Spectral Convolutional Conditional Neural Processes
Authors: Peiman Mohseni
Nick Duffield
Venue: NeurIPS 2025 poster ()
Keywords: neural process
stochastic process
meta learning
fourier neural operator
Primary area: probabilistic_methods
TL;DR: SConvCNPs replace local convolutions in ConvCNPs with global Fourier-based convolutions, efficiently capturing long-range dependencies.

Abstract: Neural Processes (NPs) are meta-learning models that learn to map sets of observations to approximations of the corresponding posterior predictive distributions. By accommodating variable-sized, unstructured collections of observations and enabling probabilistic predictions at arbitrary query points, NPs provide a flexible framework for modeling functions over continuous domains. Since their introduction, numerous variants have emerged; however, early formulations shared a fundamental limitation: they compressed the observed data into finite-dimensional global representations via aggregation operations such as mean pooling. This strategy induces an intrinsic mismatch with the infinite-dimensional nature of the stochastic processes that NPs intend to model. Convolutional conditional neural processes (ConvCNPs) address this limitation by constructing infinite-dimensional functional embeddings processed through convolutional neural networks (CNNs) to enforce translation equivariance. Yet CNNs with local spatial kernels struggle to capture long-range dependencies without resorting to large kernels, which impose significant computational costs. To overcome this limitation, we propose spectral ConvCNPs (SConvCNPs), which perform global convolution in the frequency domain. Inspired by Fourier neural operators (FNOs) for learning solution operators of partial differential equations (PDEs), our approach directly parameterizes convolution kernels in the frequency domain, leveraging the relatively compact yet global Fourier representation of many natural signals. We validate the effectiveness of SConvCNPs on both synthetic and real-world datasets, demonstrating how ideas from operator learning can advance the capabilities of NPs.

Spectral Convolutional Conditional Neural Processes
Peiman Mohseni Texas M University
Nick Duffield Texas A&M University duf filffillll dng@tamu. edu
peiman. moomen. @tami..aa....... edu
Abstract
Neural Processes (NPs) are meta-learning models that learn map sets of obser- vations to approximations of the corresponding posterior predictive distributions. By accommodating variable-sized, unstructured collections of observations and enabling probabilistic predictions at arbitrary query points, NPs provide a flexible framework for modeling functions over continuous domains. Since their intro- duction, numerous variants have emerged; however, early formulations shared a fundamental limitation: they compressed the observed data into finite-dimensional global representations via aggregation operations such as mean pooling. This strategy induces intrinsic mismatch with the fiinite-dimensions nature of the stochastic processes that NPs intend to model. Convolutional conditional neural processes (ConvCNPs) address this limitation by constructing dimeniioaai functional embeddings processed through convolutional neural networks (CNNs) enforce translation equivariance. Yet CNNs with local spatial kernels struggle capture long-range raage dependencies without resorting to large kernels, which impose significant computational costs. To overcome this limitation, we propose spectral ConvCNPs (SConvCNP,) which perform global convolution in the frequency domain. Inspired by Fourier neural operators (FNOs) for learning solution opera- tors of partial differential equations (PDEs), our approach directly parameterizes convolution kernels in the frequency domain, leveraging the relatively compact yet global Fourier representation of many natural signals. We validate the effectiveness SConvCNPs on both synthetic and real-world datasets, demonstrating how ideas from operator learning can advance the capabilities of NPs.
1 Introduction
Stochastic processes offer a mathematical framework for modeling systems that evolve with inherent randomness over continuous domains such as time and space. They underpin wide range of scientific applications- ffro spatio-temporal climate dynamics to biological and physical systems-theree
motivating the development of machine learning methods that learn from data generated by
such phenomena Mathieeu et al., 2021 Vaughan et al.) 2021 Allen et al., [025 Ashman et al.)
2025
Dupont et al.] [2021] Among classical approaches, Gaussian processes (GPs; Rasmussen
al. 2006) provide a Bayesian framework with closed-form inference and uncertainty quantification.
However, their cubic computational cost from matrix inversion and the difficulty of specifying suitable
eernels-especiillly in high-dimensional settings-limi their scalability.
Motivated by the success of deep neural networks in large-scale function approximation, neural network- based alternatives have emerged. Neural processes (NPs; Garnelo et al. 2018a b) exemplify this paradigm, combining ideas from GPs and deep learning within meta- meta-learing framework. By exposing the model to multiple realizations of an underlying stochastic process, each treated as distinct task, NPs learn shared structures across tasks parameterize neural mapping that directly gives an approximation of the corresponding posterior predictive distribution [Bruinsmal 2024). Once trained, the model enables efficient probabilistic predictions new tasks without further training.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Since the introduction of conditional neural processes (CNPs, Garnelo et al. [2018a] as the first class within the NPs family (Jha et al. 2022. numerous extensions have been proposed to enhance their effectiveness. One prominent line of work focuses on incorporating explicit inductive biases into CNPs in order to better capture the symmetries that commonly arise in scientific applications
[Gordon et al. 2019 Kawano et al. 2021 Holderrieth et al. 2021 Huang et al. 2023, Ashman
(ct al. 20242 bj. Another major direction seeks to move beyond the mean-field factorized Gaussian predictive distributions to which CNPs are limited. A widely adopted approach augments CNPs with stochastic latent variables, giving rise to the family of latent neural processes (LNPs; Garnelo et al.
[2018b), Louizos et al. [2019]. J. Wang and Van Hoof [2020), Foong et al.] [020] Lee et al. [2020].
Volpp et al. 2021], ,, Wang et al. 2022] J. Wang and van Hoof [022]. Kim et al. [2022] Jung and Park
[0023]] Lee et al. 2023 Xu et al. 2023 D. Complementary efforts explore autoregressive prediction
schemes [Bruinsma et al. 2023] Nguyen and Grover [022]. Gaussian predictive distributions
with non-diagonal covariances [Bruiisma et al.]| 2021 Markou al.
2022 and quantile-based
parameterizations of the predictive distribution [Mohseni et al. [2033]
This work focuses on CNPs, particularly convolutional CNPs (ConvCNPs; Gordon et al. [2019]
which were the first to endow NPs with translation equivariance. ConvCNPs introduce the convolu- tional deep set construction which characterizes a broad class of translation-equivariaan mappings over finite, potentially unstructured sets of observations as a composition of a functional embedding with a translation-equivane equivariant operator, typically realized through convolutional neural networks (CNNs; Fukushima 1980], LeCun et al. 1989. 1998)..
Despite their effectiveness, ConvCNPs can struggle to aggregate information from observations spread across large spatial domains challenge that becomes particularly pronounced in sparse data regimes. This limitation stems from their reliance on local convolutional kemels with small receptive fields, which hampers their ability to model long-range dependencies. A natural remedy is to enlarge the kernel size to extend the receptive field; however, this approach rapidly increases the number of model parameters and computational cost [Romero et al., 2021]. Alternatively, transformer-base architectures can capture long-range interactions but incur quadratic, rather than linear, computational complexity in the number of observations (Vaswani et al., 2017 Nguyen and Grover 2022].
In this work, we pursue an alternative paradigm that represents functions in the frequency domain, inspired by the well-established observation that many natural processes exhibit energy concentration in low- oowfrrequency bands [Field] 1987 Ruderman and Bialek 1993 Wainwright and Simoncelli 1999]. This property allows for efficient approximation using only subset of dominant spectral coefficients, enabling tractable computations while preserving the signal's 's global structure. By parameterizing convolution kernels directly in the Fourier domain over a finite set of frequencies and leveraging the convolution theorem, we can attain large effective receptive fields without incurring prohibitive computational costs.
While spectral methods have been extensively studied in neural operator learning for partial differen- tial equations (PDEs, Chen and Chen [1995], Kovachki al. [202T 2023)), their application within NPs framework remains relatively unexplored. To bridge this gap, we propose spectral convolutional conditional neural processes (SConvCNPs)
class of models that adopt Fourier neural operators (FNOs: Li et al.; [2020a]) to realize global spectral convolution while maintaining computational efficiency. Across suite of synthetic and real-world benchmarks, SConvCNPs perform competitively with state-of-the-art baselines, illustrating how ideas from neural operators can enhance the flexibility and performance of NPs.
2 Preliminaries
2.1 Fourier Neural Operators
Neural operators [Chen and Chen 1995 Li et al. 2020a Kovachki et al. 2023 are neural network architectures designed to learn mappings between function spaces rather than fiite-ddimensional vectors. Like conventional feed-forward networks, they consist of stacked layers that alternate between operator- baaed transformations and pointwise nonlinearities. A common transformation is an integral operator with kernel A : X x X + y, acting on an input function v: X' + y as
<[ul(x) = k(2, 8) u(s) ds,
2

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where X' = = and y = R are Euclidean spaces with de. dy E N. While this work focuses on operators linear in v, nonlinear formulations have also been explored, including continuous formulations of softmax attention [Ashman et al. 2024a Calvello et al. 2024]. When the kernel stationary-thal is, a(a, 8) = K(x - - the operator reduces to convolution, KC[u] = R. v, connecting neural operators to CNNs where & is parameterized by learnable weights.
Convolutional kernels are typically spatially local with limited receptive fields [Luo al. 2016 Peng et al. 2017 Wang et al. 2018]. Modeling long-range dependencies thus requires large kernels, sub- stantially increasing parameter count. The Fourier Neural Operator (FNO; Li et al.
2020a addresses
this limitation by exploiting the convolution theorem [Bracewell and Kahn 1966 Oppenheim 1999
which expresses convolution as
Klb](x)= = (E))) 6(E) (x),
where f ; F[f] denotes the Fourier transform of f. and FF denotes the inverse Fourier transform. Rather than parameterizing in the spatial domain, the FNO learns its Fourier representation directly. When e is approximately band-limited-iee. ({E) ) has negligible energy for ||]|| > 6o 5o-high- frequency components can be truncated with minimal information loss. This property, observed in
many natural signals [Field 1987 [Ruderan and Bialek, 1993 Wainwright and Simoncelli 1999). allows setting R({) = O outside the retained band without significant degradation.
In practice, functions are accessible only through discrete samples, requiring the discrete Fourier transform (DFT) for domain transitions. Given samples of on uniform grid C X, the FNO applies the DFT via the fast Fourier transform (FFT: Cooley and Tukey [1965]. Frigo and Johnson (0055). The resulting spectrum is truncated a finite set of frequency modes C 12 assumed to capture most of the signal's ' energy. For each retained frequency & E E the Fourier kernel is parame- terized by learnable complex weights We E Cdy, so that R(E) = We This formulation implicitly imposes spatial periodicity, as the kernel is represented using discrete harmonics (equivalently, a Dirac comb in frequency space). After pointwise multiplication frequency space, an inverse FFT returns the operator output to the spatial domain.
2.2 Neural Processes
Let P()**) denote the space of probability measures over measurable functions f - Y, which we interpret as stochastic processes indexed by X. We assume an unknown data-generating process y P()*), from which latent functions drawn. A task consists of finite number of noisy input-output observations generated from a realization f ~ μe, partitioned into a context set and
query set:
D= (De, Dy); D. = (Ite, Rc.
D, = ((
(2)
Observations are generated according to
ye,k = f( + Co,ks
Dq = fIx + Cgg
where cg Eq.i lid N(0, oi) and o. > 0 denotes the observation noise standard deviation.
Neural processes (NPs; Garnelo et al. 2018a bl) constitute class of models that use neural networks
to learn a mapping 7 S(X x D) P() where S(X x x)) denotes the collection of all finite subsets of X x y. Given a context set De., outputs stochastic process intended to approximate the Bayesian posterior over functions induced by μt and conditioned on D.. Typically, this process is specified implicitly through its
marginals [Garnelo et al., 2018b Bruinsma Thy ct al. 2021 Bruinsma 2024 Mathieu et al. 2023). Let X,, = (x and y. = (1g,)) We denote by (x;((x the hntte-dimensional distribution of the process nDD.] evaluated at Xg. and by
Px, (. (Da) the corresponding finite-dimensional marginal of the true posterior process. Informally,
the NP approximation aims to satisfy
r/De: x. 1.) x Px, (- Dz).
for all x, and D. [Bruinsma et al. 2021 Bruinsma 2024]. With slight abuse of notation, we write PPP x2) (Ya, for the density of this citeedimmensinall distribution evaluated at yg-
MMore generally, a matrix-valuce parameterization We X is used, where is the number of output channels, enabling joint mixing of input channels and projection to different dimensionality.
3

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In this work, we focus on conditional neural processes (CNPs; Garnelo et al. [2018a] which restrict these sniee-dimensional distributions to mean-field Gaussians, i... D, x,1 I n n/Pei Fg where each marginal n[Dr; is Gaussian. Fach predictive marginal is typically parameterized via a two-stage decoder architecture [Bruinsmal 2024 Ashman et al. [024a] 2025]. The encoder &e S( x y) - H maps the context set 'Dd to a latent representation, while the decoder sp H + O maps this representation to a function that assigns, to each query Zg parameters (Ta,1) E O of the predictive distribution n/ Zq,]]-
The vanilla CNP encoder summarizes De using a crrmmutattoo-invarrinn architecture Q et al. 2017 Zaheer et al., 2017]. Each context pair (Ie, Vc.kk) E De is independently mapped by a shared network hinit- dimensionnnii embedding Ec. which are then aggregated- -typicllyy via mean pooling-ing into single representation e.. Notably, this encoding is independent of the query locations. The decoder then combines & with each J.g.1 to parameterize the Gaussian predictive distribution.
Although sum- sum-pooling aggregation provides universal approximation guarantees (Zaheer et al. [017] Bloem-Reddy and Tch 2020], NPs employing such mechanisms often exhibit underfitting in practice Kim et al. 2019). Prior works have partly attributed this phenomena to two primary factors (Xu
al. 2020 (I) the limitation of summaries with prespecified finite dimensionality in representing context sets of arbitrary size [Wagstaff el al. 2019), and (2) the shortcomings of simple sum or mean
pooling operations to effectively capture rich dependencies between different points [Xu et al. 2020
Nguyen and Grover 2022).
Since NPs address inherently functional learning problems, it is natural to consider embeddings that themselves take the form of functions. Gordon et al. 2019) introduce a framework for translation- equivariant prediction maps over sets, satisfying
n (1 + N.yo) (Ies yo) E Dol:xn] -n(D:; -T]
for all translations т, context sets De, and collections of query locations Xg. where subtraction by acts pointwise, i.e., Xg = (Egg -- They show that a broad class of such maps can be written n[Dr] = ya pp [Dr], where the functional embedding is defined by
Pe (e(
Σ o(vc) (x - -xe).
(3)
(so.Ne)ED.
Moreover, yu : H + C) is a translation-equiivarinn decoder acting on a function space H. Co(X. )) denotes the space of bounded continuous functions fromx' to }, d(y) (1, and Ve : X + R is a continuous strictly positive-definite kernel, typically Gaussian.
In convolutional conditional neural processes (ConvCNPs). the functional embedding y [P.] is evaluated on a uniform grid G C covering the joint support of x, and Xg. yielding the discretized representation (fe (P((D)((zg))) (x,))),, eg. This representation is processed by Sa, and predictions at query locations are obtained via kernel interpolation:
9011
Σ (vea ve) Val -gg);
1= 1, Pg9
(4)
F,E9
where vha is another strictly positive-definite kernel. This interpolation step may be viewed as part of the decoder, preserving the overall encoder-decoder abstraction.
3 Spectral Convolutional Conditional Neural Processes
The decoder S in Conv CNPs is typically parameterized using standard CNNs such as U-Net [Ron neberger al. [2015) or ResNet [He et al. 2016). These architectures employ discrete convolutional kernels finite sets of learnable parameters that define localized filters operating over neighboring grid points. The kernel size, fixed priori, determines the receptive field of each convolution (Ding
et al. 2022 and is generally much smaller than the overall extent of the input signals in the physical domain Romero et al.] 2021 Knigge al., 2023). This locality constraint fundamentally limits model's 's ability to capture long-range dependencies and to integrate information from observations
2 'More generally, σ(v) = (1, y. V.....," M), where M accounts for repeated Inputs.See Gordon et al. 2019
and Bruinsma 2024 for details.

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distributed across large spatial or temporal domains [Peng et al. 2017 Wang et al. 2018 Ramachan dran al. 2019 Wang al., 2020]. The issue becomes particularly pronounced when handling sparse or irregularly sampled data, where effective global reasoning cannot emerge solely from local convolution operations. Although enlarging the convolutional kernel increases the receptive field, it results in rapid escalation of both parameter count and computational cost. Transformer-baas based architectures mitigate this issue by enabling explicit global interactions [Vaswani et al., 2017]: how- ever, they incur quadratic, rather than linear, complexity in the number of inputs, rendering them impractical for large context sets unless one resorts to approximation schemes [Nguyen and Grover
2022
Feng et al. 2022 Ashman et al. 2024 2025
To overcome this limitation without relying on prohibitively large filters or expensive transformers, we exploit the Fourier representation of signals. This is motivated by the well-established observation that many natural signals are approximately band-limited (see Section 2.1. implying that their Fourier representation offers a more compact encoding relative to its physical-domai counterpart. Concretely, we instantiate ye via spectral convolution modules based on equation | This substitution effectively enlarges the receptive field, enabling the model to capture global structures from sparse or irregularly sampled data without incurring a parameter count explosion. We refer to the resulting models as spectral convolutional conditional neural processes (SConvCNPs).
Computational Complexity. The computational cost of SConvCNPs comprises three parts: ()
O(DD. [[]) to compute the discretized functional embedding on the grid G (equation 3); (ii)
O( log [G]) for the FFT-based spectral convolutions (equation 1 and (iii) ( [ to in-
terpolate grid embeddings at the query locations (equation 4. Overall, the total complexity is
(IG1(DD) + log g + (D)))) matching the ((g)) [ + ++ (D,1) complexity of ConvCNPs up
to logarithmic factors. Both architectures therefore scale linearty with task size. A key limitation, however, is that [9] grows exponentially with the input dimension dz, restricting these methods to low-dimensional domains. In contrast, transformer-basee
NPs (TNPs; Kim al. [2019] Nguyen and Grover 2022). Feng et al. [2022] Ashmma et al. (2024a) ) avoid gridding entirely and thus scale more gracefully with input dimensionality. Instead, their computational cost scales quadratically with the task size, typically as DD.I + 12,1),, with minor variations depending on the specific
implementation (see Section B.1.1..
Positional Encodings The convolution operator preserves translation equivariance under the Fourier transform (see equation[]p. However, practical FNOs implementations often include explicit positional
information to improve predictive accuracy (Li et al. 2020a Tran et al.
202 Gupta et al. 2021
Rahman et al. 2022b Helwig al. 2023 Tripura and Chakraborty 2023 Liu al. 2023a Li et al.
2024
Accordingly, we augment the functional embedding &e [D. (x) with positional information:
(x) = (p. x))
While this augmentation explicitly breaks translation equivariance, we empirically observe perfor- mance improvements (see Section B.5). An interesting direction for future work is to investigate relative positional encodings [Shaw et al. 2018 Su et al. 2024), which can provide spatial context while preserving translation equivariance.
Discretization Sensitivity of DFT. Unlike the continuous Fourier transform, the DFT-and there- fore the FFT-depends inherently on the grid G on which [P][Dr] (or (D.) )) is represented. This dependence arises from both the grid resolution and its physical extent: changing either alters the resulting Fourier coefficients and can lead to inconsistent behavior in the outputs (see Section A). Sensitivity resolution is not unique to DFT; CNNs exhibit analogous issues [Raonic et al. 2023 Bartolucci et al. 2023). For example. Conv CNPss mitigate this effect by fixing the grid resolution. While spatial CNNs become stable once the resolution fixed, DFT-based methods remain sensitive unless both the resolution and the physical range are controlled. Accordingly, we fix both, choosing domain sufficiently large to cover all context and query inputs across tasks. When this is impractical, the domain can instead be divided into (possibly overlapping) fixed-size patches, with spectral convo- lutions applied independently to each patch and the outputs aggregated- -mirroring the mechanism of standard convolution layers. Related ideas in efficient transformer architectures suggest that this is promising direction for future work [Beltagy et al. 2020 Zaheer et al. 2020 Liu et al.) 2021 Ding et al. 2023
5

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Context
Query
Model
Ground truth
GP Ur--Maimm.S 512
GP- OP-Periadic
Santooot
Square
TE-TNP
ConvCNP
SConvCNP
Figure 1: Example predictions on synthetic data. For each model, the blue curve denotes the predictive mean, and the shaded region corresponds 2 standard deviations of the model's Gaussian predictive distribution. In the first two rows, where the data are generated from Gaussian processes, the ground- truth distribution is shown in purple: the dash-dotted dotted curve indicates the true mean, and the shaded band represents =2 standard deviations around it. Black points denote context observations, and the red point indicates the queries.
Experiments
We evaluate our framework on four regression benchmarks and compare its performance against several members of CNPs family. Specifically, we include the original CNP [Garnelo al. 2018a].
the Attentive CNP (AtCNP: Kim et al. (2019)), the Convolutional CNP (ConyCNP: Gordon et al. 2 the diagonal variant of the Transformer Neural Process (TNP: Nguyen and Grover [2022). ) and the momnslaatoo--uivariian Transformer Neural Process (TE-TNP: Ashman
ATE--ANN et al. [2024a]). For
each experimental setting, all models trained using four random seeds. Performance evaluated using the final aggregated log-likelihood and root-mean-squaree error (RMSE), and we report the mean + standard deviation across runs. Additional details regarding datasets, architectures, and training procedures are provided in Appendix B, Our implementation and experimental code are publicly available at
4.1 Synthetic 1-D Regression
We begin by evaluating models on four synthetic benchmarks generated from distinct stochastic processes: GP with a Matérn-5/2 kernel, a GP with periodic kernel, a sawtooth-wy generator, and square-wave generator. For each benchmark, the parameters of the generative process-kerne hyperparameters for the GPs, frequency and direction for the sawtooth generator, and frequency and duty cycle for the square-wave generator are sampled randomly. Complete experimental details are provided in Appendix B.I Table reports the average evaluation metrics over 1,000 test batches, each

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Table 1: Comparison of predictive performance across methods synthetically generated tasks. Lower RMSE and higher log- likelihood indicate better performance. For each metric and experimental setting, boldface denotes the top-two performing models.
Model
Metric
Data
CNP
ArCNP
TNP
TE-TNP
ConvCNP
SConvCNP
Matérn 5/2
-0.54
0.322..00
0.29
0.28 20.00
-0.30
0.29
Periodic
-1.20=0.00 oo
00.7
-0.76+0.03
0.57
20.02
0.773
00.66
Log-likelihood
Sawtooth
-0.90 oo
20.00
2..00
0.90
20.30
0.82
Square
-1.39 00
-1.41
1.33+4.02
-..17 =0.06
1.35
-1.18
Matérn 5/2
0.50
0.45 20000
0.4400.00
0.44
0.45 10,00
0.45
Periodic
0.81
0.65
00002
+0.01
0.57+0.00
0.53
RMSE
Sawtooth 0.58
0.58
0.58 0.00
0.58
0.40
0.24
Square
0.98
1.00
0. 0.93
0.81
0.89
0.79
containing 16 tasks (each task defined in equation 2). For every task indeeindennn of batch-
Te and Ty are sampled independently from 3,333. For each test batch, the number of context points, shared by all tasks within that batch, is drawn independently according to n. ~ U[, 25). The number of query points, however, is fixed at n, = 256 for all the test tasks. As shown, SConvCNP consistently outperforms or closely matches strong baselines, including TE-TNP and ConvCNP, which represent the current state of the art. Figure provides qualitative comparisons of predictive maps produced by SConvCNP, ConvCNP, and TE-TNP.
For the sawtooth wave benchmark, we were unable to successfully train either TNP or TE-TNP. Across the configurations we dnteepteddincludde increased model capacity and multiple random- ized initializations- -their predictions consistently collapsed to zero. We hypothesize that this failure mode stems from spectral bias, the tendency of neural networks to favor low-frequency structure over high-fiegghhee frequency components Rahaman et al. 2019 Ronen et al. 2019 Basri et al. 2020 Tancik cet al. 2020 Fridovich- Keil et al. 2022]. Recent findings by Vasudeva al. 2025 further suggest that this bias is exacerbated in transformer architectures compared with convolutional networks. This interpretation aligns with observations by Nguyen and Grover 2022). who originally introduced TNPs. They report degraded performance on GP samples drawn from a periodic kernel-reflecter in poor log-liielihood scores despite strong results experiments on periodic-kerne GP tasks (Table T do not replicate this limitation. key distinction is that we employ the Efficient Query TNP (EQTNP) architecture Feng et al. (2022). rather than the original TNP design (Nguyen and Grover 2022 (see Appendix 1. for architectural details). Finally, although sawtooth and square -quare-wave wave functions share similar discontinuity and figh-rreeeeccy characteristics, Table [ shows that TNP and TE- ENP do not collapse when trained on square-wave signals. Investigating the source of this discrepancy falls beyond the scope of this work but represents an intriguing direction for future study.
Mlttrn-kernee tasks. Interestingly,
4.2 Predator-Prey Dynamics
We next assess performance simulated trajectories from stochastic variant of the LolterVolterra predator-prey system [Lotka] 1910, Volterra 1926). following the formulation in BBuinssaa et al. [023] Let U, and V; denote the prey and predator populations time respectively. Their dynamics evolve according to the stochastic Voltera system
d;= = dt BU,V, dt + oUy dB(),
dV, UU dt + SU.V dt + oV, dB!")
where B(a²) and B(2)) are independent Brownian motions.In the deterministic component of the dynamics, U, grows exponentially rate a, while Vn decays at rate y. The bilinear interaction terms 3U,V and 8U,V, model predation and the corresponding transfer of biomass from prey to predators. To account for stochastic fluctuations commonly observed in empirical population counts, the dynamies are augmented with multiplicative noise terms oUH d(()) and a V/' d2))) Here, a controls the noise magnitude, while y determines how the variability scales with population size.
7

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Table 2: Comparison of predictive performance across methods tasks constructed from the Lotka- Volterra system simulation. Lower RMSE and higher log log-liiklihood indicate better performance. For each metric and experimental setting, boldface denotes the top-twe performing models.
CNP
AACNP
TNP
TE-TNP
ConvCNP
SConvCNP
Log-tikelitood
0.27 20.01
20.00
0.1220.00
0. 0.16
0. 4.14
0.14
RMSF.
0.37
0.31 20.00
0.31 40.00
0.30
0.31440.00 40.00 0.30
Ground muth
Context
Model
Prey
Predatox
TE-TNP
ConvCNP
SConvCNP
Figure Illustrative predictions on simulated Lotka-Volten Volterra predator-prey trajectories. Black markers denote the context observations. The blue curve shows the predictive mean, while the shaded region corresponds to the 22 standard deviation interval the Gaussian predictive distribution. Ground-truth population trajectories from the simulator are plotted in red.
We simulate trajectories over a dense uniform grid on / E [-10, 100), discarding the initial 10 years burn-in. For each task, context points v. and query points 2i are drawn independently and uniformly from the retained interval [0, 100). Population values (corresponding to [U, Vel) and time inputs (corresponding to x t) are rescaled by factors of 0.01 and 0. respectively, before being fed to the models. A complete description of the experimental protocol is provided in Appendix B.2
Table reports average evaluation metrics over 1,000 test batches, each containing 16 tasks. For each batch, the number of context the number of query points is fixed ng 256. SConvCNP achieves log-likel likelihood performance comparable to ConvCNP, and both are competitive with TE-TNP. In terms of RMSE, SConvCNP matches TE-TNP, indicating that its predictive uncertainty intervals slightly wider. Qualitative comparisons of predictive maps for the top three models appear in Figure 2
shared across all tasks-is sampled as n. ~ (5 25). and
4.3 Traffic Flow
For our third experiment, evaluate on the California traffic-flow dataset from LargeST [Luu et al. 2023b]. This dataset comprises five years (2017-2021) of traffic measurements recorded every minutes by approximately 8,600 loop-detector sensors deployed across California's 's highway network. We focus on the year 2020, where traffic patterns are expected to exhibit heightened variability due to the abrupt onset of the COVID-19 pandemic. For each sensor, segment its year-long time series into non-overlapping 14-day windows and downsample each window by factor of (from 5-minute 30-minute resolution). Each window is treated dense trajectory from which individual tasks are constructed. Additional details of the experimental setup are provided in Appendix B.3
Table Comparison of predictive performance across methods on tasks constructed from California traffic flow measurements. Lower RMSE and higher likeliied indicate better performance. For each metric and experimental setting, boldface denotes the top-two performing models.
CNP
AICNP
TNP
TE-TNP
ConVCNP
SComvCNP
Log-likelihoed f
1.73
1.80
1.93
1.7240.02
1.98 10.02
2.05
RMSE
0.05
0.05
0.04
0.05
0.04 10.00 0.0440.0

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Ground tuth
Context
Model
TE-TNP
ConvCNP
SConvCNP
Figure 3: Illustrative predictions on the California traffic-flow dataset. Black markers denote the context observations. The blue curve shows the predictive mean, while the shaded region corresponds to the +2 standard deviation interval of the Gaussian predictive distribution. Ground-truth traffic measurements are shown in red.
Table reports evaluation metrics averaged over 66,586 test tasks, constructed from 2,561 held- out test sensors and partitioned into batches of size 32. For each batch, the number of context points-sharee across all tasks-is drawn as ne ~ 215, 25), and the number of query points is fixed at n, 50. Across all metrics, SConvCNP achieves the best performance, attaining the highest log-likelihood and lowest RMSE among the baselines. Figure provides qualitative comparisons of the predictive maps produced by the three models.
4.4 Image Completion
For our final experiment, we evaluate model performance on an image-completion task for- mulated as spatial regression problem, where the model maps 2D pixel coordinates to their corresponding intensity values. We use images from the Describable Textures Dataset (DTD; Cimpoi et al. (2144), and construct each task from a processed x 64 subsampled crop of an original image. Because each task con- tains a large number of context and query pix- unable to fit TE-TNE within computational budget-even after reducing its size-so exclude from this experiment. Additional experimental details are provided in Appendix [B.4] Table reports results aged over 1,880 test tasks, evaluated in batches of 16. For each batch, the number of context points-shared across all tasks-is sampled as as
els,
Ground Truth
Context
TNP
ConvCNP
SComvCNP
Figure 4: Illustrative model outputs for image com- pletion on the DTD dataset. Gray pixels indicate query regions, while the remaining pixels serve as context observations. For each model, query regions are completed using the mean of the pre- dictive distribution.
n. (5, 1024): all remaining pixels query points, so that ny 4096 nie- Among the baselines. SConvCNP achieves the highest log- lideiiilloood and the second-lowen RMSE. Figure presents qualitative comparisons of the predictive means produced by the top three models.
Table 4: Comparison of predictive performance across methods on image-completion tasks con- structed from the DTD dataset. Lower RMSE and higher likeliodddood indicate better performance. For each metric and experimental setting, boldface denotes the top-two performing models..
CNP
ArCNP
TNP
ConvCNP
SConvCNP
Lng-likellhhod f
0..7
1.39
1.39
1.48
1.50
RMSE
0.14
0.08
0.06+0.00
0.08
0.08

---

5
Related Works
The high computational cost of traditional numerical PDE solvers has moti- vated the development of more efficient alternatives based on deep learning Gupta and Brandstetter
Neural PDE Solvers
[0022] Among these, neural operators [Li et al. 2020b Kovachki al. 2023 and in particular
Fourier neural operators (FNOs. Li et al. 2020a). have become popular. Numerous extensions have
since been proposed: Helwig al. [2023] introduced rotation- and reflection-equivarian FNOs;
Cupta et al. [2021] developed multiwavelet neural operators by projecting kernels onto predefined polynomial bases; and Tran al. [222]] reduced model complexity through separable Fourier repre-
sentations. Xiao et al. [024] proposed the Amortized FNO (AM-FNO), which substantially cuts
parameter count by amortizing the Fourier kernel parameterization, while Qin et al. (2024] analyzed FNOs through the lens of spectral bias. KKoshizuaa et al. (2024) provided a mean-field theoretical perspective, and Zheng al. (2024) introduced the Mamba Neural Operator, which couples global
context via state-space model to achieve linear complexity and representation equivalence, [Bar. tolucci et al. [2023] examined continuous-die discrete consistency, showing that it holds for convolutional neural operators |Raonic et al., [023]. Additional works include orthogonal-attention eigenfunction
methods for operator learning Xiao et al. 2023) and hierarchical transformers with frequency-awaar
priors for resolution-invariant super-resolution [uo et al. 2024
Function Space Inference In non-parametric Bayesian modeling, GPs and deep GPs (Damianou and Lawrence 2013) provide flexible function-space space priors with well-calibrated uncertainty estimates, but their computational cost becomes prohibitive on large datasets. This challenge has motivated the development of Bayesian neural networks (BNNs; MMc [1992], Hinton and Van Camp [1993]. Neal 2012)), which combine neural network expressiveness with Bayesian uncertainty quantification. However, specifying meaningful priors over network weights remains notoriously difficult. Recent work therefore reframes Bayesian inference in neural networks as learning a posterior over the
functions induced by stochastic weights [Wolpert] 1993 Qiu al.) [003]. Variational implicit
processes (VIPs: Ma et al.] (2019). [Santana al. [202]]. Ortega et al. [2022] generalize GPs by defining implicit stochastic processes through latent variables, while functional variational BNNs (IBNNs: Sun et al. (2019) enforce alignment between BNN-induced priors and target priors by minimizing a functional KL divergence though this objective can be difficult to compute and, in some cases, ill- iol-posed (Burt et al. 2020]. Subsequent work has aimed to address these limitations
[021]]udner Rudner et al. 2022, Wild et al. 2022 Rudner et al.] 2023 Wu et al. 2024]. Orthogonally, energy-based models have been explored for representing stochastic processes in function space (Yang et al. 2020 Lim et al.
[Ma and Hernández-
2022]. More recently, several approaches
have extended diffusion and flow models to function spaces [Phillips et al. [2022 Dutordoir et al.
2023 Mathieu et al. 2023
Franzese et al. 2023 Lim et al. 2023 Kerrigan et al. 2024 Zhang and
Scott 2025)
6 Conclusion
In this work, we introduced spectral convolutional conditional neural processes (SConvCNPs), class of conditional neural processes that incorporates ideas from operator caamingg-prttiuune Fourier neural operators- intt the neural processes framework, with emphasis convolutional conditional neural processes. Empirical evaluations on a collection of synthetic and real-world datasets indicate that SConyCNPs perform comparably to strong baselines. These results suggest that integrating techniques from operator learning into neural processes is a viable direction for probabilistic function modeling and warrants further investigation.
Acknowledgments
We thank Arman Hasanzadeh and Jonathan W. Siegel for their insightful feedback and constructive suggestions. We also acknowledge the Texas A&M High Performance Research Computing facility for providing the computational resources used in this study. Finally, we thank the anonymous reviewers for their valuable comments and suggestions, which helped improve the quality of this work.
10

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A Discretization Sensitivity of the DFT
To illustrate the sensitivity of the discrete Fourier transform (DFT) the underlying discretiza- tion ioo--and how changes in grid geometry can degrade predictive performance- -eeeeeeeeeweeeeee--e--eeee we consider simple one-dimensional example. Let A < denote fixed discretization resolution, interpreted as the spacing between adjacent grid points. Define the grid
G = (Zo..... Lmi 1)
as uniform discretization of the interval 00, 11. where my = [1/A] and the points are ordered such that In < In+1 for all E {O..... m_ 2). Let h = Ge[De] denote the encoded latent function, and let its sampled values on 91 be
(h(z))
(h(zo)..... ................................ ((((o))).,,,(mm--) h(zm--1))
The DFT of on this grid is given by
my m1-1
((()))) DEg, (1)
Σ h(In)e
12
k E {O..... "......... m1 -1.
n=0
This yields Fourier coefficients the normalized frequencies
E
As discussed in Section 2.1 the Fourier Neural Operator (FNO) implicitly ties its kernel parameteri- zation the specific frequency set E encountered during training. Now suppose the trained model is evaluated on a larger spatial domain, e.g. [0, while maintaining the same resolution A. The corresponding grid
G2 (z/- 2'
has size m2 [2/A], and the associated DFT produces coefficients normalized frequencies
E =
Since my 2 2m,, we have | 2 1. and, crucially, the frequency locations themselves differ. Let I kmax S m, denote the number of Fourier modes retained during training. Even when restricting attention the lowest kmax modes, the frequencies {k/m²) 1umxs do not coincide with (k/m: k= -max=1 This misalignment induces spectral mismatch between training and evaluation, illustrating how changes in discretization-evv at fixed resolution-can disrupt the learned Fourier parameterization and lead to degraded generalization.
B Experimental Details
implementations are written in PyTorch (Paszke et al., 2019] and publicly available at We used single NVIDIA A 1100 GPU with 40 GB of memory for all the computations. Our code is based on the implementations Ashman et al. [2024a] and Bruinsma et al. 2023
B.1 Synthetic 1-D Regression
B.1.1 Model Architectures
This section details the architectures of all CNPs variants used in our experiments. Each model outputs Gaussian predictive distribution parameterized by mean and a pre-softplus [Dugas et al. 2000) scale. The pre-softplus value passed through a softplus transformation, and minimum noise term of 10-5 6 is then added to the resulting scale. The Gaussian factorizes across query points-as as is standard for CNPs (see Section 2.22annd for multidimensional outputs, across output dimensions as well. Thus, predictions assume independence both across query locations and across components of each output vector. Unless otherwise noted, all nonlinearities use ReLU activations [aair and Hinton 2010).
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Conditional Neural Process (CNP): The CNP encodes each context pair (Ze,k, Je.k) E De using separate input and output pathways. The imput Fe,k and output Be,k are first passed through distinct MLPs with two hidden layers each of dimension 256, producing 256-dimensional representations ec.k and (u) cckk' Their concatenation e..: - (s(4] (v) c.kk is then processed by a deeper MLP with six hidden layers of width 256 and final 256-dimensional output layer, yielding an embedding for each context pair. Context embeddings &c, are averaged to form a single aggregated representation e, of dimension 256. For prediction, each query input Iy concatenated with the context representation E, and passed through decoder MLP consisting of six hidden layers of dimension 256. The decoder's s final output layer has dimensionality 2d,, parameterizing the mean and the log-scale parameters of the Gaussian predictive distribution.
Attentive Conditional Neural Process (AtCCDP). The AttCNP implementation follows the de- terministic architecture introduced by Kim et al. [201)) The initial encoding stage mirrors that of the CNP: each context pair (Ze,ks Hc,k) D. is processed by two separate MLPs (each with two hidden layers of width 256), producing embeddings ec, and (9) These are concatenated to co,k'
form cc, |s [ which is subsequently passed through an additional two-layer MLP with hidden width 256. Departing from the CNP, the AtCNP applies self-attention to the set of context embeddings. We use two layers of multi-head self-attention, each with 8 heads (head dimension 32), a feedforward subnetwork with hidden width 256, residual connections, and pre-layer normalization for both the attention and feedforward blocks. The input and output dimensionality each attention layer is fixed at 256. Query representations Eq, are computed using single layer of multi-head cross-attents attention between the (attended) context embeddings and the query embeddings, with the same configuration as above (8 heads of dimension 32, feedforward width 256, residual connections, and pre- nonnoaizazzaii... The decoder is an MLP with six hidden layers width 256. Its final layer outputs 2dy units, corresponding the mean and log-scale parameters of Gaussian predictive distribution.
Transformer Neural Process (TNP). To apply the TNP, we first construct token representations for both context and query points. For each context pair (Ie,ks ye,k) De, form the token
[Te,ks Yo.k: 1],
where the final singleton "I" serves as density flag indicating that the observation Ve,k at Te.k is available. For each query location Egg.. we instead construct
[xng11 0, 0),
where the dummy zero-vector matches the shape of ye. & and the final "0" indicates the absence of an observation [Nguyen and Grover 2022 Ashman et al. 2024a All tokens are then passed through a shared two-layer MLP (width 256), producing initial embeddings Ec, and Eq.l-
In the original TNP architecture [Nguyen and Grover 2022 context and query embeddings are processed jointly by transformer encoder. An attention mask prevents (i) interactions among queries and (ii) context->query attention, ensuring that context representations remain query-independent, while queries may attend to contexts. This design incurs quadratic complexity in the total sequence length.
O([Dol 1P,)²).
Because queries never attend to one another, Feng et al. [022] introduced the Efficient Query TNP, which we adopt here. This variant first applies self-attention context tokens produce updated context embeddings, then processes each query using cross-attention over these updated contexts. The resulting two-branch structure (contexts processed twice, queries once) reduces the overall complexity to
O(D.. 2 (Pyl
Our model uses six transformer layers with eight attention heads (head dimension 32), each followed by a feedforward subnetwork of width 256. We use the standard pre-norm architecture, applying layer normalization before both the attention and feedforward modules. All attention operations share a 256-dimensional embedding dimension. At each layer. we first apply context self-attention and then query-contery cross-attention. Notably, the same multi-head attention parameters are used for both operations; the distinction arises only from which embeddings serve as queries, keys, and values (yiclding self-attention when they coincide, and cross- cross-attention otherwise). This is not
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modeling cettirement-sspaaaeee parameter sets could be used, as demonstrated in Appendix 1.4.1 The final-layer query embeddings are passed to a decoder MLP with two hidden layers (width 256). whose output parameterizes a Gaussian predictive distribution via 2dy units corresponding to the mean and log-scale.
Translation- Dallation-Equivaiiant Transformer Neural Process (TE-TNP). The token construction in the TE-TNP closely follows that of the TNP, with a crucial modification to ensure translation equivariance: input locations are excluded from the token representations. Specifically, for each context pair (Ie,k) Vc.k) E De, we construct the token
We,,: 1].
while for each query location T we use the token
[0, 0].
All tokens are passed through shared two-layer MLP with width 256 to produce initial embeddings E, and Eq.i
The TE-TNP replaces the standard multi-head attention mechanism with the munslaaiio-equivarian attention proposed by AAshman et al.] [2024a]. For each attention head, we first compute the matrix of pairwise scaled dot products between token embeddings, analogous to conventional attention but independent of absolute input locations. In parallel, we compute the matrix of pairwise differences between token locations. We then concatenate the scaled dot-product scores from all heads with the corresponding pairwise location differences. This augmented pairwise similarity representation is processed by an MLP-acting acting an implicit kernel with two hidden layers of width 256 and an output dimension equal to the number of attention heads, yielding the final translation translation-equivarinn equivvriint attention logits.
Apart from this modified attention module, the remainder of the architecture and computational pipeline follows that of the TNP.
Convolutional Conditional Neural Process (ConvCNP): For the ConvCNP, we begin by deter- mining, for each input dimension, the minimum and maximum coordinates observed across both the context and query sets. These extrema are expanded by small margin of 0.1 in each dimension. The interval between the expanded minima and maxima is then uniformly discretized a resolution of 64 points per unit. When necessary- for example, to satisfy the CNN's minimum grid-size requirements-tthe the discretization range is further enlarged while maintaining the same resolution. The resulting one-dimensional grids are combined via a Cartesian product to yield a uniform grid G.
The functional embedding in equation 3 is evaluated on this grid. The Gaussian kernels used in this embedding are initialized with per-dimension length scales set to twice the grid resolution, i.e., 2/64. For multi-dimensional inputs and outputs, the model uses separate length scales for each dimension. Following Bruinsma et al. 2023). the embedding is further normalized using a density channel, defined as
Density(z) = Σ
th (x (xe).
(At. ED.
The functional and density channels are concatenated to form the final grid representation
Density(zg)) E(so. W.)D, d(ye) ( Te)
(5)
Density (5rg)
EgEG
Each grid point is processed independently by an MLP with two hidden layers of width 128. The resulting features are passed to CNN based a U-Net architecture [Ronneberger et al. 2015). consisting of six residual convolutional blocks (kernel size 11, stride 2, 128 channels) in the encoder, followed by symmetric sequence of six transposed convolutional blocks in the decoder, with skip connections following the design of Bruinsma et al. 2023 Since the U-Net downsamples the spatial resolution by factor of 64, we ensure that the constructed grid size is divisible by 64. For the -in-dimensional benchmarks considered in this work, this is achieved by symmetrically enlarging the discretization interval when necessary that the CNN output aligns exactly with the input grid.
To obtain predictive distribution parameters at query locations- may not lie on the grid G- an interpolation scheme analogous to equation S] applied, except that the weighted sum over the
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feature-man map values is not normalized. For this interpolation, we employ a separate Gaussian kernel with learnable length-scale parameters, distinct from the kernel used in the functional embedding. The resulting query-specific embeddings are then processed by a decoder MLP with two hidden layers (width 128), whose output consists of 2dy units parameterizing the mean and log-scale of Gaussian predictive distribution.
Spectral Convolutional Conditional Neural Process (SConvCNP) ConvCNP's U-Net backbone with U-shaped Fourier Neural Operator (FNO) architecture [Li al.
The SConvCNP replaces the
2020a Rahman al. 2022a A residual Fourier block denoted
F(Ca Cout, Sins Sout: m2);
where Gin and Cout are channel dimensions, Sin and Sout are spatial sizes, and my is the number of retained Fourier modes.
The representation from equation 5 augmented with positional encodings (grid coordinates), is first processed by an MLP with hidden layer of width 64 and GELU activations (Hendrycks and Gimpel 2016]. Its output is then passed through h the following sequence residual Fourier blocks:
* L1 F(64, 128, (g], [|9]/2]. 32)
L2 = F(128, 128, [|9]/2], [[9]/4], 32)
. L3 F(128, 256, ||9|/4] /4], 32)
T F(256, 128, ||]|/2] 32)
Ls F(256, 128, |g], 32)
Layers L, - L2 - L. - L,₃ form the contractive extrnvivv path. The final block L5 receives the channel-wise concatenation of the outputs of L4 and L1, yielding the U-shaped skip connection. Its output concatenated with the initial MLP features and processed by final MLP (one hidden layer, width 128, GELU) to produce the SConvCNP representation for the decoder.
of learnable parameters for all models used in Parameter Count. Table 5 summarizes the number our ID synthetic regression experiments.
Table Learnable parameter counts for all models used in the 1D synthetic regression experiments.
CNP
AuCNP
TNP
TE-TNP
ConvCNP
SConvCNP
Number of parameters (million)
1.3
2.1
2.6
3.1
3.8
3.7
Forward run time. Table 6 reports the forward- pass runtime (in seconds) for a batch of 16 tasks during both training and validation across all models evaluated in our 1D) synthetic regression experiments. As described in Appendix [B.1.2] the number of query points is fixed to 256 during validation, whereas during training it is sampled uniformly from U(5, 25).
Table 6: Average forward-pass runtime (in seconds) for a batch of 16 tasks during training and validation across all models in the ID synthetic regression setting.
CNP
AuCNP
TNP
TE-TNP
ConvCNP SConvCNP
Train
0.003
0.004
0.007
0.014
0.009
0.009
Validation
0.006
0.009
0.013
0.023
0.033
0.030
B.1.2 Data and Experimental Setup
We evaluate our methods on four families synthetic 1D regression tasks, each defined by distinct stochastic generative process: Gaussian process (GP) with Matérn-5/2 kernel, a GP with periodic kernel, a sawtooth- wave generator, and a square-wave generator. GP tasks are sampled using GPyTorch Garrdner et al. 2018). All processes include independent Gaussian observation noise.
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Generative Processes. For each task family, process specific hyperparameters are sampled inde- pendently at the task level.
GP with Matérn-5/2 kernel. Functions are sampled from f GP(O, kims/2 + +oi)). where
2 2-1.5
v5 -
v5 x'1)
kmss (2,
22.5
[(2.5)
X
and K2.5 is the modified Bessel function of the second kind. The lengthscale is sampled as
À ~ (0.25, 1). Observation noise standard deviation is do = 0.1.
GP with periodic kernel. Functions are sampled from f GP((, kip + +o/)) where
2sin 2 (n x'l/p)
(1, x²) exp
x2
with period p 2(0.5,2) and lengthscale À 4(0.25, 1). Observation noise standard deviation is do 0.1.
Sawtooth wave. Functions are sampled from GP oi1), where the mean function is
Maaw (x) 2 ((E (ux c)) mod 1) -
with frequency & ~ (400.5,5) 5), direction u E {+1,-1} (sampled uniformly), and phase offset ((00 Observation noise standard deviation is do 0.05.
Square wave. Functions are sampled from ~ (PPGGP(mmmm 1) where the mean function is
May (z) 2 (((E=-c) mad 1)<D) -
with frequency & ~ (0.55 5). duty cycle D ~ 240.25, 0.75), and phase offset c ~ 2[(0, 1).
Observation noise standard deviation is σg 0.05.
Each model is trained for 500 epochs using AdamW Loshchilov and Hutter 2017 with a learning
rate of 10-4 We apply gradient clipping with maximum norm of 0.5 Pascanu al. 2013).
Each epoch consists of 1000 iterations, and every iteration processes batch of 16 tasks, yielding total of 8 million on-the-fly sampled training tasks. For each batch, the numbers of context and query points are independently drawn n, ~ 21, 25) and n, Ld[5, 25). These values are shared across all tasks in the batch. Input locations are sampled uniformly and independently from [-3, 3) for each task.
For validation, we use fixed of 4.096 tasks, organized into 256 batches of 16 tasks. In these tasks, the number of query points is fixed at 256, while the number of context points is sampled using the same procedure as during training. Testing follows the same configuration as validation, except that we evaluate on 16,000 test tasks. Unlike the dynamically generated training tasks, the validation and test remain fixed across all experiments and runs.
B.2
Predator-Prey Model
B.2.1 Model Architectures
The model architectures and parameter counts closely follow those described in B.I.I The only modification concerns the discretization of the functional embedding in the ConvCNP and the SConvCNP. Specifically, we first expand the range defined by the minima and maxima of all context and query coordinates by margin of 0.5 in each dimension. The resulting interval is then uniformly discretized at a resolution of 48 points per unit length.
B.2.2 Data and Experimental Setup
We generate data from stochastic variant of the Lotka- Volterra predator perey system LLtka 1910 Volterra 1926), following the formulation of Bruinsma et al. (2023). Let U, and V2 denote the prey and predator populations time t. Their dynamics evolve according to
dU. aU, dt BU.V, dt oU? dB(2).
dV, -U- dt + SU.V, dt oV, dB
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where B(d) and B, are independent Brownian motions. The deterministic drift recovers classical predator-pre prey behavior: prey grow exponentially at rate a in the absence of predators, while predators decline rate y without prey. The interaction terms B V and 8U, Vú model predation and predator reproduction. Stochasticity enters through the multiplicative noise terms oUı and oV,", where a controls noise intensity and determines how fluctuations scale with population size. Setting 1 yields noise proportional to population level, while v < 1 and y > 1 induce sublinear and superlinear scaling, respectively.
Table Parameter distributions for the stochastic Vootera equations.
Parameter
Distribution
Initial condition U-10
4(55, 100))
Initial condition V-10
U(55, 100])
4((10,,500
u(0004,, 0.08))
Y
u(11002.00
5
4(0.04 0.08))
((0.5, 10))
u([1, 5))
Fixed at 1/6
We simulate trajectories using the parameter distributions in Table 7 over total of 110 years, discarding the first 10 years as burn-in to reduce sensitivity to initial conditions. Integration is performed using the Buler-Maruye
method with time step At 0.022, producing 5000 solver steps over the interval & E [-10, 10, 100). Although the solver runs at this finer resolution, we record states only a uniform grid with spacing 0.05, yielding approximately 2200 selected time points spanning 1-, 100).
To construct task from simulated trajectory, input locations are sampled uniformly and indepen- dently from the recorded time points in [0, 100) and paired with the corresponding values (Uz, Vt). We rescale time by factor of 0.1, mapping the 100-year interval to j0, 10), and rescale population sizes by multiplying the values by 0.01.
Models are trained for 500 epochs using AdamW with learning rate of 10 We apply gradient clipping with maximum norm of 0.5. Each epoch consists of 1,000 iterations, each processing batch of 16 tasks, yielding total of 8 million on-the-fly sampled training tasks. For every batch, the numbers of context and query points are drawn independently as n. ~ 25, 25) and ng 2/ (5, 25): the sampled values are shared across all tasks within the batch.
For validation, we use fixed set of 4,096 tasks, arranged into 256 batches of 16 tasks. In these tasks, the number of query points is fixed at 256, while the number of context points is sampled using the same procedure as training. Testing follows the same protocol validation, except that we evaluate on 16,000 test tasks. Unlike the dynamically generated training tasks, the randomization used construct the validation and test tasks fixed across all experiments and runs.
B.3
Traffic Flow
B.3.1
Model Architectures
The model architectures follow those described in Section B.I.I functional embedding in both the ConvCNP and the SConvCNP, we use an expanded margin of 0.5 and resolution of 48 grid points per unit length.
For the discretization of the
B.3.2 Data and Experimental Setup
We use the California traffic-flow dataset from LARGEST Liu et al. 2023b large-scale benchmark for traffic forecasting, available at htttps://wwa kaggle com/ datasets matasets//luu////arress
The dataset contains traffic-flow measurements from 8,600 loop-detecte sensors deployed across
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California's highway network, collected at 5-minute intervals between 2017 and 2021. In our experiments, we restrict attention data from the year 2020.
As preprocessing step, we discard sensors with more than 50% missing values. For the remaining sensors, missing entries are filled via linear interpolation between observed measurements, with leading and trailing gaps completed using forward and backward propagation, respectively. Sensors are then randomly partitioned into training, validation, and test sets using a 6:1:3 split.
Each sensor's time series is segmented into non-overlapping, continuous 14-day windows (4,032 time steps). yielding 26 windows per sensor. Within each window, timestamps are reset to start at and increase in 5-minute increments. We subsequently downsample each window by a factor of 6, resulting in temporal resolution of 30 minutes. Time indices rescaled to units of days by dividing by (60 x 24). Finally, traffic-flow values are normalized to the [0, 1] range using min-max statistics computed from the training set.
The processed dataset contains:
Training: 5,119 sensors + 133,094 windows
Validation: 853 sensors + 22,178 windows
Test: 2,561 sensors + 66,586 windows
Models are trained for 100 epochs using AdamW with learning rate of 10-4, We apply gradient clipping with a maximum norm of 0.5. Each epoch consists of approximately 4.160 iterations, each processing a batch of 32 tasks. For every batch, the numbers of context and query points are drawn independently as ne ~ 245, 25) and ny ~ 25, 25); the sampled values are shared across all tasks within the batch. A task is formed by independently and uniformly sampling ne + My time steps from a window without replacement. Training tasks are sampled on-the-fly from training windows. For validation and testing, we pre-generate fixed sets of tasks:
Validation: 22,178 tasks (batched in 32), with ny 256 and n. sampled as in training.
Test: 66,586 tasks (batched in 32), with ng = 50 and n. sampled as in training.
B.4
Image Completion
B.4.1
Model Architectures
All models output Gaussian predictive distributions parameterized by mean and a pre-softplus standard deviation. Because pixel values are normalized to [0, 1], the predicted mean passed through sigmoid. The predicted standard deviation is obtained by applying a softplus to the network output, scaling the result by 0.99, and adding a minimum noise of 0.01.
Conditional Neural Process (CNP). The CNP architecture follows the design used in previous experiments (see Section B.I.I, with the only modification being increase in the MLP hidden size from 256 to 512.
Attentive Conditional Neural Process (AttCNP). The ArCNP architecture is likewise based on the design used earlier experiments (see Section B.I.I]. The MLP hidden size is increased from 256 to 512. For the self-attention applied to the set of context embeddings, we increase the per-head dimension from 32 to 64, set the input/output dimension of each attention layer to 512, and increase the feedforward subnetwork width from 256 to 512. The same modifications apply to the multi-head cross-attention used to update query embeddings (eight heads of dimension 64 and feedforward width 512).
Transformer Neural Process (TNP). As with the other models, all MLP hidden sizes are increased from 256 to 512. We adopt the Efficient Query TNP design described in Section B.I.I which employs a two-branch structure in which context embeddings are processed twice and query embeddings once. In contrast to previous sections, we allocate separate parameters for each branch: at each layer, the multi-head self-attention over contexts and the multi-head cross-attention between queries and contexts no longer share parameters. For each multi-head attention module, the feedforward width increased from 256 to 512, and all input/output embedding dimensions are set to 512. All other configurations follow Section B.I.I
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Convolutional Conditional Neural Process (ConyCNP)
Because the data lie on regular grid,
we use the on-the-grie -ethe--rid implementation of the ConvCNP Gordon et al.] 2019]. eliminating the discretization step and improving computational efficiency. Let denote an incomplete image, with unobserved pixels filled with dummy values, and let M2 be the binary mask indicating observed (context) pixels. For multi-channel images, the mask is broadcast along the channel dimension.
As in Section B.I.I the output of the convolutional deep set module has two components: (i) a density channel capturing the spatial distribution of context pixels, and (ii) a kemnel-mmoothed representation. The kernel is implemented via a 2D convolutional layer (kernel size 11, dy input channels, 128 output channels. no bias) with nonnegativity constraint enforced by taking absolute values of the learned weights during the forward pass [Gordon et al., 2019).
The density channel is obtained by convolving this modified filter with the mask M The kernel- smoothed component is computed by first multiplying elementwise with M (setting non-context pixels to zero) and then applying the same modified convolution. Following Gordon et al. (2019) we omit normalization by the density channel, as it did not provide empirical benefits and occasionally introduced instability; however, we retain the positivity constraint on the kernel.
The resulting representation is processed pointwise by an MLP with two hidden layers of width 128. These features are then fed into a ResNet-style CNN [He et al.) (0166 consisting of six residual convolutional blocks (kernel size 11, 128 channels), following the implementation of Bruinssa et al.)
2023 The embeddings corresponding to query pixels are finally passed to a decoder MLP with two hidden layers of width 128.
Spectral Convolutional Conditional Neural Process (SConyCNP) The SConvCNP mirrors the on-the-grid ConvCNP architecture described above, with the distinction that the ResNet backbone is replaced by a U-shaped FNO. To construct positional information, we uniformly discretize the interval [-1, 1] along each spatial axis into grids whose sizes match the corresponding image dimensions. Their Cartesian product yields the 2D positional encoding, which is concatenated with the output of the convolutional deep set. This concatenated tensor is then passed pointwise through an MLP with one hidden layer of width 128 and GELU activations.
Let sh and Sw denote the image height and width, respectively. Using the notation introduced in Section B. I.I the operator consists of the following sequence of residual Fourier blocks:
L F(512, 512, (sh, Sw:, (sh, Sw), 32)
. L2 F(512, 512, (8h, Sw), ([sx/4] 4J, /sw/4]] 8)
- La F(512, 512, ([s,/4), [sw/4]), (Lsn/16) 16], [ (1./1)), ,) 2)
L1 = F(512, 512, ([sn/16], [sw//1]]), [[sn/16]. [sw /1]]). ). 2)
L5 F(512, 512, (ls,/16), [s.. (6]], [[sp/4], (smm/44) 8)
Le F(512, 512, ([s,/4), [Sve/ [sm//4]), (sk, Sw), 16)
. L7 F(512, 512, 8w), (8h, 8w),, 32)
The forward pass follows a standard U-shaped pattern. Layers L1 + L2 + Ly + L4 are applied sequentially. On the upward path, Lş receives the channel-wise concatenation of the outputs of L, and Lg. Similarly, Lg receives the concatenation of the outputs of Lş and L2, and L, receives the concatenation of the outputs of La and L, The output of L, is then concatenated with the features from the initial MI. and fed into a final MI.P (one hidden layer, width 128, GELU) to yield the SConvCNP representation consumed by the decoder.
Parameter Count. summarzzss the number of learnable parameters for all models evaluated in our image completion experiments.
Table 8: Learnable parameter counts for all models evaluated in the image completion experiments.
CNP
ArCNN
TNP
ConvCNP
SConvCNP
Number of parameters (million)
5
8.4
13.7
12.2
14.2
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B.4.2 Data and Experimental Setup
For our mage-completion experiments, we use the Describable Textures Dataset (DTD: (imp:iimpoi et al. (2014). We adopt the standard DTD split, which contains 1,880 images each for training, validation, and testing. All images are RGB, with heights ranging from 231 to 778 pixels and widths from 271 to 900 pixels. During training, validation, and testing, we sample from each image a random 192 x 192 crop, which we then downsample to 64 x 64. Pixel coordinates of the resulting 64 x 64 grid along each axis are linearly mapped to [-1, 1], and pixel intensities are normalized to [0, 1] independently across channels.
Models are trained for 500 epochs using AdamW with learning rate of 10- We apply gradient clipping with a maximum norm of 0.5. At each epoch, the model processes batches of 16 tasks. For each batch, we draw the number of context pixels as n. 25 1024), and treat all remaining pixels as queries, i.e. n, = 64 x 64 ne. A single sampled value of n. is shared across all tasks within the batch.
B.5 Ablation Studies
We conduct a series of ablation studies to assess the contribution of three core design choices in the SConvCNP:
1. the number of retained Fourier modes,
2. the discretization resolution of the functional embedding, and
3. the use of positional encodings.
Experiments are conducted on two families of one-dimensional functions: samples drawn from a GP with Matémn-5/2 kernel, and deterministic sawtooth waveform. Training follows the protocol described Section B.T with the only difference being the use of a smaller SConvCNT model (3.1M parameters, 32 Fourier modes). Results are reported over 8,096 test tasks, and the main findings summarized below.
Table Predictive log- likeliihoodd of SConvCNP for different numbers of Fourier modes m.
m
8
16
m
Matérn 5/2
00.99 10.00
0.299...00
-0.29 10.00
Sawtooth wave
0.14
0.20+0.02
0.80+6.03
Number of Fourier modes. Increasing the number of retained Fourier modes yields substantial improvements for the sawtooth signal, while having a negligible effect on Matérn-5/2 functions (). This behavior aligns with the spectral properties of the underlying functions. The sawtooth wave exhibits Fourier amplitudes that decay as I/C, leaving significant energy at high frequencies and necessitating a large number of modes for accurate reconstruction. In contrast, Matémn-5/2 samples have power spectra that decay as 1/5", resulting in extremely weak high- figh-frequency content; consequently, small set of fow-frequency modes suffices to capture nearly all signal energy.
Table 10: Predictive log-likelihood of SConvCNP for different discretization resolutions (number of points per unit).
16
32
64
Matérn 5/2
-0.30
-0.29+0.00
-0.29+0.00
Sawtooth 0.03
0.26
0.20+1.02
Discretization resolution. A finer discretization improves performance for the sawtooth signal but has little impact on Matémn-5/2 samples (Table 10 The slowly decaying spectrum of the sawtooth
implics high effective Nyquist frequency, requiring dense sampling to resolve sharp transitions. By
27

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contrast, the spectral mass of Matérn-5/ 5/2 functions is concentrated at lower frequencies, so coarser discretizations can adequately capture the relevant structure.
Table 11: Predictive log-likelihood of SConvCNP with and without positional encoding.
with positional encoding
without positional encoding
Matém 5/2
-0.2940.00 0..01
-0.00
Sawtooth wave
0.67
Positional encoding. Incorporating positional encodings consistently improves predictive perfor- mance, with particularly pronounced gains on the sawtooth tasks (Table [T]. This suggests that
explicit location information helps disambiguate high-frequency and non-smooth patterns that are not fully captured by fully translation equivariant features alone in the SConvCNP.
28

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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The abstract and introduction clearly state that introduce Spectral Con- volutional Conditional Neural Processes (SConvCNPs), explain the motivation for using spectral methods to address limitations in ConvCNPs, and accurately describe the scope of our experimental validation. The claims match the actual content and results presented in the paper.
2. Limitations
Question: Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification: Section 3.1 discusses the limitations of our approach, including the trade-off
between positional encodings and translation equivariance, and the discretization sensitivity of the FFT.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?
Answer: [NA]
Justification: This paper primarily presents methodological contribution with empirical validation rather than thcoretical results requiring formal proofs.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed reproduce the main perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
The paper provides detailed descriptions of the model architecture, data generation process, and experimental setup, ensuring that the main results can be reproduced. In addition, all implementations are publicly available in our code repository.
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Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
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Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: The paper provides detailed descriptions of the model architecture, data generation process, and experimental setup
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Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Answer: [Yes]
Justification: All experimental results in Tables 1-3 report standard deviations across seeds, providing a clear measure of the statistical variability in results.

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8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
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30

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31


Reviews:
[{"review_id": "LiWirCw5xZ", "reviewer": "Reviewer_jGd3", "summary": "Convolutional Conditional Neural Processes are modified by replacing the standard convolution with a spectral convolution. The authors show that this is advantagous for modeling cyclic and long-range dependencies by plugging Frequency Neural Operator (FNO)-style spectral convolution layers into the ConvCNP architecture, replacing the original local, spatial-domain convolutions.\nSpectral convolution enables the model to learn global, task-adaptive kernels in the frequency domain.\nThe authors introduce truncated frequency filtering for efficiency and regularization. \nPositional encodings plus a sliding-window FFT mitigate discretisation artefacts.\n\nThe authors evaluate the model on synthetic data and real-world data. \nComparison methods are CNP, Attentive CNP, TNP, TETNP, and ConvCNP. The results show that the spectral convolution is able to capture long-range dependencies, especially for cyclic dependencies.\n\n2 minor errors:\n- LargeST [?] (reference is missing.)\n- This frequency truncation is the key to *reduces* the computational complexity.", "questions": "The comparison of predictive performance reported in Table 1 and Table 2 and Table 3shows that the proposed method provides results that are either statistically equivalent or better than the comparison methods. However, looking at the numbers, the paper reports numerical values with extremely high precision in significant digits and reports very small (often exactly 0) standard deviation on RMSEs and log-likelihoods. Given that 4 seeds where used for generation in each experiment, the reported standard deviations may be underestimated, or the generated data is (close to) deterministic.\n\nWhat is the runtime of the method vs. dataset size in comparison to other related methods?\n\nHow do the discussed limitations and provided fixes effect robustness of the method and the results?", "limitations": "Given that there are a few design experiments discueed in the paper (discretisation, sliding window, positional encodings, etc.), I would have liked to see a more detailed analysis of the design choices and their impact on robustness of the models and the results, e.g. using a more detailed ablation study.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Incorporating spectral convolutions into the ConvCNP framework is conceptually elegant and practically valuable, as it enables global receptive fields without increasing kernel size.\nHowever, while effective, it is a relatively targeted architectural change, rather than a broader theoretical or methodological contribution.\n\nThe experimental results show that the spectral convolution is able to capture long-range dependencies, especially for cyclic dependencies on par or better than the comparison methods.\n\nLack of assessment of uncertainty calibration. The fact that the model outputs uncertainty estimates are a key benefit compared to simpler models.\n\nGiven that there are a few design experiments discueed in the paper (discretisation, sliding window, positional encodings, etc.), I would have liked to see a more detailed analysis of the design choices and their impact on robustness of the models and the results, e.g. using a more detailed ablation study.\n\nRuntime experiments are not provided in the paper, especially as the method is motivated (amongst other things) to avoid cubic runtime complexity of GPs.\nIn fact, GPs are missing from the comparison methods comparison. As these are the most common methods for probabilistic modeling of functions, it would have been interesting to see how the proposed method compares to them.", "was_revised": true, "final_justification": "I thank the authors for taking on some of my recommendations, promising ablation studies and runtime analyses. As only limited results have been shown during the rebuttal phase, I these did not significantly change my review.", "rebuttal": "We thank the reviewer for the thoughtful and constructive feedback. We are pleased that you found our spectral convolutional modification \"conceptually elegant and practically valuable,\" and appreciate your recognition of the model’s strong empirical performance on long-range and cyclic dependencies. We address your comments below.\n\n**(1) Uncertainty Calibration**\n\nWe agree that uncertainty quantification is a key strength of neural processes (NPs). While we acknowledge that additional metrics—such as calibration error—could provide a more comprehensive assessment, we followed standard practice in the NP literature [Gordon et al., 2019; Bruinsma et al., 2023; Lee et al., 2023; Ashman et al., 2024] by using log-likelihood as our primary evaluation metric. As a proper scoring rule, log-likelihood directly measures the quality of probabilistic predictions and remains a widely accepted benchmark for assessing uncertainty in generative models [Lakshminarayanan et al., 2017].\n\n**(2) Ablation Studies on Design Choices**\n\nThank you for the suggestion. Our discretization and hyperparameter settings follow prior work [Bruinsma et al., 2023], and positional encodings were adopted from UNO [Rahman et al., 2022] using Cartesian grid coordinates. While we did not employ the sliding window in our current experiments (as our domains were large enough), we mentioned it as a potential extension for future applications (e.g., forecasting). That said, we will expand the ablation study in the revision to examine the effects of discretization density and positional encoding.\n\n**(3) Runtime and Efficiency Analysis**\n\nWe agree that runtime comparisons are crucial, especially given that one of our core motivations is the efficiency of global convolution. Our method scales linearly with the size of the context and target sets, in contrast to attention-based models and Gaussian Processes, which exhibit quadratic and cubic complexity, respectively. While our approach employs the FFT as part of the FNO architecture—resulting in a complexity of $ \\mathcal{O} ( | \\mathcal{ G } | \\log | \\mathcal{G} | ) $ for processing the functional embedding over the grid $ \\mathcal{G} $—this is only slightly higher than the $ \\mathcal{O} ( |\\mathcal{G}| ) $ cost of spatial convolutions used in ConvCNPs. In practice, however, the FFT implementation in PyTorch is highly optimized, introducing minimal computational overhead. To support our claims, we will include runtime vs. dataset size plots, along with benchmarks of wall-clock time and memory usage.\n\nAverage forward runtime for an image completion task:\n- SConvCNP: 0.042 s\n- ConvCNP: 0.022 s\n- TNP: 0.07 s\n- CNP: 0.005 s\n\n**(4) Comparison to Gaussian Processes (GPs)**\n\nThank you for pointing this out. We focused on NP baselines to highlight architectural advancements within the same framework. Comparing GPs to NPs is not straightforward: GPs are typically fit independently per task (with no inter-task learning), while NPs amortize across a distribution of tasks [Kim et al., 2019]. Moreover, while GP kernels like Matern or periodic are compatible with some of our synthetic tasks, no analytic kernel is known to (e.g., sawtooth or square waves) such that the induced GP coincides with the used processes for generating data.\n\n**(5) Precision and Standard Deviation Reporting**\n\nYou are right to question the very low standard deviations. While each experiment used 4 random seeds and multiple model inits, the small variance arises from: (1) deterministic model components, (2) large numbers of training tasks, and (3) a fixed test task set. This phenomenon is also observed in prior work [Gordon et al., 2019; Bruinsma et al., 2023].\n\n**(6) Corrections**\n\nThank you for catching these:\n- \"LargeST\" will be corrected to \"largest\" or otherwise revised for clarity.\n- \"reduces the computational complexity\" will be revised to \"reduce the computational complexity.\"\n\n**Summary**\n\nWe thank the reviewer again for their constructive feedback. In response, we will:\n- Expand the ablation study on design components.\n- Include runtime and memory benchmarks.\n\n**Refrences**\n\n- Rahman, Md Ashiqur, Zachary E. Ross, and Kamyar Azizzadenesheli. \"U-no: U-shaped neural operators.\" arXiv preprint arXiv:2204.11127 (2022).\n\n- Lakshminarayanan, Balaji, Alexander Pritzel, and Charles Blundell. \"Simple and scalable predictive uncertainty estimation using deep ensembles.\" Advances in neural information processing systems 30 (2017).\n\n- Bruinsma, Wessel P., et al. \"Autoregressive conditional neural processes.\" arXiv preprint arXiv:2303.14468 (2023).\n\n- Gordon, Jonathan, et al. \"Convolutional conditional neural processes.\" arXiv preprint arXiv:1910.13556 (2019).\n \n- Ashman, Matthew, et al. \"Translation equivariant transformer neural processes.\" arXiv preprint arXiv:2406.12409 (2024).\n\n- Lee, Hyungi, et al. \"Martingale posterior neural processes.\" arXiv preprint arXiv:2304.09431 (2023)."}, {"review_id": "gLMciKTYd2", "reviewer": "Reviewer_RBsp", "summary": "This paper introduces Spectral Convolutional Conditional Neural Processes (SConvCNPs). \n- This model builds on Convolutional Neural Processes (ConvCNP) in which the representation of conditional data points is translated to function space. \n- The SConvCNP replaces the CNN in the ConvCNP with a Fourier Neural Operator (FNO) which the authors claim improves the ability of the model to capture long range dependancies. \n- They benchmark their method on various 1D meta-learning regression tasks.", "questions": "- Can you clarify why the TNP figures for the square wave  look terrible, while the performance reported in Table 1 indicates equal performance to SConvCNP?  \n- Can you comment on your choice of synthetic functions from the perspective of bandwidth limited functions? The sawtooth and square wave are where your method performs better than the alternatives, but both of these have FTs which drop with 1/frequency, which I believe is a much slower decay than the GP examples. This seems to contradict your earlier discussion of frequency cutoff in line 63-65, so I am interested in your interpretation of the results. \n- Can you comment on how your proposed method scales with the number of input dimensions? \n- In line 98 does \"complex\" refer to the weights being a complex number (as is commonly used in FFTs), or just a statement that the weights can have arbitrary form? In line 102 you mention $W_r$ as being the real weights, making me thing $W_c$ are complex numbers, but I don't see why you would need complex numbers from this application. Additionally  $W_r$ appears to not be used. \n- Can you explicitly define the following symbols: \n\t- $C_b$ on line 122? \n\t- $\\theta$ on line 125? I assume that this is the mean and variance of the prediction, but would like this confirmed. \n\t- $t$ in equation 4? I assume this should be $t \\in \\mathbb{X}$.", "limitations": "The authors do briefly discuss the limitation of thier method in section 3.1. I would like a discussion of how this method scales in input dimension, as mentioned in my question, as I believe that this method would scale poorly with that.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 1, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths \n- The approach combines Neural Operators and Neural Processes, which is an exciting approach which I believe is currently under-explored. I thought this was a really interesting approach and was a paper which I enjoyed reading and thinking about the ideas they presented here.  \n- The results they present look promising, although I do have some issues with the current benchmarking which I mention below. \n\nWeaknesses \n- The referencing in this paper was sloppy, resulting in  concerning  errors (missing .bib entry on line 216, incorrect citation for ConvCNPs line 118, incorrectly referencing both Garnelo2018a in line 104).\n- I would say that the current selection of benchmarks is weak:\n\t- The authors restrict their experimentation to only be one-dimensional examples. My understanding of this method is that it should generalise to higher dimensions than this but it is not explored. \n\t- The authors use only one real-world data example, with section 4.1 and 4.2 both being synthetic. \n\t- The authors do not provide the code. I am very surprised by the poor performance of the TNP in figure 1 (see also the question on this), and would have appreciated the ability to check the code implementation. \n- The introduction of the model in Section 3 is missing details. I would have liked a restatement of the method, with the proposed changes, in an equation. At a minimum the writing needs to be tightened up significantly. For example, the authors refer to \"$\\Phi$ in equation 4\" when $\\Phi$ does not appear in equation 4, but does appear in equation 5.  \n- (Minor point.) I would disagree with the framing of the problem NPs are solving as \"learning the covariance structure implicitly\" on line 38, as my understanding is that NPs are actually capable of learning more complex dependencies than just covariances.", "was_revised": true, "final_justification": "I think that the research is interesting, and think that the idea is certainly interesting enough to merit publication. However, I feel that the authors have some issues with their evaluation which prevent me from recommending acceptance. \n\nThe experimental are carried out with their own method having far more parameters than alternatives, making it hard to establish if the improved performance is due to the architecture, or just having a larger model. \n\nAdditionally, the lack of ablation studies of their method, especially with respect to number of frequencies in the FNO,  make it hard to evaluate which features of their model are important.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and constructive feedback. We appreciate your recognition of the novelty in combining neural operators with neural processes, and we are pleased that you found the ideas in our paper engaging. Below, we address each of your comments in detail.\n\n**(1) Referencing and Citation Errors**\n\nThank you for identifying these issues. We will correct:\n- The missing `.bib` entry on line 216\n- The incorrect citation for ConvCNPs on line 118\n- The ambiguous reference to Garnelo et al. [2018a]\n\nThese will be resolved in the revised manuscript to ensure accurate and consistent referencing.\n\n**(2) Benchmark Scope and Dimensionality**\n\nWe agree that extending to higher-dimensional tasks is important. However, scaling ConvCNPs—and by extension, SConvCNPs—is challenging unless the data lies on a grid or one invests heavily in compute. This is due to the need for a fine discretization of the input domain before applying convolutions, as discussed in [Ashman et al., 2024].\n\nFor example, in our synthetic 1D tasks, the domain [-3, 3] is discretized at 64 points/unit, yielding 384 grid points. Extending this to 2D with similar resolution leads to 384^2 points, which quickly becomes computationally prohibitive. Moreover, this resolution is often necessary for good performance.\n\nThat said, we did include a real-world 2D image inpainting task using the Describable Textures Dataset [Cimpoi et al., 2014], demonstrating the model's performance on 2D real-world data. The loglikelihood results are:\n\n- CNP: 0.66 ± 0.00  \n- TNP: 1.70 ± 0.01  \n- ConvCNP: 1.68 ± 0.05  \n- SConvCNP: 1.74 ± 0.00\n\n**(3) Code Availability and TNP Performance**\n\nWe used the official codebase from [Ashman et al., 2024] to train all transformer-based models (TNP, TETNP), adhering strictly to their training and hyperparameter protocols. The poor performance of TNP in Figure 1 appears to result from the *spectral bias* of transformers—i.e., their tendency to prioritize smooth, low-frequency signals.\n\nThis is consistent with findings in [Nguyen and Grover, 2022], where TNP underperforms even vanilla CNPs on GP data with periodic kernels. Additional studies (e.g., [Vasudeva et al., 2024]) support this, highlighting transformers' difficulty in modeling high-frequency components. We retrained TNP and TETNP with larger capacities, and the performance remained unchanged, reinforcing this hypothesis.\n\n**(4) Inconsistency Between Figure 1 and Table 1 for TNP on Square Wave**\n\nThank you for catching this error. Upon rechecking, we found that the TNP results for the square wave in Table 1 were inadvertently copied from another experiment. We have re-run the benchmark and confirmed that the numerical results match the poor qualitative performance in Figure 1. For TNP on squarewave, the log-likelihoods averaged over 4 seeds was $-1.39 \\pm 0.00$. This correction further highlights the limitations of TNP on high-frequency signals.\n\n**(5) Missing Equation and Notation Issues in Section 3**\n\nWe will revise Section 3 to provide:\n\n- A clearer restatement of our method with explicit equations comparing SConvCNP and ConvCNP  \n- Corrections to referencing errors (e.g., \"Equation 4\" should be \"Equation 5\")  \n- Clarified notations, including:  \n  - Line 122: C_b(X, Y) is the space of bounded continuous functions from X to Y  \n  - Line 125: θ denotes parameters of the predictive distribution (i.e., mean and variance for Gaussian outputs)  \n  - Equation 4: t ∈ X (analogous to x); we will unify notation for clarity\n\n**(6) Line 98 – \"Complex Weights\"**\n\nIn line 98, \"complex\" refers to complex-valued weights, which are standard in frequency-domain methods using FFT. These allow flexibility in modeling kernels without assuming real-valued spectra. By contrast, W_r (line 102) are real-valued weights used for pointwise channel mixing, akin to residual projections in CNNs or transformers. We will clarify these design choices in the revised text.\n\n**(7) Function Choice and Bandwidth in Synthetic Benchmarks**\n\nWe appreciate this nuanced observation. Indeed, sawtooth and square waves have slower spectral decay, and our method’s strong performance on them stems from two key factors:\n\n1. **Periodic alignment**: The FFT assumes periodicity, which aligns well with these synthetic functions, enabling accurate spectral representations  \n2. **Energy distribution**: On a fixed grid, the FFT returns fixed frequency bins. For a GP with a Matérn-5/2 kernel, most energy is concentrated in low frequencies, while periodic functions like sawtooth and square waves spread the same energy across a wider band. As a result, the set of these fixed FFT modes is more likely to capture informative components for periodic functions\n\nWe will include a discussion of these points to clarify how function class influences performance.\n\n**(8) Scaling with Input Dimension**\n\nAs noted, SConvCNP inherits ConvCNP’s scalability challenges due to the discretization bottleneck. While FNOs improve parameter efficiency and receptive field size compared to CNNs, the cost of discretizing the entire input space (especially in 2D or 3D) still grows exponentially with dimension.\n\n**(9) Formatting and Presentation**\n\nWe appreciate the detailed formatting feedback. We will:\n\n- Fix the `.bib` entry on line 216  \n- Ensure consistent significant figures in Table 1  \n- Bold the best performance values in all result tables\n\n**Refrences**\n- Cimpoi, Mircea, et al. \"Describing textures in the wild.\" Proceedings of the IEEE conference on computer vision and pattern recognition. 2014.\n- Gordon, Jonathan, et al. \"Convolutional conditional neural processes.\" arXiv preprint arXiv:1910.13556 (2019).\n- Nguyen, Tung, and Aditya Grover. \"Transformer neural processes: Uncertainty-aware meta learning via sequence modeling.\" arXiv preprint arXiv:2207.04179 (2022).\n- Ashman, Matthew, et al. \"Translation equivariant transformer neural processes.\" arXiv preprint arXiv:2406.12409 (2024).\n- Vasudeva, Bhavya, et al. \"Transformers Learn Low Sensitivity Functions: Investigations and Implications.\" arXiv preprint arXiv:2403.06925 (2024)."}, {"review_id": "aiH7K3XXol", "reviewer": "Reviewer_Bbzq", "summary": "The paper introduces a new architecture based on Conditional Neural Processes (CNPs), named Spectral Convolutional Conditional Neural Processes (SConvCNPs). This model replaces the standard local convolutions in the spatial domain with global convolutions in the spectral domain, inspired by Fourier Neural Operators (FNOs). The authors argue that this modification helps address the limitations of CNPs, particularly in capturing long-range dependencies and handling sparse or irregularly sampled data. The method is evaluated on several synthetic and real-world datasets, where SConvCNPs perform comparably or better than existing baselines.", "questions": "1.  There seems to be something wrong with Eq 4, where in the second term in the parentheses, the two $\\psi_{e}(t, x_i)$'s in the top and bottom can be cancelled out? I would guess it shoud be $\\frac{\\sum \\psi_e(t, x_i) y_i}{\\sum \\psi_e(t, x_i)}$ instead. Also, the citation of ConvCNP seems to be wrong, it should be J. Gordon et. al. .\n2.  Line 122: What does $C_b(\\mathbb{X}, \\mathbb{Y})$ mean? And Eq. 5 is not clear to me, could you explain it?\n3.  Line 145: \"the operator $\\Phi$ in Equation 4 ...\": I believe it's Eq. 5? Is it a typo?\n4.  Line 150: \"replaces ... as defined in Equation 3\": I could not see how to plug Eq. 3 into Eq. 4 or 5, could you explain more?\n5.  Line 159: \"Althought $\\tilde{E}$ breaks strict translation equivariance ...\": Does it mean then the contextual points must maintain a specific spatial order? If not, how would you deal with it?\n6.  Line 165 - 168: I do not quite understand what do you mean by \"keeping the window's range and resolution constant\". Is the window's range the kernel size? Which resolution are you referring to?\n7.  Line 216: Empty citation?\n8.  I am curious about the performance of SConvCNF in high-dimensional data, e.g., images, as the neural operators have great potentials in resolution-free inference. It would be nice to see the applications in super-resolution or image inpainting tasks.", "limitations": "The authors did not discuss the potential social impact of the work in the submission.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Quality**\n\nThis is a generally sound paper that introduces a new CNP-based model, SConvCNP, designed to improve global contextual conditioning in Conditional Neural Processes. The model demonstrates improved performance in the presented experiments, supported by detailed metric evaluations and comparisons with related models. Sufficient details of implementation are provided and the used codebase is promised to be released.\n\nHowever, the theoretical grounding of the proposed method is limited. For example, it is unclear how the spectral convolution in Eq. 3 is integrated into the overall CNP framework. In addition, the main motivation of the paper is to address global and irregular conditioning. Yet, the experiments do not clearly reflect these scenarios, making it difficult to assess whether the proposed modification effectively addresses the stated challenges.\n  \n**Clarity**\n\nThe paper is generally easy to follow. The contribution is clearly stated in the introduction, and the necessary preliminary background is reviewed in Section 2. The experimental section provides sufficient details, along with comprehensive benchmarking tables and clear, readable figures, followed by a review of related work.\n\nHowever, as noted earlier, the theoretical presentation of SConvCNP in Section 3 is oversimplified. The core component $\\tilde{E}[\\mathcal{D}] = (E[\\mathcal{D}], t)$ is not clearly explained and may cause confusion. Additionally, Section 3.1 lacks an explicit discussion of the limitations of the standard CNPs that the proposed method aims to address (see questions below).\n  \n**Significance**\n\nThis work makes a useful step toward improving CNPs, but it would benefit from further validation, both in theory and in practice.\n  \n**Originality**\n\nWhile SConvCNP presents a new architectural design, its two main components—CNP and FNO—are already well-established. Therefore, the combination, though practical and effective, may be viewed as incrementally novel rather than conceptually new.", "was_revised": true, "final_justification": "During the discussions, the authors provided additional clarifications on the main theoretical constructions and agreed to reorganize the text with more details, which should help improve the quality of the paper. Therefore, I have decided to raise my score to positive.", "rebuttal": ""}, {"review_id": "pcSzKCbvTY", "reviewer": "Reviewer_c7aT", "summary": "The paper introduces Spectral Convolutional Conditional Neural Processes. It is a novel extension of Convolutional Conditional Neural Processes (ConvCNPs) that leverages spectral methods to address limitations in capturing long-range dependencies in irregularly sampled data. Note tha Traditional ConvCNPs use local discrete kernels, which struggles with global patterns due to finite receptive fields. SConvCNPs replace these with global convolutional filters parameterized in the frequency domain, inspired by Fourier Neural Operators (FNOs). This approach efficiently models stationary stochastic processes by focusing on low-frequency components, reducing computational costs while improving performance.\nIn terms of evaluation, The authors validates SConvCNPs on synthetic 1D regression tasks and real-world benchmarks, e.g. Lotka-Volterra dynamics, traffic flow data. Results show superior performance over baselines in log-likelihood and RMSE, particularly for periodic functions.", "questions": "See Weakness part.", "limitations": "See Weakness part.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\nCompared with ConvCNPs, the prorponsed SConvCNP benefits from (1) It considers the Global dependency modeling. It addresses ConvCNPs' limitation of local receptive fields by leveraging spectral convolutions. (2) It presents outstanding empirical results: SConvCNHP outperforms baselines on synthetic and real-world tasks, especially for periodic patterns. Spectral truncation reduces computational overhead compared to large spatial kernels. (3) There constitutes theoretical insights. Builds on well-established frameworks (FNOs, CNPs) with clear motivation. \n\n**Weakness:**\n\n(1) I think FFT-based methods depend on grid resolution, which could affect robustness in highly irregular data. This part can be better analyzed. (2) For nonstationary cases, I am wondering SConvCNP’s performance over existing methods. If not, it would be better to include the limitation section. (3) Scalability issue in optimization objective. This work narrows the scope into ConvCNP. In terms of optimization objective, can the mentioned Fourier neural operator be easily integrated into vanilla NPs, e.g., approximate ELBO [1], importance weighted NP methods [2-3] or mixture-of-expert methods [4]? This can be discussed. (4) Some typos, such as missing reference number in Line 216 LargeST [?]\n\n**References:**\n\n[1]Garnelo, Marta, et al. \"Neural processes.\" arXiv preprint arXiv:1807.01622 (2018).\n\n[2]Wang, Qi, Marco Federici, and Herke van Hoof. \"Bridge the inference gaps of neural processes via expectation maximization.\" arXiv preprint arXiv:2501.03264 (2025).\n\n[3]Foong, Andrew, et al. \"Meta-learning stationary stochastic process prediction with convolutional neural processes.\" Advances in Neural Information Processing Systems 33 (2020): 8284-8295.\n\n[4]Wang, Qi, and Herke Van Hoof. \"Learning expressive meta-representations with mixture of expert neural processes.\" Advances in neural information processing systems 35 (2022): 26242-26255.", "was_revised": true, "final_justification": "After reading the rebuttal, my concerns are addressed. The work contributes a novel spectral convolution for neural processes, which is meaningful in the field.", "rebuttal": "We thank the reviewer for their positive assessment of our work and for highlighting the strengths of our approach—including the motivation for global dependency modeling, promising empirical performance on periodic tasks, and the theoretical grounding in established operator learning frameworks.\n\n**(1) Robustness to Irregular Sampling and Grid Resolution**\n\nWe appreciate the reviewer’s insightful comment. It is true that FFT-based methods typically rely on regular grids, which may raise concerns in the context of irregularly sampled data. To address this, we adopt the approach used in ConvCNPs, wherein context observations are first projected onto a fixed-resolution uniform grid via a learned interpolation kernel. Spectral convolutions are then applied to this gridded representation. This step allows our model to benefit from global Fourier structure while retaining robustness to the irregularity of inputs.\n\n**(2) Scalability and Integration with Other NP Objectives**\n\nWe agree that extending SConvCNP to the class of latent NP families is a promising direction. While our current focus is on deterministic ConvCNP-style architectures, the proposed spectral module is compatible with latent-variable extensions. These extensions would enhance uncertainty modeling and enable richer predictive distributions.\n\n**(3) Errors and Typos**\n\nWe thank the reviewer for noting the typo on line 216 (“LargeST [?]”). This will be corrected in the revision. We will also perform a comprehensive proofreading pass to address any remaining typographical issues and ensure clarity and consistency throughout the manuscript."}]

Author rebuttal:

Meta-review: All reviewers found the core idea of combining Fourier Neural Operators (FNOs) with Convolutional Conditional Neural Processes (ConvCNPs) to be novel, interesting, and conceptually elegant. There was a consensus that the proposed method demonstrates strong performance, in particular in periodic signals and irregular signals (e.g. sawtooth).

The initial main concerns were about:
1. Experimental scope and validity: missing important ablation studies.
2. Model size differences in the comparison: The proposed model is larger and therefore it is not clear where the gain is coming from.
3. Clarity and errors: The description of the method was not detailed enough and some errors were found.

After these points were discussed with the authors, most reviewers were satisfied with the provided answers and additional results. 

The final recommendations of the reviewers are: 3 reviewers recommend acceptance, and 1 reviewer is willing to accept the paper but is more reluctant due to the large amount of updates required after the discussion period. 

After reading the reviews and the discussion, I believe this paper should be accepted, provided that the authors make all the updates mentioned in the discussion.
Decision: Accept (poster)