wiilx
(8)
Substituting the original blocks in (3) the neural operator No with modified blocks from results in DINOZAUR architecture that we denote as No.7, with all non-diffusion parameters lumped together in 0 and separated from T.
(8)
Universal approximation Despite the seeming simplicity of the diffusion multiplier, we claim that our formulation of neural operator exhibits universal approximation properties: see Proposition D Proposition 1. Let Q C Rd be bounded domain with Lipschitz boundary and such that the closure Ω (0,27) 2x)² For given integers s, > O, let N C (5); Rdh) - C (D) be continuous operator, and fix compact set A C. C (?); Rd Then there exists continuous, linear operator Σ: (9, - C (Td. TR such that Ela]la for all C (S); Rda). Furthermore, for any E > 0, there exists DINOZAUR such that
sup [|W[[]] (No,r ([a))lallo> Se. a
where No.- Q No. P. (TT,R Cd (T..R..).
We provide the proof in the Appendix
3.3 Bayesian inference for diffuuson multiplier
Now that we established the overall architecture, will describe the Bayesian inference scheme to obtain UQ An advantage of the proposed diffusion multiplier as message passing scheme the ability impose meaningful priors on the diffusion-tima parameters Doing recasts DINOZAUR as a Bayesian neural operator, with the source uncertainty coming from the integral transform. In method, other model parameters (8, o) are treated deterministically. The conciseness of our multiplier's 's parametrization allows for tractable inference and alleviates the need to grapple with extremely high-dimensional distributions. We define our conditional generative model for any output signal un given an input signal a, as
In ~ (ppriory prioc):: = 1,..... M; c= =1=1.... 1,...,de Uh (-)lan()) N(No,r (-),0)))))))
(9)
where f(-) denotes functions evaluated any finite number of points in the domain Ω N denotes normal distribution, and Mprioo; o ppior hyperparameters of the prior distribution. We collect all
5

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time variables from each channel and block i for notational convenience, 7 (17)) ) M
See Figure 2 aa for the corresponding graphical model. We also collect training data in a set of discrete observations, D { aank } K., k=1 {ung 1رm1 n=1
with possibly different sets of function evaluation
points the input and output.
We seek to recover the approximate posterior for the diffusion time, (r)) according to (9p which yields posterior for the diffusion multipliers, Rr, and the entire neural operator No,-. This can be used to produce a posterior predictive distribution over the output for a new input function a, (see Figure b))):
Po.- (u. D) f D
(10)
where we marginalize
to reflect epistemic uncertainty over the possible operators that fit the data, as well as aleatoric uncertainty from the assumed normal noise with fitted a. Note that while the noise is point-wise independent for a fixed т, the dependence spatially correlates predictions.
( (T))) is intractable, so we resort to VI to recover an approximate posterior. We assume that it can be adequately approximated by distribution q(T: Φ) parametrized by o. We choose q(T; o) such that it factorizes over the i = ,,...,, M blocks of the NO but keeps correlations between diffusion times of channels in a block:
Variational inference As mentioned, the posterior pa.a
M II oz)=
M II N(In T3 Ps. E.).
((;;6)=
(11)
where 14, E Rdd, and Σg E Rdexde We collect the parameters of the approximating factors, c [ = {(u,, 1. for notational convenience. Due to their parsimonious and mechanistic role in controlling message propagation, diffusion time parameters avoid the redundancy and symmetries that challenge VI for typical neural network weights, allowing for better identifiability and justifying full-rank approximation within each block.
We then seek to find o that minimizes the Kullback-Leibler divergence (KL) from q(T: ø) to the true posterior. As this objective is still intractable [53), we instead maximize lower bound (ELBO):
CELBO (0, o; の) (xg(() [In pe.o (D]r)] - DKL (q(;; ()]p((})),
(12)
which allows to jointly learn the network parameters (0, or) and variational parameters ø. To predict for new instance, we replace the true posterior Po.a with its approximation g(T: () in (10)
4 Experiments
We evaluate the performance of the DINOZAUR architecture on several standard benchmarks. Se
results for the deterministic variant of DINOZAUR. In Section 4.2 we enable the Bayesian version and assess its UQ capabilities, comparing to established UQ methods for neural operators. Ablation studies and computational performance comparisons are detailed in Section 4.3 All models are implemented in PyTorc [54] and trained on single NVIDIA A 100 GPU for 150 epochs with AdamW (55) and One-Cycle scheduler (56) AdamW Schedule-Free (57] with learning rate (in most cases) set to 10-3 See the Appendix B for details. All experiments were repeated three times with random initialization of trainable parameters. Reported results include the mean and standard deviation over all runs.
Benchmarks We evaluate our method on two regularly sampled datasets from 9 and two datasets with irregular domains used in [10]. For the 2D Darey flow, we learn mapping from a scalar permeability field of resolution 2412 to scalar pressure field, applying standard scaling to both. The
2D Ntvier-Stokes (NS) dataset contains 642 spatial points and 20 time steps of simulations with viscosity v = 10-5 on a uniform grid. We learn to map the first time steps of vorticity to the last 10, without applying data scaling. For further dataset and PDE details, we refer to [58] The irregular-domain benchmarks include two datasets for predicting surface pressure on 3D meshes:
ShapeNet car (59) and Ahmed bodies [60]. Following the protocol of Li et al. [10), evaluate the

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signed distance function (SDF) of each mesh on a 64 uniform grid and apply standard scaling to the outputs during training. We learn a map from SDF values to scalar pressure on mesh vertices, with inlet velocity added to input in the Ahmed dataset. A summary of the benchmark datasets is provided the Appendix
Baselines In all baselines, used "double skip" implementation of the FNO block introduced
in 12]. On uniform benchmarks, we compare DINOZAUR to the FNO architecture from Li et al.
9 with 4 blocks. We set the width de = 32 and modes kmax [12, 12] for Darcy and d. 64,
kmax [16, 16, 4] for NS. For non-uniform domains, we follow the GINO-decoder setup from [10 with 4 FNO blocks, I GNO decoder block, and parameters de 64, kmax [24, 24, 24]. The original GINO applied low-rank Tucker factorization to the multipliers R, with a rank r 0.4.
Given that FNO/GINO, even with moderate factorization, produce significantly more parameters than DINOZAUR, we also include Tensorized FNO/GINO (TFNO/TGINO) (12) in our baselines. We choose a factorization rank r 10 for TFNO/TGINO as it yields a comparable parameter count to that of DINOZAUR in all cases. Additionally, for all models, we assess the influence of spatial gradient features by including them in the (FNO/GINO); and (TENO/TGINO); implementations, as well as removing them from our model (DINOZAUR As the main metric, we report the Relative L2:
Nuch
(x) - {z(x))
RL2
We emphasize that the only structural change our model makes the multiplier in FNO blocks.
4.1 Results: deterministic performance
Table reeporss evaluation results. DINOZAUR significantly outperforms all baselines on Darcy and remains competitive on other benchmarks. The parameter counts in the table highlight key trend: as problem dimensionality increases, FNO's parameter count grows by two orders of magnitude, whereas DINOZAUR scales only with network width. Although TFNO shares DINOZAUR favorable scal- ing, it suffers notable performance drops on uniform datasets. Interestingly, reducing the parameter count in the non-uniform setting improves GINO's performance, further underscoring the overparam- eterization issue. contrast, DINOZAUR maintains strong results across domains, demonstrating the effectiveness of diffusion-based message passing parameter-efficient alternative.
Table Comparison of DINOZAUR against the baselines in deterministic setting. reporting the RL2 values on test split. Mean and standard deviation values are computed from three trained models. The best result bold, second best is underlined
Model
# params
RL2 + (x10-2)
params (x (x10))
RL2 + (x10-2)
(x 10a)
Darcy flow
Navier-Stokes
FNO
361.4
0.90 + 0.03
12,650.8
15.90 0.20
FNO,
371.9
0.91 + 0.07
12,692.2
17.08 + 0.12
TENÓ
17.7
8.04 + 0.86
69.1
21.48 0.27
TENO,
28.2
4.16 + 0.61
110.6
19.47 + 0.58
DINOZAUR (Ours)
17.5
1.08 + 0.06
68.2
20.10 + 0.31
DINOZAUR (Ours)
28.0
0.75 - 0.10
109.7
15.68 - 0.21
ShapeNet car
Ahmed bodies
GINO
49,608.6
8.03 + 0.39
49,612.9
8.57 + 0.13
GINO, g
49,650.1
7.76 0.64
49,645.3
8.25 + 0.16
TGINŌ
149,8
7.27 + 0.16
154.0
8.19 - 0.49
TGINO,
191.3
8.30 0.38
195.5
8.23 = 0.64
DINOZAUR no-g (Ours)
138.9
8.34 0.80
143.2
9.59 = 0.62
DINOZAUR (Ours)
180.4
7.04+0.00
184.6
7.99 + 0.07

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Point 50
200
100 samples
Pressure
Ground truth
Point 40
100
Point 30
Point 20
-100
10
20
30
40
50
60
70
Point 10
Point index
Point O
Figure 3: Uncertainty predictions. Left: mean of pressure samples test mesh with geodesic path traced along the top. Right, top: ground truth and 100 sampled predictions along the same path. Right, bottom: profile with qualitative visualization of the standard deviation of pressure samples along the outline. High uncertainty is near sharp field changes, e.g., between Points 0 and 10.
We find that gradient features do not consistently improve the performance of baseline models. While general tensors in FNO are capable of implicitly learning gradient-like behavior, our experiments suggest that the addition of explicit gradient features may introduce redundancy interfere with learned representations models that already have sufficient capacity. In contrast, without gradients, DINOZAUR no-g performs poorly each benchmark. As noted earlier, this is due the radial symmetry the diffusion operation, which limits the NO's training capabilities. These observations consistent with the findings of Sharp et al. [34]. By incorporating gradient features into the diffusion block, DINOZAUR matches the performance dense FNO/GINO models.
4.2 Results: uncertainty quantification
In this section, we assess the UQ performance of the Bayesian version of our model, DINOZAUR introduced in Section 3.3 on the more challenging non-uniform data. As baselines, we consider
MC Dropout [16] (NOp) and Laplace approximation 4 (NOr). MC Dropout is widely used, UQ method, which we apply by enabling 10% dropout in the Fourier blocks' MLPs
emulate stochasticity the message passing. The Laplace approximation follows (44, modeling
uncertainty via fitted distribution over the parameters of the final MLP layer Q. The discussion
of hyperparameter choice, including Mppriors prior required for setting distributions is in the Appendix B
UQ metrics in Table are calculated by drawing 100 posterior predictive distribution samples per test data instance. The reported mean and standard deviation are obtained from three independently
Table 2: Evaluation of the uncertainty output quality of DINOZAUR and baseline models. Mean and standard deviation values are computed from three trained models. The best result bold, second best is underlined
Model
RLg & (x10-2)
NLL
MA
IS
ShapeNet car
GINOD
7.94 + 0.54
3.090 0.044
0.288 + 0.010
14.731 + 0.625
GINO,
0.29
3.106 0.023
0.283 0.007
14.709
+ 0.343
TGINO
7.18
2.996 0.010
0.278 + 0.005
13.704
TGINO,
7.73
3.071 = 0.059
0.266 0.018
14.240
0.811
DINOZAURD (Ours)
6.96
2.987 0.003
0.272 = 0.002
13.479
0.044
DINOZAUR, (Ours)
7.11
3.041 = 0.008
0.266 - 0.001
13.749
0.008 0.003
DINOZAURg (Ours)
7.49
2.767 + 0.011
0.168 0.001
11.667 0.136

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350
0.11
medepphh
memodee
1.024M
remodes
depet
-5
BINO
TENO, -00
300
0.1
swwidth
sewwithh
TPNO, 10"
BENNGZAUUR nog
Time 6883
0.00
256k
128K
20
200
150
64L
0.08
33k
Ink
50
2
16 32 64
129256
4
8 16 64 128255
Tran
Profkn
Predact
(a)
(b)
(d)
Figure 4: Scalability and efficiency of DINOZAUR, (a) Test RL2 error on ShapeNet car as width, depth, and modes are varied. (b) Model size under the same settings, (c) Average peak GPU memory usage across five train-predict cycles on synthetic data. (d) Average time under the same settings.
trained models with random weight initialization. Metrics are computed using the Uncertainty Toolbox [61]; full metric definitions are provided in the Appendix D Aleatoric noise was added at prediction time to all model outputs, using a fixed o² = e for dropout and Laplace models, and learned variance term for Bayesian DINOZAUR that converged to a² e DINOZAUR matches or surpasses all baselines, achieving lower negative log-lik ligeliidlhoodd (NLL) than dropout models or Laplace approximation. It also outperforms all baselines in terms of miscalibration area (MA) and interval score (IS). An important consideration is that the Laplace approximation requires a secondary computationally expensive step to obtain the Hessian of the loss function with respect to the network parameters after training. In contrast, DINOZAURg natively supports UQ, thereby providing accurate uncertainty estimates out of the box. See results in Figure 3
4.3 Ablation studies
To evaluate the scalability of DINOZAUR, we conduct experiments varying model width, depth, and the number of truncation modes using the ShapeNet car dataset. As shown in Figure [a). test error (measured by RL2 loss) decreases with increasing architectural hyperparameters, saturating near the values used in our main experiments. Further improvements may be limited by the input data resolution of 643. Figure bb) confirms our scaling claims: depth increases parameter count almost linearly, width affects it quadratically, while modes do not affect the model size.
We further assess runtime and memory efficiency using synthetic uniform data (6-43 resolution, input channels, 7 output channels), fixing architecture settings to 128 channels, [32, 32, 32] modes, and FNO blocks. We compare our model with and without gradients, FNO, TENO (r = 0.4), and TFNO (r 10-5, matching DINOZAUR across five rums. Memory usage is highest for FNO and drops with rank r in TFNO. DINOZAUR and TFNO (r 10-5) yield nearly identical memory profiles, while our full model uses roughly twice the memory due to gradients. Runtime varies: TENO (r 0.4) is slower than standard FNO due to additional tensor multiplications in Tucker decomposition. Our model with gradients similarly slow, limited by FFT complexity, as we concatenate gradient features before inverse FFT. Corresponding charts are in Figures c-d).
parameter count). We record VRAM and training time
5 Discussion
The recent emergence of neural operators has introduced a powerful alternative to traditional, compu- tationally expensive numerical methods for solving PDEs in downstream applications. Architectures such the Fourier Neural Operator and the Geometry-Infomee notable success in learning solution operators across various physical systems. However, these models rely on highly parameterized transformations in spectral space, which can hinder scalability and interpretability. Furthermore, FNOs and other common architectures fail to provide native support for uncertainty quantification, critical requirement for reliable deployment in scientific and engineering applications. As a result, practitioners are often forced to apply generic UQ techniques post hoc, which fail to exploit the spatio-temporal inductive biases intrinsic the underlying physical systems.
Neural Operator have demonstrated
In this work, we introduced DINOZAUR, an FNO-based model featuring a parsimonious and physically motivated parameterization of the message passing mechanism. This design stems

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from the analytic solution to the heat equation and is augmented by spatial gradient features. By presenting learned diffusion time parameters, DINOZAUR defines new diffusion multiplier that is pimeeniionalit--ndeeeedee and offers more favorable scalability than the original FNO. Through extensive evaluation on benchmarks, we showed that DINOZAUR consistently matches or exceeds the predictive performance of dense FNO and GINO architectures while using orders of magnitude fewer parameters. Leveraging this efficient parameterization, we introduce meaningful priors over diffusion time parameters, obtaining Bayesian neural operator that delivers competitive UQ performance against classical deep learning UQ methods such as MC Dropout and Laplace approximation. To the best of our knowledge, DINOZAUR is the first Bayesian NO to explicitly define distributions in the integral transform.
Limitations and future work In our investigations, we found that the primary limitation of the diffusion mechanism is its radial symmetry. We observed that diffusion alone struggles to train well. Our solution is to include gradient features, though at the cost of increased computational time. Investigating other strategies, like introducing anisotropy intrinsically, using the modifications proposed in 622. or mixing gradients with other features before applying inverse FFT overhead, promising directions for future work. We found that our Bayesian formulation sensitive to hyperparameter settings of the prior distributions. Customizing priors across network layers to capture different physical properties could improve the robustness and interpretability of the model. More broadly, exploring other parsimonious kernel parameterizations may offer both interpretability and improved generalization, especially in settings with limited data or strong physical priors. Finally, since our parameterization is independent of spatial dimensionality, it opens up opportunities for pretraining on siwer-dimensional or simpler physical systems, with the potential to accelerate learning in more complex, higher-dimensional domains through transfer learning [63]
We believe work is step forward deeper understanding of neural operators, advancing them toward becoming reliable tools for real-world scientific and engineering applications.
Acknowledgments and Disclosure of Funding
We want thank Greg Bellchambers, Bachir Djermani, Axen Georget, Pavel Shmakov, and Phoenix Tse for insightful discussions and collaboration on the implementation.
Conceptualization, Methodology. Software, Investigation, Writing Original Draft, Visualization. Sanmitra Ghosh: Conceptualization, Methodology, Software, Investigation, Writing Review and Editing, Visualization. Aamal Hussain: Software, Investigation, Writing Original Draft. James-Michael Leahy: Conceptualization, Methodology, Software, Writing Review and Editing. Michalis Michaelides: Conceptualization, Resources, Writing Original Draft, Supervision, Project administration.
CRediT author statement: Albert Matveev:
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20

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Sight-Weight Diffusion Multiplier and Uncertainty Quantification for Fourier Neural Operators: Supplementary Material
A Benchmarks
Table Summary of the benchmark datasets
Dataset
d
Input type
Output type
Nusin Noest
Darey flow
Regular [241:241]
Regular 2211:211
1024, 1024
NS 2D+T
2+1
Regular [64; 64; 10]
Regular (64; 64; 10]
1000, 200
ShapeNet car
Regular 64;66;
Irregular [3,586]
561, 100
Ahmed bodies
3
Regular (64; 64; 64]
Irregular [150,000]
500, 51
Table 4: Data transformations during training
Dataset
Inputs
Input scaler
Padded input
Target
Target scaler
Darey flow
Permeability
Standard
271;
Pressure
Standard
NS 2D+T
Vort. fort.logr.cio- [z, y,
None
64: 64; 15
Vort. Vort./nost<<2r 10211<20
None
ShapeNet car
SDF
None
72: 72; 72
Pressure
Standard
Ahmed bodies
SDF, Inlet velocity
None
[72; 72; 72
Pressure
Standard
To evaluate our model and compare it against the baselines, we selected two datasets with regularly sampled domains and two datasets with non-uniform outputs.
Darcy flow This dataset contains steady-state 2D solutions of the Darcy flow equation on the unit box, modeling the flow of fluid through porous medium [ It includes samples with spatial resolution of 241 x 241 points. The network received scalar permeability field as input and was tasked to predict scalar pressure values. To ensure non-perionn non-periodicity, we applied padding during training, increasing the resolution to 271 x 271. A standard scaler was applied to both input and output fields. Train and test splits had 1024 samples each.
Navier-Stokes The Navier-Stokes dataset includes simulations of incompressible, viscous fluid flow with constant density and viscosity v = 10-5 [9] It features a periodic 2D domain of resolution 64 64, with 20 time steps of the vorticity field evolution. We trained models predict the final 10 time steps from the initial 10, using 1000 training and 200 test samples. Inputs were composed as 3D tensors containing 10 time steps for every spatial point. Padding was introduced only to the time dimension, making the input to be of 64 x 64 x 15. We appended spatial coordinates uniformly sampled [0, 2x) and time coordinate with an integer index = [0, 1,...9) 9). which resulted in input channels that were mapped to one channel.
ShapeNet car This benchmark contains 661 meshes paired with Reynolds- Averaged Navier-Stokes (RANS) simulations [59]. We focus on predicting the scalar pressure field at the mesh vertices, which are fixed in size 3,586 points per mesh. The dataset is split into 561 training and 100 test samples. Target pressures were standardized during training. Following Li et al. [10] inputs constructed by sampling a fixed-resolution 64 grid over the global bounding box encompassing all meshes. A signed distance function (SDF) is computed per mesh. Grids are padded to 72 before being passed to the network.
Ahmed bodies The Ahmed bodies dataset 60 follows a similar setup to the ShapeNet car but
introduces varying inlet velocities as an additional signal. This constant is appended to the SDF input, resulting two input channels. Meshes vary in size, containing between 90,000 and 200,000 vertices. The grid resolution, padding, and target scaling match those used in the ShapeNet car.
21

---

The GINO-decoder architecture [00] used for both ShapeNet and Ahmed datasets requires precom- puted fixed-radius neighborhoods map grid samples mesh vertices. We use the radii from the original paper: 0.05 for the ShapeNet car and 0.035 for Ahmed bodies.
For models trained non-uniform data, additionally apply Transformer-style positional encod- ing [64] to the input features, grid locations, and mesh vertices.
Results in the main text are reported on the unscaled targets. Raw dataset specifications provided Table and data transformations are detailed Table 4
B Configurations of trained architectures
Table 5: Baseline configurations. GF indicates Gradient Features
Dataset
Model
de
Modes
Rank
GF
Extra parameters
Darcy
FNO
[12, 12]
False
flow
FNOg
32
[12, 12]
True
TFNO
32
[12, 12]
4
False
TFNO,
32
[12, 12]
10-4
True
DINOZAUR no-g
32
[12, 12]
No
False
DINOZAUR
32
[12, 12]
No
True
NS 2D+T
FNO
[16, 16, 4]
False
FNOg
64
[16, 16, 4]
True
TENO
[16, 16, 4]
10-4
False
TENO,
64
[16, 16
10-1
True
DINOZAUR
64
[16, 16
No
False
DINOZAUR
[16, 16, 64
No
True
ShapeNet
GINO
64
24, 24]
0.4
False
car
GINO,
64
22, 24, 24]
0.4
True
TGINO
[2, 24]
10-4
False
TGINO
64
[24, 24, 24]
10
True
DINOZAURED-
64
[24, 24, 24]
No
False
DINOZAUR
64
[24, 24]
No
True
GINOD
64
[24, 24, 24]
0.4
False
TGINOD
64
[24, 24, 24]
10-4
False
D D 22= =
DINOZAURD
64
[24, 24, 24]
No
True
GINO,
64
[24, 24, 24]
0.4
False
o = c
TGINO,
64
[24, 24, 24]
10-4
False
C 500,
DINOZAUR,
64
[24, 24
No
True
a = 108
N (In (In (-
DINOZAUR
64
[24, 24, 24]
No
True
Ahmed bodies
GINO
64
[24, 24, 24]
0.4
False
GINO
64
[24, 24, 24]
0.4
True
TGINO
64
[24, 24,
10-4
False
TGINO 3
64
[24, 24]
10-4
True
DINOZAUR, no-g
64
[24, 24]
No
False
DINOZAUR
64
[24, 24,
No
True
All results reported our paper are from experiments conducted specifically for this study to ensure fair and consistent comparison across models. Architectural settings for the baselines are summarized Table All models featured FNO blocks with GELU activations. The channel width d, and number of modes were selected to align with the original configurations of FNO 9 and GINO 10 with the exception of Sivier-Stokes data. We increased modes and channels to reflect the complexity of this benchmark. For models employing Tucker factorization, the rank was either adopted directly
22

---

from the original implementation (GINO) or chosen to match the total number of parameters in DINOZAUR.
For all models incorporating dropout, the probability for a weight element to be zero was set to D = 0.1 during training.
For the Laplace approximation, we the Hessian the Negative log-likelihood loss (assuming homoscedastic noise with o exp(-4)) with respect to the parameters 0e of the final (linear) layer Q each model. Following Ritter et al. 65 we scale the Hessian and regularize it by adding
multiple of the identity matrix, yielding
8 log (D) 00)
Hc, = CE
+ tal
002
where the expectation approximated via Monte Carlo samples, and C and are hyperparameters selected on validation set. In our experiments, we set C 500, 10t
set the prior distribution parameters for each In and specify the initial values for the approximate posterior In T. D: we initialize this distribution with a full- rank covariance matrix within each NO block. Distribution parameters informed by the time distributions the deterministic experiment on ShapeNet car (Figure
DINOZAURg requires
Table 6: Baseline optimization parameters
Dataset
Model
Optimizer
epochs
Scheduler
LR
Darcy
FNO
AdamW
150
One-Cycle
10 10-3
flow
FNOg
AdamW
150
One-Cycle
10
TFNO
Adam W Schedule-Free
150
None
10-3
TFNO
AdamW W
150
One-Cycle
10
DINOZAUR Ro-g
Adam W
150
One-Cycle
10
DINOZAUR
Adam W
150
One-Cycle
10-3
NS 2D+T
FNO
AdamW W
150
One-Cycle
10
FNOg
AdamW W
150
One-Cycle
10-3
TENO
AdamW W
150
One-Cycle
10-3
TFNO g
Adam W
150
One-Cycle
10
DINOZAUR
AdamW
150
One-Cycle
10 10-2
DINOZAUR
AdamW
150
One-Cycle
10-2
ShapeNet
GINO
AdamW Schedule-Free
150
None
10 10-3
GINO,
AdamW Schedule-Free
150
None
10-3
TGINO
Adam W Schedule-Free
150
None
10-3
TGINO
AdamW
150
One-Cycle
10 10-3
DINOZAUR
AdamW
150
One-Cycle
10-3
DINOZAUR
AdamW
150
One-Cycle
10
GINOp
AdamW Schedule-Frre Free
150
None
10
GINO,
AdamW Schedule-Free
150
None
10-3
TGINOp
AdamW Schedule-Free Free
150
None
10-3
TGINO,
Adam W Schedule-Free
150
None
10-3
DINOZAURD
Adam W
150
One-Cycle
10-3
DINOZAUR
Adam W
150
One-Cycle
10-3
DINOZAUR
AdamW
200
One-Cycle
10-3
Ahmed bodies
GINO
AdamW Schedule-Free
150
None
10
GINO, g TGINO
Adam W Schedule-Free
150
None
10-3
Adam W
150
One-Cyck
10-3
TGINO
AdamW W
150
One- One-Cycle
10
DINOZAUR nno-g
Adam W
150
One-Cycle 10
DINOZAUR
AdamW
150
One-Cycle 5 x 10-3
Optimization hyperparameters are detailed in Table All models were trained using either AdamW Schedule-Free [57] or AdamW combined with One-Cycle scheduler [56), weight decay set 10-
23

---

Both options allow for faster convergence with fewer epochs and reduce sensitivity to hyperparameter choices. The final setup was selected by comparing both options and reporting the results for the configuration that achieved the best performance. A limited learning rate sweep was conducted in the range [10-1 10-3), with early termination applied underperforming runs. For the One-Cycle scheduler, the learning rate in the table refers to the max LR parameter.
C Distributions of diffusion times
Dataset
ShapeNet
Ahmed
2
Block
3
4
5
0.001
0.01
0.1
T
Figure 5: Diffusion times gathered from FNO blocks of deterministic DINOZAUR trained non- uniform datasets.
Having trained the deterministic DINOZAUR on the non-uniform datasets ShapeNet car and Ahmed bodies, we analyze the behavior of the learned diffusion times across the network. empirical distributions of the 64 diffusion times extracted from each block of the neural operator. We observe two main trends. First, diffusion times generally decrease with network depth. This aligns with the findings of Sharp et al. [34 and suggests that deeper blocks capture finer-scale features, whereas earlier blocks encode broader, more global structures. Second, diffusion times learned on Ahmed bodies dataset consistently larger than those learned the ShapeNet We attribute this to the presence of varying inlet velocities in the Ahmed bodies dataset, which increases the variability in the data and may require coarser diffusion scales to capture the broader range of feature patterns effectively.
shows the
For the Bayesian version of our model, we initialize all blocks with the same prior distributions for 7 and likewise set identical initial guesses for the corresponding variational posteriors. This initialization allows each block adapt its uncertainty about diffusion time scales through learning. To visualize this, we draw samples from the approximate posteriors and estimate their empirical location and scale parameters, characterizing the learned diffusion behavior in the probabilistic setting. Figure 6 illustrates these posterior distributions alongside the prior. We find the same consistent pattern: posteriors systematically shift toward shorter diffusion times as depth increases. and their spread becomes tighter. This mirrors the trend observed in the deterministic setting (Figure (5). reinforcing the interpretation that deeper blocks specialize in capturing high- -ree-freeeenc, localized features. The prior has a strong influence on the resulting posteriors, forcing diffusion times to be lower (closer to the prior mean, which was chosen promote shorter ranges for message passing).
Finally, Figure 7 we provide the full covariance matrices for each of the posteriors that were learned by the network. Analyzing covariances, we note that they are dominated by the diagonals with few spots of strong correlation off-diagonal. This indicates a good separation of information within features.
24

---

Prior
Block
Block
Block
Block
0.005
0.01
0.015
0.02
0.025
0.03
Figure 6: Prior distribution set according to I 0.01, 1) and approximate posteriors gathered from the NO blocks of DINOZAURg trained on ShapeNet car dataset through CELBO maximization.
Black
Block
0.5
20-
20-
20:
-05
1.0
40
60
20
40
60
20
40
Figure 7: Lower diagonal full-rank learned covariance matrices of the posterior distributions. Each matrix represents a de x de (64 x 64) covariance matrix of the multidimeensionll distribution of diffusion times within each block.
D Metrics definitions
Given element index = 1,..., Nuest and point evaluation index 1,..., Jus let unf denote the ground truth observation of the target function for element at point j. and let ūny denote the corresponding neural operator prediction. For baseline models that support uncertainty quantification via sampling. ūny represents the empirical mean over samples, and ony denotes the empirical standard deviation. In all probabilistic experiments, we generated 100 samples per prediction.
The RL2 loss is normalized metric that quantifies the discrepancy between predicted outputs and ground truth values, scaled by the squared norm of the ground truth. This normalization allows for fair comparisons across datasets tasks with varying magnitudes and units.
Relative L2
Neant
=== (Wnj ūnn)
RL2
Nrest
Jm ビe=
22
Negative log-likelihood NLL reflects the mean log-likelihood of the true values under the predicted Gaussian distribution. Lower NLL indicates better uncertainty calibration and fit to the data.
News
In(ōn, V2T) + (Unj Tnnj )2
NLL
N test
252 nj
This averaged formula used only for reporting metrics for probabilistic models.
25

---

Miscalibration area The MA is scalar metric that quantifies the discrepancy between predicted confidence levels and the actual observed frequencies of events, providing measure of uncertainty calibration. It captures the area between the model's empirical calibration curve and the ideal diagonal (perfect calibration). Lower values indicate better uncertainty calibration. Let π = K for O,.... K, denote the expected coverage levels. For each test sample n, the observed coverage at level (kk is given by:
(k)
Jn
Unn
Uns! Onj
--11
( (k) 2
where 4- the inverse the standard normal cumulative distribution function. Then, the miscali- bration area across all elements of the test set is given by averaging the area estimates for each of the elements:
Nuwut K
(T+1)) - 7(1)
((()) ((6) + + (((++1) n [
MA
Ntest
2
n= k=0
Interval score IS is another metric for evaluating predictive uncertainty. It balances the width of the prediction interval with a penalty for ground truth values falling outside the interval. For a chosen coverage level E (0, 1), define the lower and upper predictive bounds:
Unj Fnj + nj
[p]
(1-2) 2
and ((P) njj
Unj + ōnj क
+ 2
where - is the inverse of the standard normal cumulative distribution function. Then the interval score at level p for prediction at point j in sample n is:
( m5
b nj
2
(((h) -u.) (u, < ((
2
-1 b(()) 1 Uu > (1)
b("))
nj
1
To compute the overall interval score, average predefined set of P equally spaced coverage levels (p- = [0.01, 0.02, ...99)......)):
Nies
IS
(P)
Nuest
P
L J
j=1
E Elaboration on the universal approximation
Here, we formulate the statement of Proposition ! reported in the main text, more strictly.
Following the exposition of Lanthaler et al. (66], let Q C Id be bounded domain with Lips- chitz boundary and lct A(2 (Rd) (12, (1,2,111,)) "., VS2, R4)) denote Banach space functions on Ω. DINOZAUR architecture defines mapping
No.r A(2, Rd)) R4..
which can be written composition of form:
No Q Ne
Ne, P.
consisting of lifting layer P. hidden blocks No..,, i = 1,..., M, and a projection layer Q. Given channel dimension d,, the lifting layer P is given by mapping:
P A(2, Rd + V(S2 Id),,
a(x) p()(r),
where P: Rdd x X + de is a learnable neural network acting between knite-dimension Euclidean spaces. For = 1,.... M, each block No. of the form
G F (exp(_4x7211 (1)2T) @ (1)) 1)
F [exp((4424 11|) @ Flo](t)]
[b]()] := ( ((x) + b, W.mix
26

---

Each hidden block defines mapping No..r. ((2,R4.) + ((, For i = 1..... M. the matrices Wakipp W gddxxde W gd.x2de diffusion times T, Rd. and bias Rd are learnable parameters. The non-linearity a R + P acts element-wise and is assumed to be smooth, non-polynomial, and Lipschitz-continous. Finally, the projection layer 2 is given by mapping:
@ V(2, R + (S2, "),,
v(xr) + (()()),)),
where Q : R x Ω + R is also a learnable neural network acting between finite-dimensional Euclidean spaces.
From now on, we will focus on the case of a single hidden block:
[o, (2 No,. (x)))
Now, we re-state the Proposition
D
Proposition 1. Let Ω C R& be bounded domain with Lipschitz boundary and such that the closure Ω (0,27) 2x) S For given integers 8, s O, let N C (S2; Rd.) + C' (Ωt; (Rd))) be continuous operator, and fix compact set A C (S2; Rda). Then there exists continuous, linear operator 8: 09;;; Rd)) - C (T4, (R such that Ela||a a for all C* Rda). Furthermore, for any
8 > 0, there exists DINOZAUR such that
sup [|[[a]] (No,r o @[aa)aallo* (0:344) S,
a
where No.- Q No P: C* (T4, - Ct (T..
Proof. First, consider periodic torus T Without loss of generality, since the weights of the network are free to be updated, we let the matrices Winix be of the block structure
mmmxx
[T
with I E Rd being the identity matrix and Rd.xd, a matrix of all zeros, which renders hidden block to:
\o)x= ( (wrphip (yzkipp(x)= + tb +
1 [T
1-p(-44711411+ Flo)()))(s)
F
By letting diffusion times r + 0o, we get
1up(-4721|<111 = 1, ||*||| = O, exp( exp( (-4721 - 0, ||||| > O,
which, in the limit, results in the block:
No [w](x) = a
Wokip u(x) +b [ F [F(](())
Woskip (xx) + by + ff v(y)dy)
making full network No. Q No,
averaging neural operator in the sense of Equation 2.3 in [66] on the torus T By Corollary 3.3 [66) NN is universal approximator on
the torus. Here, we note that the lifting and projection operators defined in [66] have an explicit positional dependency, and thus, one should understand them as periodized MLPs.
Now consider a general domain C. (0, 2d By Lemma 41 by Kovachki et al. 667 and Lemma A1 in [66], there exists a continuous, linear extension operator E : C* (Ω2; Rd + (TT, (Rd=) such that E[a]la and E[a] is periodic on [0. 2 for all E C*22;; Rd)). We note that the conditions of the Theorem 9 in [67 are satisfied. By following the proof of this theorem, we apply the extension operator & and use the universal approximation on the torus, establishing universal approximation on Ω
NNote that Theorem 2.1 in 66 it is formulated, applies to bounded Lipschitz domains. Thus, the proof of Corollary 3.3 (66] not immediate. One must either follow the proof of Theorem 2.1 to that applies the torus, or first formulate on domain (0, 2xr" invoke the Theorem 2, and then argue that holds when restricts to periodic functions (0, 2m)
27

---

Broader impact
This study presents neural operator tailored for the efficient and scalable solution of partial differen- tial equations, intending to advance computational modeling in scientific and engineering disciplines. By enabling more essource-efficien simulations, our method has the potential to accelerate research in various domains where PDEs play a central role. Furthermore, the integration of uncertainty quantification in our model supports more robust becision-making, which is particularly valuable in high-stakes applications.
While work is primarily theoretical and presents low immediate risk, we recognize that advances in machine learning for scientific computing may influence decisions in sensitive contexts. As neural operators become more widely adopted, it will be essential ensure that they are used responsibly, particularly in applications where predictive reliability and interpretability are critical. Ongoing dialogue among machine learning researchers, domain experts, and the general public is essential to ensure that the development and deployment of such tools remain aligned with societal values and serve the public interest.
28


Reviews:
[{"review_id": "CP3tlM9qUQ", "reviewer": "Reviewer_HQ9U", "summary": "The present paper proposes DINOZAUR, a lightweight variant of the Fourier Neural Operator that introduces a diffusion-multiplier architecture to replace dense Fourier multipliers. By doing so, it reduces the parameter count significantly while preserving predictive accuracy. Furthermore, DINOZAUR embeds a Bayesian uncertainty quantification mechanism directly into the operator which results in spatially correlated posterior predictive uncertainty. The authors present experiments across multiple PDE benchmarks and demonstrate favorable trade-offs in accuracy and UQ compared to various baselines.", "questions": "See above and:\n- What is the overhead of variational inference and sampling during  predictions of the framework ?\n- A brief empirical test of anisotropic alternatives ( for instance directional kernels) would strengthen the paper.\n- The authors should also discuss diffusion-based generative models as some support UQ intrinsically and could offer competitive alternatives (see e.g. \"Generative learning for forecasting the dynamics of high-dimensional complex systems\" or \"Generative ai for fast and accurate statistical computation of fluids\")", "limitations": "Clearly mentioned and discussed in the paper", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The paper provides a diffusion multiplier grounded in the heat kernel that is a compact alternative to the dense multipliers used in FNOs.\n- The probabilistic formulation directly over diffusion times is original and compelling as most UQ efforts in this domain are  currently post hoc.\n- The empirical evaluation is well-done and the paper is well written. DINOZAUR outperforms competing methods for the experiments presented.\n\nWeaknesses:\n- As mentioned by the authors DINOZAUR without gradient features performs poorly as the isotropic diffusion alone is insufficient. While adding gradient feature fixes this, it somewhat undermines the initial idea of simplicity and raises computational cost.\n- UQ is only evaluated on a single dataset. From the reviewers' point of view, a broader UQ evaluation would better demonstrate generalizability of the Bayesian formulation.\n- Tables 1 or 2 only contain evaluation of a single run of the framework. The reviewer acknowledges that the performance trends are clear, but still would have preferred to see results averaged over multiple seeds or initialization to strengthen the results. Reproting a standard deviation or similar would strengthen the paper.", "was_revised": true, "final_justification": "All my concerns have been addressed and I keep my score of accept.", "rebuttal": "We thank the reviewer for their careful and constructive review. We are pleased that the reviewer found the diffusion-based architecture and the probabilistic formulation of DINOZAUR to be both original and compelling. We’re glad that the diffusion-based formulation and the direct probabilistic treatment of diffusion times were found to be original and meaningful contributions. It is also encouraging to see that the reviewer considered our experimental results and overall presentation as clear and well-executed.\n\nWe discuss some of the points raised by the reviewer that were not addressed in the main text.\n\n**UQ is only evaluated on a single dataset**\n\nThat is a fair criticism, although the dataset we chose allows us to demonstrate the spatially correlated nature of our uncertainty estimates to the fullest extent. The intent was to show the applicability of our Bayesian diffusion multiplier for geometric problems. \n\n**Tables 1 or 2 only contain evaluation of a single run of the framework**\n\nWhile performance trends were stable across runs, we agree that reporting results averaged over multiple seeds would add robustness and clarity to the comparisons.\n\n**What is the overhead of variational inference and sampling during predictions of the framework?**\n\nDuring the prediction time, it is required to draw multiple diffusion times from the posterior and produce full inference for each sample. We chose 100 samples for reporting the final metrics in our tables. We note that it is straightforward to parallelize sampling in batch-processing fashion, by drawing a batch of time samples from the posterior and propagating them with the batch of duplicated inputs. We also point out that the same process is required for both alternatives we consider — MC Dropout and Laplace approximation. Our computational overhead is therefore similar to other UQ methods, although we did not benchmark this overhead precisely. \n\n**The authors should also discuss diffusion-based generative models as some support UQ intrinsically and could offer competitive alternatives**\n\nWe appreciate reviewer’s suggestion to consider diffusion generative models. Although we acknowledge the importance of developing generative models for physics applications, we argue that the referred papers either solve different problems compared to ours, or take a substantially distinct approach to providing the UQ estimates. Our method follows the classical paradigm of creating surrogates for physics modelling, and provides a physically-motivated uncertainty. Nonetheless, generative diffusion models represent an interesting complementary line of work and could offer valuable insights."}, {"review_id": "8ocLGf5MOG", "reviewer": "Reviewer_KxCZ", "summary": "The manuscript introduces DINOZAUR, a new diffusion-based neural operator designed to address scalability and uncertainty quantification challenges in solving partial differential equations (PDEs). While Fourier Neural Operators (FNOs) are widely used, they face issues like overparameterization and lack of native uncertainty quantification. DINOZAUR improves upon this by replacing the dense tensor multiplier in FNOs with a more efficient diffusion multiplier that reduces parameters and memory usage, while maintaining predictive performance. It also incorporates Bayesian formulation of the diffusion times, allowing for uncertainty quantification and spatially correlated outputs. The method demonstrates competitive or superior results on PDE benchmarks while providing effective uncertainty estimates.", "questions": "Would the authors suggest using the diffusion kernel for complex PDE including shock and bifurcation phenomena? \nCould the authors better clarify their last claim: \"Finally,\nsince our parameterization is independent of spatial dimensionality, it opens up opportunities for\npretraining on lower-dimensional or simpler physical systems, with the potential to accelerate learning\nin more complex, higher-dimensional domains through transfer learning\"? How such a super-resolution strategy would help?", "limitations": "The authors identified that the main limitation of the diffusion mechanism lies in its radial symmetry and correctly underlined that relying solely on diffusion kernel does not train effectively. To address this, they incorporated gradient features, though this comes with the trade-off of increased computational time. Additionally, they rightfully observed that their Bayesian formulation is sensitive to the hyperparameter choices of the prior distributions, although  tailoring priors for different network layers to better capture varying physical properties could be tricky.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths\nIntroducing UQ explicitly in the NO formulation is a smart idea and very useful one. \n\nWeaknesses\nThe paper should better identify the intuitive reasons why a diffusion kernel is adapted to learn complex non linear PDEs of any kind. The lack of anisotropy, despite being mitigated by the introduction of the gradient features, might introduce some overconstraint.", "was_revised": true, "final_justification": "The paper is solid and original. There is potentiality to improve and adjust such a method to tackle complex physics phenomena.", "rebuttal": "We are grateful for the reviewer’s careful assessment and recognition of our contribution to integrating uncertainty quantification into the neural operator framework. The comments regarding both the limitations of our approach and the potential of diffusion-based kernels in complex PDE settings raise important points. Below, we address these questions in detail.\n\n**Would the authors suggest using the diffusion kernel for complex PDE including shock and bifurcation phenomena?**\n\nWe agree that simplistic nature of the diffusion kernel is counterintuitive. However, there exists an extensive literature on the universal approximation capabilities of neural operators. While the isotropic nature of the kernel might appear overly restrictive, our design is supported by theoretical results: building on the work *“Nonlocality and nonlinearity implies universality in operator learning”*, we have established a similar universality result for our diffusion multiplier in the space of continuous functions (a more detailed and rigorous proof would be included in camera-ready submission). We also believe that it will be fairly straightforward to generalize that theorem to Sobolev spaces. \n\nAlthough we acknowledge that theoretical universal approximation results are rarely useful in practice, we believe that this results serves as a basis for applicability of our kernel. The lack of anisotropy is indeed a core limitation of our current design, and we admit that exploring more expressive or anisotropic kernels is a natural next step and an exciting direction for future work.\n\n**Could the authors better clarify their last claim: \"Finally, since our parameterization is independent of spatial dimensionality, it opens up opportunities for pretraining on lower-dimensional or simpler physical systems, with the potential to accelerate learning in more complex, higher-dimensional domains through transfer learning\"? How such a super-resolution strategy would help?**\n\nThank you for this request for clarification. In this passage, we aim to highlight that our architecture **does not change** with varying dimensionality of a problem (1D, 2D, …). This property allows for exploring pretraining strategies across dimensions. For instance, we imagine that pretraining our model with 2D Navier-Stokes flows could expedite convergence for 3D flows. The underlying hypothesis is that structural or physical priors learned in lower dimensions may accelerate convergence in higher-dimensional settings. At the moment, there is no empirical evidences for that, but our architecture allows to conduct such experiments. \n\nWe wouldn’t refer to such experiment as super-resolution, but rather as transferring some general knowledge about physics from lower dimensional problems."}, {"review_id": "PB3UYptiUh", "reviewer": "Reviewer_CUNK", "summary": "This paper introduces DINOZAUR, a novel approach to operator learning for solving partial differential equations (PDEs). DINOZAUR leverages a Bayesian formulation to enable uncertainty quantification (UQ) – providing not just a solution, but also a measure of confidence in that solution. The core innovation lies in replacing the learnable kernel of traditional Fourier Neural Operators (FNOs) with a precomputed diffusion multiplier, motivated by solutions to the heat equation. This dramatically reduces the number of trainable parameters while maintaining competitive performance on operator learning benchmarks.  \n\nBy utilizing a fixed kernel, DINOZAUR achieves significant parameter reduction, and the inclusion of gradient features overcomes limitations imposed by the radial symmetry of the diffusion mechanism. While acknowledging limitations in demonstrating scalability to higher-dimensional operator learning problems and a sensitivity to hyperparameter choices, the paper presents a promising advancement in neural operator design, offering a balance between efficiency, accuracy, and the ability to quantify solution uncertainty for a range of PDE applications.", "questions": "- Have you explored any parallelization strategies to accelerate training or inference?\n- Could you elaborate on the choice of the variational inference method and whether you considered other VI techniques or approximate Bayesian computation methods?\n- Beyond the ablation studies presented, have you investigated the impact of different activation functions or network architectures within the FFN used for gradient features mixing?", "limitations": "Yes.", "rating": 6, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "**Strengths**\n\n- The dimensionality-independent nature of the parameterization hints at scalability.\n- The proposed method achieves performance comparable to or better than baseline models on standard benchmarks.\n- Inclusion of gradient features effectively addresses the limitations of radial symmetry in the diffusion mechanism, improving performance.\n- The Bayesian formulation allows for principled uncertainty quantification, providing not just solutions but also estimates of confidence.\n- The use of a precomputed diffusion kernel motivated by PDE theory is a novel approach to kernel design in neural operators.\n- Significant reduction in the number of trainable parameters compared to traditional FNO and GINO architectures.\n\n**Weaknesses**\n\n- The lack of experiments in higher dimensions (beyond 2D/3D) don't allow comparison of scaling in $d$.\n- The use of a standard normal prior in the Bayesian formulation is relatively simple and could potentially be improved with more informed priors tailored to the specific PDE being solved.\n- The UQ performance wasn't comprehensively compared to a wide range of state-of-the-art UQ methods.\n- While parameter efficient, the quadratic scaling of parameters with channel width may become a limitation for very high-resolution problems.\n- The paper focuses on a single kernel motivated by the heat equation; exploring other physically motivated kernels could broaden the applicability of the approach.\n- A minor point, but the FFN in Figure 1 needs to be defined.\n- There is minor symbol reuse (e.g., $\\sigma$).", "was_revised": true, "final_justification": "Thank you to the authors for the thorough response. I have no further comments or questions, and I maintain my recommendation of \"strong accept.\"", "rebuttal": "We greatly appreciate the reviewer’s thoughtful and positive evaluation of our work and the recognition of DINOZAUR’s key contributions, including the use of a diffusion-inspired kernel for improved efficiency, the integration of gradient features to address radial symmetry, and the principled Bayesian formulation for uncertainty quantification. We also value the reviewer’s acknowledgment of our method’s strong performance on standard benchmarks, its dimensionality-independent design, and the significant reduction in trainable parameters. Several of the weaknesses raised were already acknowledged in the paper and discussed as future directions. Below, we address the remaining points in more detail.\n\n**Experiments in higher dimensions.**\n\nWe agree that conducting experiments with higher-dimensional data is an exciting investigation that would emphasize the scalability of DINOZAUR. However, we would like to note that widely accepted benchmark datasets in the neural operator literature rarely include PDEs in dimensions higher than 3D as it is the most common setting for practical applications.\n\n**The UQ performance wasn't comprehensively compared to a wide range of state-of-the-art UQ methods**\n\nWe acknowledge the importance of careful benchmarking — in our work, we included two widely used and accessible baselines, MC Dropout and Laplace approximation. These methods were selected because they are among the most straightforward and broadly adopted in the neural operator and broader machine learning literature. While there exist more elaborate UQ approaches, our goal was to compare against methods that strike a balance between implementation feasibility and relevance to operator learning settings. \n\n**While parameter efficient, the quadratic scaling of parameters with channel width may become a limitation for very high-resolution problems.**\n\nThank you for raising this point. We'd like to clarify that the quadratic parameter scaling is not a consequence of the diffusion multiplier, but of the MLP components used within the architecture. We also note that changing width of the network is not directly linked to the resolution (although it definitely may be beneficial for high-resolution data).\n\n**A minor point, but the FFN in Figure 1 needs to be defined**\n\nWe will introduce FFN definition in the camera-ready submission in the case of acceptance. Thank you for pointing this out.\n\n**Have you explored any parallelization strategies to accelerate training or inference?**\n\nNo explicit parallelization was introduced neither for deterministic nor Bayesian training. As for the model prediction, we found that sampling from the posterior for diffusion times can be treated in batch-processing fashion, by drawing multiple samples for parameters and applying them to a batch of duplicated inputs. This helps to slightly reduce the overhead of sampling during inference of Bayesian DINOZAUR, but our investigation didn’t go further than that.\n\n**Could you elaborate on the choice of the variational inference method and whether you considered other VI techniques or approximate Bayesian computation methods?**\n\nWe chose ELBO minimization as our variational inference objective, as it is the most widely adopted approach in Bayesian neural networks and is a natural choice for our framework. While MCMC methods (e.g., NUTS-HMC) are often considered the gold standard, they are prohibitively expensive in our setting due to the need for many model evaluations across large datasets. The same limitations apply to simple Approximate Bayesian computation, though more advanced variants may offer improvements. Other alternatives such as expectation propagation are less common in this context but could be worth exploring.\n\nIn our implementation, the inference was over global model parameters $\\tau$ of a fixed size, so we used a common mean-field approximation with the assumption that the joint distribution factorizes between layers, where a multivariate normal is used to allow for intra-layer (between channel) correlations in $\\tau$. Though more expressive approximations are possible, our empirical results suggest this formulation provides effective uncertainty estimates. Our method does not advocate for a specific inference method, but focuses instead on the way a Bayesian mechanistic NO model can facilitate good UQ.\n\n**Beyond the ablation studies presented, have you investigated the impact of different activation functions or network architectures within the FFN used for gradient features mixing?**\n\nOne of the goals we had in mind during writing the paper was to isolate the influence of our proposed multiplier — this resulted in having a robust comparisons where all the architectural choices are shared across all the baselines and the proposed method. While we did not conduct an exhaustive ablation over FFN configurations, we agree that exploring alternative activations and architecture designs could yield interesting performance trade-offs."}, {"review_id": "Tj4RL4F1KE", "reviewer": "Reviewer_oyPk", "summary": "This paper introduces a new model, called DINOZAUR, within the Neural Operator (NO) family. DINOZAUR modifies the Fourier Neural Operator (FNO) in three ways.\n1. First, rather than use a convolution within Fourier space, DINOZAUR has a scalar diffusion multiplier for each channel that is directly inspired by the analytic solution to the heat equation. This drastically reduces the number of parameters compared to a FNO with the same number of channels.\n2. Because the heat equation is isotropic, the authors double the number of channels. The second half of the channels apply a gradient operator to the output of the first half. This allows for additional flexibility.\n3. DINOZAUR introduces uncertainty quantification, Each of the diffusion multipliers has a normal prior associated with it, and solutions are sampled from the posterior distribution. There is also aleatoric uncertainty parameter $\\sigma$ at each point. DINOZAUR is trained using variational inference by minimizing the evidence lower bound.\n\nIn experiments, the authors show that their model outperforms FNO on Darcy Flow and Navier-Stokes while using signfiicantly fewer parameters. For irregular domain problems where FNO doesn't apply, DINOZAUR has good performance compared to GINO while using over 200x fewer parameters.\n\nThe authors show that DINOZAUR performs well in some measures of UQ (miscalibration area, interval score) compared to other UQ methods. However, for negative log likelhood (NLL), DINOZAUR does not do as well as Laplace approximation.\n\nFinally, the authors show that DINOZAUR uses significantly less VRAM than the FNO (Figure 4).", "questions": "1. Does the presense of the aleatoric uncertainty matter? Can it be removed and the only source of uncertainty come from the diffusion parameters?\n2. Would it be possible to modify the aleatoric uncertainty to vary throughout the domain and depend on the input? For example, you could train another neural operator that takes the input and returns $\\{\\sigma_n\\}_{n=1}^N$.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Overall, I think this is a solid paper that merits acceptance. The paper is clearly written and solves a well-motivated issue with FNO. While the model performance doesn't seem to blow the competition out of the water, the DINOZAUR method uses significantly fewer parameters than FNO and GINO and as a result has much better VRAM performance. I also think incorporating UQ by assigning priors to the diffusion parameters is sensible and it appears to do well in capturing areas of high uncertainty (Figure 3).\n\n### Strengths\n- The architectural changes for DINOZAUR are clearly explained and motivated by a physical model (heat equation).\n- Uncertainty quantification is added in a sensible way.\n- The experiments are extensive and cover all aspects of model performance (relative MSE, UQ, computation).\n- Even thought the architecture is much smaller and arguably simpler, DINOZAUR delivers comparable performance to much larger models. This is important for scaling. Also, from Figure 4, FNO uses 40 Gb of VRAM for training on ShapeNet while DINOZAUR  uses about 12 GB. The latter can fit on a modern consumer GPU, while the former requires a professional card.\n\n### Weaknesses\n- The model performance is not dramatically improved across the board. It would be nice if DINOZAUR could be scaled up to the same number of parameters as FNO/GINO and delivered more outperformance.\n- It doesn't make sense that the aleatoric uncertainty (controlled by $\\sigma$ would be uniform across the whole domain and for different inputs. It also weakens the spatially-correlated uncertainty from the diffusion priors shown in Figure 2 (See my question below).", "was_revised": true, "final_justification": "I had no issues in my original score that needed to be addressed. There were no major concerns that I missed that were brought up by other reviewers.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and constructive feedback. We appreciate the recognition of our contributions, including the physical motivation behind DINOZAUR’s architecture, the integration of uncertainty quantification, and the breadth of our experimental evaluation. We are pleased that the reviewer found our approach to be clear and well-motivated, and that the efficiency of DINOZAUR—in both parameter count and VRAM usage—was acknowledged as a meaningful strength.\n\nBelow, we respond to the reviewer’s concerns and questions.\n\n**The model performance is not dramatically improved across the board. It would be nice if DINOZAUR could be scaled up to the same number of parameters as FNO/GINO and delivered more outperformance.**\n\nIndeed, we see the model’s scalability as one of the main strengths of our method. While it is certainly possible to scale up DINOZAUR by increasing the capacity of its internal components (e.g., by deepening the MLPs or increasing the number of hidden channels within MLPs), we deliberately chose to keep the model architecture closely aligned with that of the baselines to ensure that the observed performance gains stem from our design changes. Our primary goal was to keep the architectural components — particularly the block structures — as comparable as possible to those used in the baseline models. \n\nAs the second weakness concerning the aleatoric uncertainty is mainly expressed through the questions, we will answer them hoping that would clarify the reviewer’s concern.\n\n**Does the presense of the aleatoric uncertainty matter? Can it be removed and the only source of uncertainty come from the diffusion parameters?**\n\nThe scalar $\\sigma$ in our probabilistic formulation (Equations 9–10) models aleatoric uncertainty, which accounts for inherent data noise. It plays a role by shaping the loss through the Gaussian negative log-likelihood (as part of the ELBO), so it is needed for stable training. Even if the data contains little noise, assuming a $\\sigma$ bridges the gap between model limitations and the true data-generating process.\n\nIn general, Bayesian inference distinguishes aleatoric and epistemic uncertainty. This distinction allows us to produce predictive distributions that incorporate either only epistemic uncertainty (eq. 10 with $\\delta(\\mu) = \\lim_{\\sigma \\to 0} \\mathbf{N}(\\mu, \\sigma)$), or both epistemic and aleatoric uncertainty (eq. 10 as written). We expect inference to be improved when aleatoric noise assumptions match the true process. \n\nWe assume there is little noise in the examples presented, and that this noise, if present, is not spatially correlated, which motivates modelling aleatoric uncertainty with independent Gaussian noise at every vertex with common variance. It should be noted that this iid, finite-variance assumption is commonly employed, albeit implicitly, by any model that uses an MSE loss for training. Under this assumption, $\\sigma$ is expected to be small in our datasets, and that is the case we observe in our trained network: the learned $\\sigma$ is around 0.008. Because $\\sigma$ is negligible, we omit it during prediction and estimate predictive uncertainty solely via posterior sampling of the diffusion parameters, which capture epistemic uncertainty.\n\nIn summary, while $\\sigma$ shapes the loss landscape and stabilizes training, it does not affect test-time uncertainty reported in our results.\n\n**Would it be possible to modify the aleatoric uncertainty to vary throughout the domain and depend on the input? For example, you could train another neural operator that takes the input and returns ${\\sigma_n}_{n=1}^N$.**\n\nYes, in principle, this is possible. One could introduce an additional output from the final MLP layer to model spatially dependent $\\sigma_n$ values. However, in our experiments, we found that a global scalar $\\sigma$ was sufficient to enable effective training and yielded good empirical results."}]

Author rebuttal:

Meta-review: This paper introduces DINOZAUR, a lightweight variant of the Fourier Neural Operator that replaces dense Fourier multipliers with a diffusion-multiplier architecture and integrates a Bayesian uncertainty quantification (UQ) mechanism. The diffusion multiplier, motivated by the heat equation, reduces parameter count while preserving predictive accuracy, and the Bayesian formulation produces spatially correlated posterior uncertainties. Experiments on multiple PDE benchmarks demonstrate favorable trade-offs in accuracy, memory usage, and UQ compared to FNO and GINO baselines.

The work is well-motivated and clearly presented, with a physically grounded architecture and principled uncertainty estimation. Strengths include significant parameter reduction, competitive predictive performance, efficient VRAM usage, and the effective use of gradient features to mitigate isotropic diffusion limitations. Weaknesses noted by reviewers include limited UQ evaluation, reliance on single-run results without reported variance, and some minor presentation issues.

Overall, the contributions in efficiency, scalability, and integrated UQ are significant, and I recommend ACCEPT.
Decision: Accept (spotlight)