Efficient Dilated Squeeze and Excitation Neural Operator for Differential Equations
Authors: Prajwal Chauhan
Salah Eddine Choutri
Saif Jabari
Venue: Accepted by TMLR ()

Abstract: Fast and accurate surrogates for physics-driven partial differential equations (PDEs) are essential in fields such as aerodynamics, porous media design, and flow control. However, many transformer-based models and existing neural operators remain parameter-heavy, resulting in costly training and sluggish deployment. We propose D-SENO (Dilated Squeeze-Excitation Neural Operator), a lightweight operator learning framework for efficiently solving a wide range of PDEs, including airfoil potential flow, Darcy flow in porous media, pipe Poiseuille flow, and incompressible Navier–Stokes vortical fields. D-SENO combines dilated convolution (DC) blocks with squeeze-and-excitation (SE) modules to jointly capture wide receptive fields and dynamics alongside channel-wise attention, enabling both accurate and efficient PDE inference. Carefully chosen dilation rates allow the receptive field to focus on critical regions, effectively modeling long-range physical dependencies. Meanwhile, the SE modules adaptively recalibrate feature channels to emphasize dynamically relevant scales. Our model achieves training speed of up to $\approx 20\times$ faster than standard transformer-based models and neural operators, while also surpassing (or matching) them in accuracy across multiple PDE benchmarks. Ablation studies show that removing the SE modules leads to a slight drop in performance.

Published in Transactions on Machine Learning Research (02/2026)
Efficient Dilated Squeeze and Excitation Neural Operator for Differential Equations
Prajwal Chauhan
pc33770nmn.edu
Engineering Division
New York University Abu Dhabi
Abu Dhabi, UAE
Salah Eddine Choutri
sc8101 @nyu. edu
NYUAD Research Institute
New York University Abu Dhabi
Abu Dhabi, UAE
Saif Eddin Jabari
sef7Onyu.edu
Engineering Division
New York University Abu Dhubi
Abu Dhabi, UAE
Reviewed on OpenReview: https: openreview. net/ forum? id= X1942THUU
Abstract
Fast. and accurate surrogates for physics-driven partial differential equations (PDFs) are essential in fields such as aerodynamics, porous media design, and flow control. However, many transformer-basee models and existing neural operators remain parameter-heavy. re- sulting in costly training and sluggish deployment. We propose D-SENO (Dilated Squeeze- Excitation Neural Operator), lightweight operator learning framework for efficiently solv- ing wide range of PDEs, including airfoil potential flow, Darcy flow in porous media, pipe Poiseuille flow, and incompressible Savier-Stoker vortical fields. D-SENO combines dilated convolution (DC) blocks with squeczo-and-excitatio (SE) modules to jointly capture wide receptive fields and dynamics alongside channel-wise attention, enabling both accurate and efficient PDE inference. Carefully chosen dilation rates allow the receptive field focus on critical regions, effectively modeling long-range physical dependencies. Meanwhile, the SE modules adaptively recalibrate feature channels to emphasize dynamically relevant scales. Our model achieves training speed of up to R 20x faster than standard transformer-based models and neural operators, while also surpassing (or matching) them in accuracy across multiple PDE benchmarks. Ablation studics show that removing the SE modules leads to a slight drop in performance. Code: github.com/pjl10 pj1911
1 Introduction
Partial differential equations (PDEs) dictate many natural phenomena, from synoptic-scale weather systems to winglip vortices, to subsurface multiphase flow, and many more. Conventional numerical solvers such as finite difference, finite volume, spectral, and finite element schemes deliver high fidelity, yet their cost explodes with mesh resolution, geometric complexity, and parametric swcops Quarteroni Valli (2008). To close this computational gap, the focus increasingly shifting to machine learning surrogates, which reduce the cost of high-resolution data in training and then provide accurate predictions at lower cost.

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Recent neural PDE solvers range from physics-informed regressor to fully data-driven operator learners, each occupying a different point on the spectrum of accuracy, flexibility, and computational efficiency. At the physics-heavy end of the spectrum, physics-informe neural networks (PINNs) embed the governing equations, imitial conditions, and boundary conditions directly into the loss via automatic differentiation, allowing them to train on sparse or even entirely unlabeled solution data. However, their stiff multiscale residuals make optimization delicate, and each change in geometry or forcing profile usually requires an expensive full retraining or at least substantial fine-tuning [Raissi et al. (2019). To reduce training and inference costs, recent work has focused on operator learning, where single neural network approximates the solution operator, mapping input functions to their corresponding PDE solutions across a family of problems.
The first learainr- model, DeepONet, Lu el al. (2019) represents the mapping with two sub-networks: a branch net that processes the input function and a trunk net that evaluates basis functions at the query points. Its mesh-free design and small paramcter count make it attractive for paramotric problems, yet running both nets at every spatial point slows training and inference as the grid becomes large Lu et al.] (2021). To mitigate this scaling bottleneck and capture interactions that extend across the entire domain, subsequent work turned to architectures with built-in global couplings.
The Fourier Neural Operator (FNO) addresses these goals by exchanging, to certain extent, spatial locality for spectral mixing Li et al. 2020 : Fast Fourier transforms (FFT) reduce complexity to O(N log N) and grant mesh invariance, but the resulting all-to-all convolutionn can blur sharp discontinuities and inflate GPU memory on large three-dimensional domains such as wind-tumnel flows. To enhance spatial locality, some variants perform FFTs over overlapping windows or decompose the integral kernels onto wavelet bases Cuppta et al. 2 Tripura & Chakraborty (2023 While, these approaches partially recover local intcractions, the necessary window overlap or boundary padding adds computational overhead and can introduce artifacts that complicates training.
In fixed Cartesian grids, U-NO Rahman et al. [002] combines global Fourier-based integral layers with a U-Net-style encoder decoder and multiscale skip connections. whereas F-FNO Tran et al. (0011) employs separable spectral (Fourier) layers factorized across dimensions alongside residual shortcuts. Like FNO, both U-NO and F-FNO inherit its core limitations: they depend on global FF'S,, making them naturally suitable for periodic and uniform grids. They also share FNO's high memory and computational costs, especially in higher dimensions. Their spectral kernels require deeper architectures or windowed variants to capture fine-scale locality, which further increases overhead and can destabilize training.
Transformer- based neural operators have recently advanced the state of the art by leveraging multi-head attention to capture long-range dependencies. However, standard self-attention scales quadratically with the number of spatial grid points O(V2), making high-resolution problems computationally demanding without aggressive sparsification or patching strategics Cao 2021 Guibas et al. 2021).
Among these models, Transolver Wu et al. (2024) introduces physics-aware clustering mechanism that groups flow-aligned regions before applying intra- and inter-cluster attention. This approach yields statc-of- the-art accuracy across several PDE benchmarks, outperforming FNO, UNO, and prior transformer-based baselines. Despite its accuracy gains, Transolver incurs substantial computational overhead: its iterative clustering, multi-stage refinement, and large hidden dimensions lead to training and inference times an order of magnitude higher than many neural operator architectures.
Our Contribution. We introduce D-SENO, a lightweight neural operator architecture designed for fast and accurate surrogate modeling of partial differential equations (PDEs). While previous convolution- based neural operators (e.g., CNO Raonic et al. (023) rely fixed receptive fields and uniform filter patterns, D-SENO advances this paradigm by effectively integrating two key components within a unified francwork:
1. Dilated Convolutional Blocks: We apply residual blocks with non-uniform, datasel-specific dila- tion rates across spatial dimensions. This allows the model to flexibly expand its receptive field and capture multiscale spatial dependencics, while maintaining strict locality and lincar computational complexity. In contrast the Dilated Convolution Neural Operator (DCNO) Xu al. (2025). which employs all dilation rates in every process block, our approach customizes and selects single,
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dataset-specific rate per block and connects them via residual links for improved expressiveness and adaptability.
2. Convolutional Channel Attention: erations, we incorporate Squeezo-and-Excttane (SE) blocks Hu et al. (2018) using pointwise convo- lutions. These blocks dynamically recalibrate feature channels based on global context, improving relevance to physical patterns without incurring the overhead of transformer-style attention.
To enhance channel-wise expressiveness without global op-
This architecture remains strictly local and fully convolutional, requiring neither Fourier transforms nor self-attention mechanisms. All layers preserve spatial resolution and scale linearly with input size, making D-SENO particularly efficient for high-resolution PDE inference.
We validate our model on diverse PDE benchmarks, including airfoil potential flow, Poiseuille pipe flow, heterogeneous Darcy flow, and Navier-Stokes vortices. D-SENO achieves high accuracy while offering up to a 20x speedup in training and inference over state-of-the-art neural operator and wranformer-basee surrogates.
The remainder of this paper is organized as follows: we begin by reviewing the relevant prior work, followed by discussing the core components of our approach, then detail the architecture and training protocol of D-SENO, and finally evaluate its accuracy, efficiency, and generalization across multiple PDE benchmarks.
2 Related Work
While the literature contains wide array of neural PDE surrogates, focus our discussion representative models that are most relevant to our approach: the Fourier Neural Operator (FNO), the Convolutional Neural Operator (CNO), and the Dilated Convolutional Neural Operator (DCNO). We briefly summarize their core methodologies and highlight their limitations in the context of our work.
Fourier Neural Operator (FNO). The Fourier Neural Operator (FNO) Li et al. 2020) was among the first neural operators to approximate mappings between function spaces using the Fast Fourier Transform (FFT). Instead of learning convolutional kernels in physical space, FNO projects the imput function onto a spectral basis via FFT, applics learned complex-valued weights to a subset of low- now-frequency modes, and then returns the physical domain using an inverse FFT. This produces global convolution-like behavior over the spatial domain.
At each layer, the update rule combines spectral convolution (in frequency space) with a pointwise convo- lution, enabling the model to capture long-range dependencies. Since FFTs operate globally, FNO exhibits resolution invariance and has shown strong performance structured and periodic PDE problems. How- ever, this global nature introduces limitations. FNO may struggle with problems involving localized features (e.g,, sharp shocks), complex or irregular boundary conditions, or uneven spatial behavior due to the lack of spatial localization in spectral kernels Chauhan et al. (2025).
In contrast, our method replaces global spectral operations with spatially localized dilated convolutions, allowing receptive fields to grow in controlled manner. Combined with channel-wise recalibration via siieeze-and-xxiitatien (SE) blocks, this leads to lightweight architecture better suited for capturing localized physical patterns.
Convolutional Neural Operator. The Convolutional Neural Operator (CNO) Raonic et al. (2023 is fully convolutional architecture for learning resolution-specife mappings arising in PDEs, designed to be robust across varying discrctizations and noise levels. It builds on standard convolutional layers arranged in an encoder encoder-decoder decoder (U-Net-like) fashion, with residual blocks and anti-aliased downsampling to maintain spatial fidelity, while avoiding the use global Fourier or attention-based mechanisms. The model empha- sizes locality and generalization, but does not incorporate direction-dependent non-uniform convolutions for data-dependent receptive field shaping, nor explicit input-adaptive feature mixing through channel wise recalibration mechanisms. Our approach addresses these limitations by integrating non-uniform dilated
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a(x)
DS Block (d1)
DS Block (d2)
DS Block (dm)
u(x)
v(x)
D.Conv2d
Gelu
D-Conv2d
Global Average Pooling
Conv2d
Gelu
Conv2d
Sigmoic
Gelu
SE Block
DC Block
Figure 1: D-SENO architecture
convolutions for multiscale spatial context and Squeexe-and-Fxxitatiinn blocks for input-aware channel recal- ibration, yielding improved adaptability in both space and feature selection.
Dilated Convolution Neural Operator. The Dilated Convolution Neural Operator (DCNO) Xu et al. (2025) introduces a hybrid operator learning architecture that combines spectral and spatial mechanisms. Specifically, it inserts dilated convolutional blocks between Fourier layers, aiming to capture both high- frequency local details and low-frequency global behavior. The use of dilated convolutions allows for an expanded receptive field without additional parameters, while the Fourier layers provide resolution-invarit global context. Although DCNO leverages FNO components to preserve operator generalization across discretizations, the added convolutional modules improve efficiency and local feature representation. While this design improves expressiveness, it also introduces several limitations. First, the reliance on FFT makes DCNO best suited for periodic domains and structured grids, limiting its applicability in problems with irregular or non-periodic boundaries. Second, although dilated convolutions are lightweight, the interleaved Fourier layers increase modcl's complexity and reduces it's flcxibility. Third, the fixed dilation schedule may not generalize optimally across different datasets spatial scales. These trade-offs afTect both runtime performance and adaptability across diverse PDE regimes.
3 Background
Deep neural networks have demonstrated remarkable effectiveness in modeling spatially structured data across wide range of domains. To handle complex patterns that vary across spatial scales, modern ar- chitectures often incorporate mechanisms that expand receptive fields and adaptively recalibrate internal representations. This section reviews two such components: Dilated Convolutions (DC) and Squeeze-and- Excitation (SE) network blocks.
Dilated Convolutions. Dilated convolutions, also referred to atrous convolutions, are variant of the standard convolutional operation that introduces a spacing factor between kernel elements. This modifi- cation enables the filter to cover a larger receptive field without increasing the number of parameters or roducing resolution. Originally explored in wavelet analysis Holschneider al. (1987): Shensa (1992), di- lated convolutions were later adopted in deep learning for tasks such as semantic segmentation Yu & Koltun 2015), and recently, operator learning Xu et. al. (025)

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Let f: I² +R be discrete imput function and Ωt - R convolutional kernel with support size
(2r + 1)² E Zzo- The conventional 2D convolution is given by:
(x)) = E f(s)- k(t).
(1)
s+IE
A dilated convolution introduces a dilation factor E N. We denote the I-dilated convolution by *t, defined as:
(1)))(z)== Σ f(8)-A(t).
(2)
This effectively stretches the kernel by inserting -- zeros between adjacent filter elements along each axis. When 1 = 1, the operation reduces to the standard convolution. Higher dilation rates allow the network to aggregate information across wider area of the input without pooling or strided operations. In our case, we allow different dilation factors along each spatial axis, denoted by l2.1, E N, yielding a direction- dependent dilated convolution *((,,,,) This results non-uniform receptive field expansion along the spatial dimensions, cnabling the operator to capture clongated and orientation-spoeifcc structures by aggregating spatial context at different rales along the horizontal and vertical directions while preserving resolution.
Dilation rate
Dilation mile
Dilaiion
Figure Top. Dilated Convolution for 3 dilation rates, Middle. Cumulative tap locations for dilation rates 1,2 3 (left to right), Bottom Receptive field growth with propagation for dilation rates 1,2,3, each applied twice (left to right)
aoneeeze-and--xciiatiiie Blocks. To enhance the representational capacity of convolutional neural net- works, Hu et al. (2018) introduced the Baneezo-and-Exciialle (SE) block, lightweight architectural unit that adaptively recalibrates channel-wise feature responses. The core idea is to model inter-channel depen- dencies explicitly, allowing the network to emphasize informative features and suppress less useful ones based on global context.
The SE block acts on the feature maps U, generated by a preceding transformation. Let U E NxXC be the output of the DC block, the SE block then recalibrates these features in two stages:

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Squeeze. In the first stage, global information is captured by compressing each channel-wise feature map E &R (for channel index e) into single scalar descriptor via global average pooling:
W ΣΣ =1
(i,))))
(3)
HXW
resulting in vector that summarizes the global statistics for each channel. In implementation, this vector is typically stored 1 x 1 x C' tensor, so that it can be broadcast back over the feature maps, but in the following we identify it with its flattened vector form E RC for notational convenience.
Excitation. In the second stage, the aggregated descriptor 2 then passed through a small channel-wise gating mechanism. Although implemented using two successive x 1 convolutions in our architecture, this operation is equivalent to applying two fully connected layers the flattened descriptor and is expressed in matrix form for clarity:
2 (Wiz + b1), (W22 + b2),
(4)
(5)
where Wi E FxC and W2 E CRC are learnable weight matrices, b E RR. and b E RC are bias terms, à is the GELU activation function, and a the element- wise sigmoid function. The reduction ratio r E Z4 (with r 1 mcaning no reduction) explicitly controls the width of the intermediate channel representation: the first layer reduces the chanmel dimension from C to C/r, and the second layer restores it from C/r back to C. The output RC provides set of channel-wise modulation weights.
Finally, the original feature maps are reweighted channel-wise using the learned modulation weights. The vector E RC, in practice, is expanded to RR 1xxxx and broadcast over spatial locations, and each channel slice u, E H of U Rx is scaled by its corresponding weight Sc:
ú.(i.) = 8. u..(ij),
(6)
yielding a recalibrated feature map Ü HXXCC which each channel's contribution adjusted based on global context.
SE blocks have been successfully integrated into various backbone architcctures (e.g., ResNet, Inception) and demonstrate consistent improvements in classification accuracy at low computational cost Hu el al.
2018
4 Method
4.1
Learning solution operator for PDEs
A (solution) operator G + 24 maps elements from an input function space A (e.g., boundary conditions, source terms, or coefficients of a PDE) to an output function space U, which typically contains the corre- sponding solutions of the PDE. The precise definition of these spaces depends on the particular PDE under consideration and the physical domain of interest.
Solving PDE can thus be interpreted evaluating
G(a), where E A is the input function and E L is the resulting solution field. A neural operator aims to approximate this mapping G in a data-driven manner by learning parametric representation Go x G, which generalizes to new inputs E A that were not seen during training.
In the following subsoction, we introduce our proposed model, D-SENO, which is specifically designed learn such solution operators for PDEs while efficiently capturing both local and global dependencies through dilated convolutions and squeeze-and-exoitatii mechanism.

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4.2 D-SENO
We develop a fully convolutional architecture that combines multiscale dilated convolutions with SE-based adaptive channel-wise recalibration. The model follows a fee-process-projeee structure: the imput field a is first lifted to a latent space, processed by a sequence of Dilated-Squecze (DS) layers, and then projectod to the output solution u. An overview is shown in Figure II
Each DS layer, denoted as B, begins with a dilated convolution block (DC) which consists of two stacked k l convolution layers (typically 3 3) that use dataset-specific dilation rates applied independently along the and y-axes. These dilation rates (for details, refer the reader to supplementary materials) allow the receptive field to grow flexibly without down sampling or increasing the number of parameters. Following the dilated convolutions, a lightweight SE block adjusts the importance of each feature channel. We modify the original squeeze-ande axxeiiitttiinn block by retaining global average pooling but substituting the fully--onnectedd layers with two x 1 convolutions, separated by a GELU activation. These layers reduce and then restore the number of channels, controlled by a reduction factor r Z+ as shown in section 3 Global average pooling used to compule channel-wise statistics, which are transformed into scaling weights and applied via element wise multiplication. A residual connection adds the DS block input to the recalibrated output, followed by a GELU activation. This allows each B; to modulate both spatial and channel information, improving expressiveness and efficiency.
The full model is composed of three stages:
1. Lifting: A pointwise convolutional layer P transforms the input into high-dimensionall latent space.
A sequence of DS blocks B Ba, each preserving spatial resolution and expanding receptive fields.
Processing:
Projection: A final pointwise layer Q maps the processed features to the output u.
All spatial operations preserve resolution; striding or downsampling is used. Global average pooling occurs only within SE blocks to compute channel-wise weights, without altering spatial dimensions.
Formally, the model approximates the solution operator g by a learned mapping Go : A + U, producing an approximate solution function:
a(.) go(a Q (B, ..... B,) P(a)().
where 0 denotes the model parameters.
To assess the impact of dilation and recalibration, conduct ablation studies and refer the reader to the supplementary materials for details on the configurations evaluated.
5
Experiments
We evaluate the proposed D-SENO on the PDE benchmarks used by previous operator learners and trans- former based models. The suite spans stutuued and regular grids, mirroring the experimental protocol of Transolver Wu et al. 2024h for direct comparability.
Benchmarks. Table presents four benchmark PDE problems: (1) airfoil potential flow, (2) Poiseuille pipe flow, (3) heterogeneous Darcy flow (these three are steady), and (4) the time-dependent incompressible Navier-Stokes equations (unsteady), brief per-datasct summaries are given below.
5.1 Airfoil
The airfoil dataset comprises transonic Euler simulations of inviscid flow around randomly deformed NACA- 0012 airfoils. The dataset (2D) consists of 1,000 training and 200 test samples of airfoil geometries, each
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Table Summary of experiment benchmarks with dimensions and mesh size.
BENCHMARK
#DIM
MESH
Airfoil
2D
221 51
Pipe
2D
129 129
Darcy
2D
85 X 85
Navier-Stokes (time)
2D +
64 x 64 x 20
discrctized on a structured mesh of size 221 51. For each case, the input tensor has shape 221 2, encoding the airfoil's structural geomelry at every mesh point, and the larget output is the local Mach number (the ratio of the local flow velocity to the speed of sound) at each point of shape 221 x 51 x 1. All shapes are generated by deforming the baseline NACA-0012 profile provided by the National Advisory Committee for Acronautics Li et al. (023b).
5.2 Pipe
The pipe-flow dalaset contains sinulattons of incompressible, viscous fluid moving through two-dimensional pipes whose centerlines are randomly curved. The dataset (2D) comprises of 1,000 training and 200 test samples of pipc gcometrics, each discretized on a structured mesh of size 129 x 129. For each case, the input tensor has shape 129 x 129 x 2, which encodes the structural geometry of the pipe each mesh point, and the target output is the horizontal velocity of the fluid at each point of shape 129 129 1. All samples are generated by varying the pipe's centerline geometry Li et al. 2023b
5.3 Darcy
Darcy flow describes the steady, viscous motion of an incompressible fluid through a porous medium. The dataset (2D) consists of 1,000 training and 200 testing samples of two-dimensional porous media, originally defined 421 x 421 regular grid and down-sampled to 85 x 85 for experiments. Each input tensor of shape 85 x 85 x 1 encodes the binary structure of the porous medium, and the target output of shape 85 x 85
gives the fluid pressure at each grid point. Different cases contain different medium structures Li et al. 2020
5.4 Navier Stokes
The Navier Stokes dataset models incompressible, viscous flow on a unit torus with constant density and viscosity 10-5 Li et al. (2020). Each trajectory discretized on a 64 x 64 regular grid. The input tensor of shape 64 x 64 x 10 contains the two velocity components over the past. 10 time steps, and the target output of the same shape predicts the velocity field for the next 10 steps. A total of 1,000 simulations with varying initial conditions are used for training, and 200 new trajectories reserved for lesting.
Baselines and Implementation Protocol. We compare D-SENO with more than ten representative models: neural operators (FNO Li et al. (2020), U-NO Rahman et al. (022), LSM Wu et al. (2023) transformer-style PDE solvers (GNOT Hao et al. (023)) Factorized FNO Tran al. (222))) Transolver Wu all benchmarks. Moreover, observe that modest tweaks to FNO such as replacing ReLU with GELU activations, removing batch normalization, preserving sufficient Fourier modes and implementing complex Fourier multiplication directly with einsum, yield a notable boost accuracy, speed and parameter counts. We refer to this refined baseline as FNO*, and it surpassos many neural operator variants introducod after the original FNO. The appendix provides a detailed breakdown of the specific changes made to FNO. Note that, D-SENO and FNO + have been trained using the same hyper parameters shown in Table 3 unless stated otherwise.
et al. (2024) represents the previous state-of-thh-art
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Table 2: Cross-representation test errors (lower is better). '/2 indicates a value not reported.
MODEL
STRUCTURED MESH
REGULAR GRID
AIRFOIL
PIPE
DARCY
FNO
Li et al.
2020
/
/
0.0108
0.1556
WMT Gupta et al.
2021
0.0075
0.0077
0.0082
0.1541
F-FNO
Tran et al. 2021
0.0078
0.0070
0.0077
0.2322
U-NO Rahman et a.. [2022)
0.0078
0.0100
0.0113
0.1713
U-FNO Wen et al. 2022
0.0269
0.0056
0.0183
0.2231
geo-FNO Li et al. 2023b
0.0138
0.0067
0,0108
0.1556
LSM Wu et all (2023
0.0059
0.0050
0.0065
0.1535
Galerkin Cao
2021
0.0118
0.0098
0.0084
0.1401
HT-Net. Liu et al.
2022
0.0065
0.0059
0.0079
0.1847
OFormer Li et al.
(002
0.0183
0.0168
0.0124
0.1705
GNOT Hao et al. 2023
0.0076
0.0047
0.0105
0.1380
Former Li et al. (2023a)
0.0071
0.0060
0.0109
0.1214
ONO Xiao et al.
(0233
0.0061
0.0052
0.0076
0.1195
FNO (Time s)
0.0057 (1.67)
0.0072 (5.41)
0.0070 (1.10)
0.1054 (5.75)
Transolver (Time s)
0.0053 (28.03)
0.0033 (41.00)
0.0057 (18.47)
0.0900 (245.10)
Ours (Time s)
0.0052 (1.99)
0.0030 (3.30)
0.0048 (0.93)
0.1391 (7.04)
All D-SENO experiments are performed on single NVIDIA A100 80 GB PCLe GPU, and performance numbers are averaged over three independent random seeds. We report the relative L2 error of predicted fields on the held out test sets as the primary evaluation metric. For additional metrics and ablation results along with the hyperparameter details for every benchmark, we refer the reader to supplementary materials.
6 Results and Discussion
Table 2 summarizes the results obtained by D-SENO the PDE benchmarks. The model consistently outperforms the prior state-of-the-art Transolver Wu et al. (2024) on three out of four datasets, showing markedly reduced training time ("Time s' in Table represents the training time per epoch) along with superior predictive accuracy, highlighting the high accuracy and lightweight design of D-SENO. Conducting similar experiments with FNO+, we find that it consistently achieves significantly higher accuracy than the original FNO in all four datasets. Moreover, it oulperforms many neural operators developed after FNO on the same benchmarks. For details on how its performance varies with the number of Fourier modes and the projection width, we refer the reader to the supplementary material.
Figure 3 compares the Mach number reconstructions around deformed NACA- NACA-0012 airfoil in latent space. Both D-SENO and FNO+ capture the overall shock and expansion pattern, but FNO shows slightly larger errors, softening those steep Mach transitions. D-SENO, in contrast, significantly resolves these discontinu- ities, closely matching the reference peak and trough values along the airfoil surface.
Mcanwhile, Figure A presents the centerline velocity profiles for Poiscuille pipe flow. Although both operators reproduce the characteristic velocity profile, FNO+ exhibits higher errors continuously along the edges of the pipe, indicating persistent under prediction of the steep velocity gradient. D-SENO, by contrast, confines its largest discrepancies to the downstream end of the pipe and at significantly lower magnitudes than FNO+, thereby preserving accurate gradient structure throughout the channel.
Figure 5 shows the results for Darcy flow, it can be observed that FNO tends to systematically over predict the pressure in regions of high permeability, resulting larger relative errors. In contrast, D-SENO exhibits minimal underestimation overall. This behavior further highlights D-SENO's superior ability to resolve sharp transitions and fine-scale heterogeneity in complex porous medía. flows
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1.25
1.25
0.02
1.00
1.00
0.00
0.75
0.75
20
0.02
0.50
0.50
0.25
0.25
-0.44
100
200
100
200
100
200
X
(a) GT
(b) Ours
(c) Err (Ours)
1.25
0.02
1.00
0.00
0.75
-0.0
0.50
0.25
-0.04
100
200
100
200
X
(d) FNO
(e) Err (FNO+)
Figure 3: Sample from the Airfoil dataset.
128
0.3
0.002
W
0.001
96
0.2
0.2
0.000
64
-0.001
0.1
0.1
32
-0.002
0.0
0.0
-0.003
32
64
96
128
32
64
96
128
32
64
96
128
(a) GT
(b) Ours
(c) Err (Ours)
0.3
0.002
0.001
0.2
0.000
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0.1
0.002
0.0
-0.003
32
64
96
128
32
64
96
128
(d) FNO
(e) Err (FNO+)
Figure 4: Sample from the Pipe dataset.
In the Navier-Strr Stokes torus dataset shown in Figure 6 FNO outperforms D-SENO. This may be due to periodic boundary conditions on the unit torus, that align naturally with FNO's Fourier-based representation, enabling it capture spatiotemporal flow patterns more effectively. Moreover, the 2D+T forecasting task leverages FNO+'s ability perform multi-step predictions in the frequency domain. In contrast, D-SENO
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0.0125
0.0125
80
0.0002
0.0100
0.0100
0.0075
0.0075
0.0000
40
0.0050
0.0050
-0.0002
20
0.0025
0.0025
0.0000
0.0000
0.0004
0
21
42
63
84
21
42
63
84
21
42
63
84
X
(a) GT
(b) Ours
(c) Err (Ours)
0.0125
0.0002
0.0100
0.0075
0.0000
0.0050
0.0002
0.0025
0.0000
0.0004
21
42
63
84
21
42
63
84
(d) FNO
(e) Err (FNO*)
Figure 5: Sample from the Darcy dataset.
Predictions Ground Iruth
Prodictions Ground
Error
Eroor
0
Figure 6:
Predicted D-SENO
results for
Figure 7: Predicted FNO+ results for Navier Stokes fields at times f. = 12, 14, 16, 18, and 20 (left. to right).
Navier-Stokes fields at times t 12, 14, 16, 18,
and 20 (left to right).
primarily performs better for static spatial deformations and does explicitly exploit temporal periodicity, which likely limits its accuracy in this dataset. Future extensions, such integrating periodic temporal kernels or expanding D-SENO's spectral receptive field may help bridge this gap.
These results collectively demonstrate that D-SENO captures discontinuities better than FNO+ not only in block-ddoinated airfoil flows and shear-layer pipe flows, but also in Darcy porous-media media flows. By using multiple rates dilated convolutions to expand its receptive field, aggregating multiscale context, and using squeezo-and-exciite excitation blocks adaptively re- re-weight channel responses, D-SENO sharpens and emphasizes steep gradients, preserving them with lower local error and resolving fine-scale heterogeneity in the solution.
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7 Conclusion
This paper introduces the D-SENO, a fully convolution surrogate that combines receptive-fildd expanding dilated kernels with lightweight squeeze and excitation channel attention. By retaining strict locality, that is, no Fourier transforms, self-attention, positional encoding, D-SENO reduces the time complexity for training the model while capturing domain-scale interactions that are essential in fluid and porous-modia flows. Extensive experiments on canonical PDE benchmarks show that D-SENO altains low relative error while delivering speed-ups of multiple orders of magnitude over transformer- beasooedee-- solvers and neural oper- ators. Beyond its empirical performance, D-SENO also offers practical benefits it relies exclusively on standard convolution operations, enabling efficient execution existing, highly optimized hardware.
Limitations. While D-SENO achieves strong accuracy and efficiency trade-offs on a range of benchmarks, several aspects merit further development. First, on the Navier-Stokes benchmark, transformer-based solvers and neural operators often attain higher accuracy, suggesting that more explicit global context aggregation may be beneficial for highly multiscale dynamics. Second, our evaluation focuses on structured-gria datasets, extending the framework to unstructured meshes and more complex geometries an important next step to broaden applicability. Third, the present study is primarily empirical, and complementary theoretical analysis of approximation, stability, and generalization would strengthen understanding. Finally, dilation schedules are currently chosen per dataset, learning or adapting dilation patterns automatically could improve robustness, with potential compute and accuracy trade-offs.
Future work. Promising avenues for future work include extending D-SENO to unstructured meshes via masked or graph-basod dilation, integrating physics-informed priors or constraints to improve gencralization and combining D-SENO with attention modules to capture rangrange dependencies even better. We envision D-SENO a simple, computationally efficient baseline and flexible building block for the next generation of neural solvers in science and engineering.
8 Acknowledgment
This work was supported by the NYUAD Center for Interacting Urban Networks (CITIES), funded by Tamkeen under the NYUAD Research Institute Award CG001. The views expressed in this article are those of the authors and do not reflect the opinions of CITIES or their funding agencies.
References
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Zongyi Li, Daniel Zhengyu Huang, Burigede Liu, and Anima Anandkumar. Fourier neural operator with learned deformations for pdes on general geometries. Journal of Machine Learning Research, 24(388): 26, 2023b.
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Md Ashiqur Rahman, Zachary E Ross, and Kamyar Azizzadenesheli. U-no: U-shaped neural operators. arXiv preprint arXic:2204. 11127, 2022.
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Table 3: Training hyperparameters for each benchmark.
Benchmark
Daraio
Bist
Width
Loss
Epochs
Batch
LR
Step
Decay
Weight Decay
Airfoil
1000
200
64
Rel. La
500
20
10
100
0.5
10
Pipe
1000
200
96
Rel. L2
500
20
1 10-3
100
0.5
1 10
Darcy
1000
200
48
Rel. L2
500
20
1 10-3
100
0.5
1 10
Navier Stokes
1000
200
Rel. L2
500
20
x 10
100
0.5
1 1x
Table 4: Model parameterization for each benchmark.
Dataset
DC Block Conv 1
DC Block Conv 2
SE Reduction
SE Block Conv
SE Block Conv
Kernel
Blas
Kernel
Bias
Kernel
Bias
Kernel
Bias
Airfoil
3 3
Yes
5 5
No
1x1
Yes
1 1xt
Yes
Pipe
3 3
Yes
3
Yes
1x1
Yes
1x
Yes
Darcy
Yes
5
No
1x1
Yes
1x1
Yes
Navier Havier-Stoles
3 x.x.3
Yes
3
Yes
1x1
Yes
Haixu Wu, Huakun Luo, Haowen Wang, Jianmin Wang, and Mingsheng Long. Transolver: A fast transformer solver for pdes general geometries. arXiv preprint ar arXiv:2402. 026:6400.0266, 2024.
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Bo Xu, Xinliang Liu, and Lei Zhang. Dilatod convolution neural operator for multiscale partial differential equations. Journal of Compulational and Applied Mathemalics, 461:116408, 2025.
Fisher Yu and Vladlen Koltun. Multi-scale context aggregation by dilated convolutions. ar preprinl arXiv:1511.07129 2015.
A Appendix
A.1 Ablation Study
Our ablation methodology addresses five architectural and data-related parameters: (i) depth (number of DS blocks), (ii) squeeze and excitation block, (iii) dilation configuration, (iv) projection width (latent channel dimensionalityy, and (v) dataset resolution (for Darey dataset only). We apply this study our model D- SENO (Dilated Squeeze-and Excitation Neural Operator), FNO, the FNO variant introduced in the main paper, and Transolver, and discuss each parameter in detail in the following subscctions. Table and Table 4 details the training configuration (optimizer type, learning rate and schedule, batch size, number of epochs, etc.) and the model configurations used for each of the datasets in the following experiments. Note that D-SENO, + and FNO use the same parameters, while for Transolver the parameters given in Wu et al.) (004)) were used unless otherwise specified. The datascts used in this study are publicly available: airfoil potential flow and Poiseuille pipe flow datasets are available at. http://hhhhhha..omm//eeaapppttrr Geo-FNO and the Darcy flow and Navier Stokes datasets are available at lntps://drrive.gogl...
Equueze-and-Exxiiattio
A.1.1 Depth Sensitivity
In this subscction, we evaluate how model depth, defined by the number of DS blocks, affects the predictive accuracy. For each of the three steady benchmark datasets (Airfoil, Pipe and Darcy dataset), we system- atically increase the number DS blocks to quantify accuracy gains as model depth grows, while keeping other architoctural and training settings fixed. At each depth, we record the complete configuration and report the time per epoch (in seconds), the number of trainable parameters (in millions,, and the relative La error; all error values are averaged multiple random seeds to mitigate variance. In addition to
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Table 5: Model performance summary on the Airfoil dataset.
Model
# DS Blocks
Time per Epoch (s)
# Parameters (M)
Relative L2 Error
Model Airfoil-A
1
0.43
0.156
0.0131
Model Airfoil-B
0.304
0.0072
Model Airfoil-C
3
0.93
0.451
0.0067
Model Airfoil-D)
4
1.16
0.599
0.0057
Model Airfoil-E:
1.43
0.747
0.0055
Model Airfoil-F
6
1.68
0.894
0.0055
Model Airfoil-G
7
1.99
1,042
0.0052
Model Airfoil-G w/o SE
1.70
0.984
0.0056
Model Airfoil-G w/o SE (PM)
7
2.05
1.042
0.0056
Model Airl
7
2.04
1.042
0.0054
Transolver
28.03
2.811
0.0053
FNO (m 8)
2.123
0.0058
FNO (m = 16)
1.51
8.414
0,0059
FNO (m 24)
1.67
18.899
0.0057
FNO Original (m = 24)
2.47
37.774
0.0060
Table 6: Detailed dilation values for Airfoil dataset showing dj and dy in each model.
Model
dz
dg
Model Airfoil-A
[16]
[6]
Model Airfoil-B
[16, [4]
[4,10]
Model Airfoil-C
[16, (16,48,22
[1,6,4]
Model Airfoil-I)
[16, 56,56,,00 56,30,2]
[1,2,10,4]
Model Airfoil-E
[1,2,10,6,1]
Model Airfoil-F
42,,66,,22,3,,,,,1
[1,2,8,12,6,1]
Model Airfoil-G
[16, 32,
[1,2,8,12,6,2,11
Model Airfoil-G w/o SE
[16 36,,42,6,,,,,24
[1,2,8,12,6,2,11
Model Airfoil-G w/o SE (PM)
166,56,42,36,32,22,,
[1,2,8,12,6,2,1]
Model Airfoil-G-alt
[2,4.8,10,8,4,2]
model, we also report the above metrics for Transolver, FNO and FNO We test FNO + by varying the number of Fourier modes, then test FNO using the modes count that gave the best results for FNO
The results for airfoil, pipe, and Darcy dataset are summarized in Tables 5 and Note that in the tables, each model variant denoted by appending successive letters (A, B, C, ...) to name, corresponding to increasing numbers of DS blocks. The results show that increasing the number of DS blocks lowers the relative L2 error, this is because the additional dilation rates expand the receptive field and yield significant accuracy gains, but this comes with increased computational cost, as can be seen by the increased number of parameters and higher training time per epoch. It can also be seen that FNO significantly outperforms the regular FNO in all cases, showing that its architectural enhancements, increase its expressive capacity to capture the underlying dynamics. Similar experiments on the time-varying Navier-Stokes dataset, shown in Table 11 were carried out with only single dilation rate, since D-SENO did not outperform FNO*. The details of the exact dilation rates used for the above experiments for each of the four dataset, given in 10 and 12 where dz gives the dilation rates along dimension and dy gives the dilation rates along the y dimension.
Table 6
A.1.2 Squeeze and Excitation impact
In this subsection, we assess the impact of the Squeezo-and-Excitn D-SENO to a variant with the SE block removed from each DS block. The ablated model is identified in the results table by the suffix "w/o SE" appended to its name. From Tables 5 observed that omitting the SE block leads to a substantial degradation in D-SENO's O's accuracy, reflected in a significant increase in the relative L2 error, which highlights the importance of the SE module for effective feature representation learning.
(SE) block by comparing the full
and II it can
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Table 7: Model performance summary on the Pipe dataset.
Model
#
Blocks
Time / Epoch (s)
Parameters (M)
Relative L2 Error
Model Pipe-A
1
0.77
0.197
0.0107
Model Pipe-B
2
1.29
0.382
0.0073
Model Pipe-C
3
1.79
0.567
0.0046
Model Pipe-D
4
2.31
0.752
0.0040
Model Pipe-E
5
2.80
0.936
0.0035
Model Pipe-F
6
3.47
1.121
0.0031
Model Pipe-G
7
4.28
1.306
0.0030
Model Pipe-G w/o SE
7
3.30
1.176
0.0030
Model Pipe-G w/o SE (PM)
7
4.01
1.306
0.0037
Model Pipe-G-alt
7
4.05
1.306
0.0030
Transolver
40.01
2.811
0.0033
FNO (m 8)
5.41
4.769
0.0074
FNO + (m = 16)
5.79
18.925
0.0075
FNO (m 32)
6.72
75.548
0.0072
FNO Original (m = 32)
4.44
9.488
0.0086
Table 8: Detailed dilation values for Pipe dataset showing dg and dy each model.
Model
d
d
Model Pipe-A A
[9]
[9]
Model Pipe-B
[23, 1]
[23, 1]
Model Pipe-C
[23, 11,
[23, 11,1]
Model Pipe-D
[23,15, 11,,711]
[23, 15,7,1,1
Model Pipe-E
123,15,9,3,1]
[23,15,9,3,1]
Model Pipe-F
[25, 125,19,11,,,3.1
[25,19,11,7,,,1)
Model Pipe-G
12,,77,3,,,,7,3.11
[23,17,13,9,7,3,
Model Pipe-G w/o SE
123,17,
[23,17,13,9,7,3,1]
Model Pipe-G w/o SE (PM)
128,17,18,,,7,8.77
123,17,18,,,,,3.11 125,19,11,9,5,3.11
Model Pipe-G-alt
[25 126,19,11,
Additionally, conduct a parameter- matchhd ablation to separate the effect of the SE mechanism from that of model capacity. In this variant, the SE block is still removed, but the parameter count is restored to match the basca D-SENO by appending two 1 x point-wise convolutions after the dilated spatial convolutions in each DC block. This adds learnable channel-mixing capacity while preserving the original receptive field and residual structure, yet does not perform explicit SE-style channel recalibration. We denote this model as "w/o SE (PM)" the results, so that performance differences relative the full model primarily reflect the functional role of SE block rather than reduced parameters. From the results shown in Tables performance than the garameter-matcheer baseline the airfoil, pipe, and Navier Stokes datasets, while the baseline obtained by simply increasing the number of parameters performs better only on the Darcy flow task. This pattern indicates that the performance gains largely stem from the SE architecture rather than more model size, and that overall our SE-based approach provides the stronger model across datascts.
and 11
that the SF-based model consistently achieves better
A.1.3 Dilation rate impact
In this study, the dilation rates are dataset-specific hyperparameters rather than learned quantities, selected via a small number of manual trials using a simple, well-spaced pattern that covers both local and global spatial scales. This is a heuristic choice, not an exhaustive grid search or an adaptive method. To isolate the influence of receptive-field enlargement strategy, we replace the adopted dilation rates in the SE- basbased model with an alternative rates for each dataset, keeping all other hyperparameters constant, this reveals the sensitivity of the model to the chosen pattern of dilation. Models employing these alternative rates are identified by the suffix "Lalt" appended to their names in the results. The exact dilation rates used given
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Table Model performance summary on the Darcy dataset.
Model
# DS Blocks
Time
Epoch (s)
Paramcters (M)
Relative L2 Error
Model Darcy-A
0.35
0.089
0.0282
Model Darcy-B
0.46
0.173
0.0247
Model Darcy-C
3
0.55
0.256
0,0084
Model Darcy-D
0.69
0.338
0.0064
Model Darcy-E
0.80
0.422
0.0051
Model Darcy-F
6
0.93
0.505
0,0048
Model Darcy-F w/o SE
0.80
0.476
0.0053
Model Darcy-F w/o SE (PM)
0.99
0.505
0.0046
Model Darcy-F-alt
0.92
0.505
0.0050
Transolver
18.48
2.811
0.0057
FNO (m - 8)
0.78
1.196
0.0086
FNO (m 16)
0.90
4.735
0.0073
FNO (m 32)
1.10
18.890
0.0070
FNO (m - 42)
1.30
32,531
0.0071
FNO Original (m = 32)
1.92
37.765
0.0089
Table 10: Detailed dilation values for Darcy dataset showing dg and dy in each model.
Model
d
dy
Model Darcy-A
[7]
[7]
Model Darcy-B
[1,19]
[1,19]
Model Darcy-C
[1,11,19]
[1,11,19]
Model Darcy-D
[1,7,13,19]
[1,7,13,19]
Model Darcy-E
[1,5,9,13,19]
[1,5,9,13,19]
Model Darcy-F
[1,3,5,9,18,19]
(1,3,5,9,13,99]
Model Darcy-F w/o SE
[1,3,5,9,13,99
[1,3,5,9,13,99]
Model Darcy-F w/o SE (PM)
[1,3,5,9,13,19] 11,,,7,11,15,21]
[1,3,5,9,13,19]
Model Darcy-F-alt.
11,3,7,11,15,211
It can be observed that, across the datasets, using alternative dilation rates results in only negligible changes in accuracy, showing the robustncss of the model.
in Table 6
S 10 and 12
A.1.4 Projection width
In this case, we vary the projection width of the inputs examine the trade-off between representational capacity and performance of the best performing model from Tables 7 and g The results are shown in F It can be observed that the relative L2 loss decreases with projection width until it reaches a dataset specific minimun. Beyond this width, further increases produce negligible improvement. Also, projection width increases, both time per epoch and parameter count grow, reflecting the additional computational cost.
A.1.5 Resolution robustness
To probe resolution robustness, used heterogeneous Darcy flow dataset. The dataset uniformly down sampled from the original 421 x 421 grid to several spatial resolutions. Each of these resolutions used to test the three models Transolver, FNO+ and D-SENO shown in Table 13 The results indicate that D-SENO oulperforms both Tra and FNO- across all resolutions. The exact rates of dilation used for D-SENO are given in Table [5] Complementing these findings, Table [4] reports a systematic analysis of how varying the number of retained Fourier modes influences performance at each resolution for FNO+. Note that the hyper parameters and error used here for Transolver are the same as mentioned in Wu et al. 2224h for Darcy dataset.
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Table 11: Model performance summary on the Navier Stokes dataset.
Model
# DS Blocks
Time / Epoch (s)
Parameters (M)
Relative L2 Error
Model NS-A
7.04
0.666
0.1391
Model NS-A w/o SE
5.94
0.600
0.1432
Model NS-A w/o SE (PM)
7.53
0.666
0.1433
Model NS-A-alt
7.67
0.666
0.1431
Transolver
245.10
11.232
0.0900
FNO* (m = 8)
5.75
2.123
0.1054
FNO+ (m 16)
5.81
8.414
0.1122
FNO (m = 32)
6.93
33.580
0.1153
FNO Original (m m 8)
13.53
4.220
0.1436
Table 12: Dilation values (dæ and dy) for models corresponding to performance results in Table 11
Model
dy
Model NS-A
1,3,, 1,, [15,25, 1,,,,,,,,,,,, 11,22,,1,,1,,9,7,88,
Model NS-A w/o SE
15,25,17,13,7,5,311,
Model NS-A w/o SE (PM)
[15,25.17,18,7,5,3,11 121,27,19,11,9,77,388
Model NS-A-alt
A.2 FNO Architecture
We introduce FNO+ as a minimally modified variant the original FNO, designed to serve as a strong and improved baseline. In its original formulation, the FNO architecture consists of a lifting layer that maps the input fields to higher-dinennsional channel space, followed by stack of Fourier layers and projec- tion layer that maps back to the target field. Each Fourier layer performs spectral convolution with the retained Fourier modes, combined with a point-wise linear transformation in physical space. The default im- layers within each Fourier block, while
plementation uses ReLU activations and includes realizing complex Fourier multiplications via separate real and imaginary linear projections. Architecturally, FNO+ has the following changes compared to the standard FNO: (i) we replace the ReLU nonlinearity with GELU in all FNO layers, (ii) we remove all nathh-oormllizttonn layers, and (fii) we implement complex- valued Fourier multiplication directly using single einsum- based operation. Beyond these changes, the layer types, block strueture, and connectivity patterns are kept identical to the original FNO design. On top of this fixed architecture, consider purely experimental variations, reported in the appendix, such runs with different numbers of retained Fourier modes and different widths in the lifting layer to probe the effect of spectral resolution and model capacity.
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Aifoll
Atfoll
Parameters (Millions)
Akfoil
Pipe
Time Epoch
Dancy
0.006
Darcy
Durey
Relative La
2-
0.004
25
50
75
100
50
100
25
50
75
100
Width
Width
Width
(a) Time/epoch
(b) Rel. L2
(c) Params (M)
Figure 8: Performance metrics projection width.
Table 13:
Performance metrics (time epoch, parameter count, and relative L2 error) for different models at various input resolutions on the Darcy dataset.
Model
32
32
64 x 64
128 x 128
256 x 256
Time (s)
Params (M)
Rel.
Time
Params (M)
Rel.
Time
Params
Rel.
Time
Params
Rel.
L2
(s)
L2
(s)
(M)
L2
(s)
(M)
L2
Transolver
12.21
2.827
0.0135
12.56
2.827
0.0069
38.95
2.827
0.0052
151.73
2.827
0.0058
FNO-
0.66
4.734
0.0159
0.74
18.890
0.0083
2.33
75.513
0.0065
9.52
302.006
0.0063
D-SENO
0.35
0.256
0.0131
0.70
0.588
0.0063
1.73
0.505
0.0042
7.96
0.588
0.0036
Table 14: 2D Darcy relative Le loss and per-epoch time number of modes across resolutions for FNO +
MODES
32 x 32
64 x 64
128
128
256 x 256
PARAMS (M)
Rel lg
TIME (s)
Rel 1.2
TIME (s)
Rel l.2
TIME (s)
Rel Lgg
TIME (s)
8
0.0157
0.65
0.0092
0.66
0.0081
1.19
0,0080
5.32
1.195
16
0.0159
0.66
0.0084
0.67
0.0070
1.23
0.0068
4.96
4.734
32
0.0083
0.74
0.0067
1.44
0.0065
5.85
18,890
0.0065
2.33
0.0064
75.513
128
0.0063
9.52
302.006
Table 15: Dilation values (d,, and dy) for models corresponding to performance results in Table 13
Resolution
dx
dy
32 32
[1,2,6] [1,3,5,7,9,13,, 11,5,9,15,21,1 21,6,7,15,23,3, 23,99,,1,,,, (1,5,7,15,,3,39,
[1,2,6] [1,3,5,7,9,13,]] [1,5,9,15,21,2]
64 X 64
128 X 128
256 x 256
19


Reviews:
[{"review_id": "4bxdseL55N", "reviewer": "Reviewer_pPAC", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes D-SENO, a convolutional neural operator for solving PDEs that combines dilated convolutions with squeeze-and-excitation (SE) blocks. The method is designed to model long-range spatial dependencies while maintaining computational efficiency by avoiding Fourier layers and self-attention.  Experiments on four PDE benchmarks—airfoil flow, pipe Poiseuille flow, Darcy flow, and Navier–Stokes—show that D-SENO achieves competitive or superior accuracy to neural-operator and transformer-based models, and demonstrates ~20× speed improvements in maximum. Ablation studies further assess depth, dilation patterns, SE impact, and resolution robustness.", "claims_and_evidence_yn": "No", "claims_explanation": "- Limited Novelty\n\nApart from combining two existing techniques, dilated convolutions and squeeze-and-excitation, this paper has limited novelty. This paper should provide a more comprehensive comparison between D-SENO and existing work (CNO and DCNO), both conceptual and empirical, to demonstrate the key differences and improvements.\n\n\n- Insufficient Experimental\n\nThe most important baselines, CNO and DCNO, are not included in the experiment. Given that they are the most related works, their absence makes it comparison unfair. Although Table 2 presented results for more than ten baselines, it is unclear which results were reproduced by the authors and which numbers were directly copied or from original papers. Also, this paper does not specify hyperparameters for these baselines, making it unclear whether the baselines are well-tuned. \n\n- Reproducibility Issues\n\nThe provided code in the supplementary material does not contain the necessary data for reproduction, e.g., `NACA_Cylinder_X.npy`.\n\n- Claims of Efficiency Lack Critical Measurements\n\nThe major contribution claim is a gain in training efficiency, but the comparison in model size and memory requirement is missing. This is critical in understanding potential tradeoffs.", "audience_yn": "No", "audience_explanation": "The degree of interest is highly moderate due to limited framework novelty and missing empirical evidence.", "requested_changes": "- Add empirical comparisons with CNO and DCNO.\n- Specify experiment details about baselines, such as hyperparameters.\n- Ensure that the supplementary codebase includes all required data\n- Report additional measurements beyond “time per epoch\", such as model size comparisons", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "HTXXO0zT4T", "reviewer": "Reviewer_kvk2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This manuscript proposes D-SENO, a neural operator learning model that combines local dilated convolutions with global squeeze-and-excitation blocks, enabling it to combine local and global information much more efficiently than current state of the art transformer or Fourier neural operators. The authors demonstrate the effectiveness of their model on four standard PDE learning benchmarks: airfoil potential flow, Poiseuille pipe flow, heterogeneous Darcy flow, and the time-dependent incompressible Navier–Stokes equations.\nshow state-of-the-art performance at a fraction of the computational cost. \n\nStrengths:\n- The methodology as presented is sound and details of the model are well explained. The contributions are contextualized well within prior relevant work, the framing of the architecture within three stages lifting, processing and projection makes it easy to understand.\n- The evaluation across four standard PDE benchmarks clearly demonstrate the effectiveness of the model, with significant gains in efficiency while still preserving state-of-the-art accuracy on all the problems.\n- The Appendix also contains a series of ablation studies showing the impact of the different design choices made by the authors. Overall, the results show that the model accuracy consistently improves with depth and is not too sensitive to the exact choice on the dilation pattern.\n- The implementation and inclusion of the improved FNO+ model is also a great contribution in itself, the authors should probably explain the improvements in better detail in the Appendix.\n\nWeaknesses:\n- No major weaknesses, though a few minor ones that can improve the quality of the paper\n- Some theoretical grounding for the approach would be welcome, it is currently not clear why the dilated convolutions and SE-blocks should lead improvements in operator learning and generalization, particularly because the appendix shows minor improvements in accuracy from the SE-blocks, unlike the claim by the authors themselves. This minor accuracy improvement can easily be justified simply the the increased number of parameters, rather than the effective exploitation of global information. Maybe some comparison of a smaller D-SENO model vs a model without SE-blocks of a similar parameter count would result in some more insights?\n- The paper mentions “dataset-specific dilation rates,” but doesn’t describe how these are selected. This makes it harder to assess whether the model can easily be tuned and generalize on new datasets.\n- The authors mention that the model's purely convolutional nature allows it to be applied on irregularly sampled data, unlike FNOs, but do not really demonstrate this claim.\n- No discussion of model limitations, beyond just the brief mention of difficulties with periodic data such as the Navier-Stokes equations problem. It would be nice to see the authors discussing other potential limitations or drawbacks of the current approach.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Overall, most of the claims made are supported by accurate, convincing and clear evidence. As mentioned before, the authors evaluate their model on a series of standard PDE benchmarks and demonstrate both state-of-the-art performance and high computational efficiency. \nThe only claim that I would say is not well supported is the claim that the inclusion of the SE block significantly improves the accuracy of the model, which I feel is not supported by the results in the appendix. However, as I mentioned the authors can perhaps do an additional small experiments to better support their claim.", "audience_yn": "Yes", "audience_explanation": "The field of ML for physics and PDE learning in particular has been increasingly popular and relevant recently, so I am sure that most TMLR readers working in this or related fields would be interested to learn of this model, given it's good performance and relative computational efficiency.", "requested_changes": "None of these changes would be critical for my recommendation, but I will sort in order of decreasing potential to improve the work:\n\n- Provide additional analysis comparing models with and without SE-blocks at matched parameter counts to verify whether performance gains stem from architectural benefits rather than increased model size.\n- Describe the procedure for choosing “dataset-specific” dilation rates — whether it’s heuristic, grid-searched, or adaptive — and discuss how this process might generalize to new PDE datasets.\n- Either add a small experiment or discussion to support the claim that D-SENO can handle irregular or nonuniformly sampled data (e.g., masked grids or unstructured domains).\n- Expand the conclusion or discussion section with a short reflection on the current model’s limitations\n- Describe the improvements made to the original FNO model in more detail in the Appendix\n- Include a brief theoretical or conceptual justification for why dilated convolutions and SE-blocks should enhance operator learning and generalization, beyond empirical evidence.", "broader_impact_concerns": "There are no broader impact concerns that I am aware of.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "MiJkIxaEnw", "reviewer": "Reviewer_HWMT", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Current PDE surrogates lead to expensive training and slow inference due to the large number of parameters in the models. Applications of PDE surrogates require models that are fast and accurate. The paper presents a lightweight operator learning framework for solving a range of PDEs while achieving up to 20x speedup over standard transformer-based models and neural operators. It also matches or improves upon the accuracy of PDE solving. \n\nThe paper identifies limitations of prior work that are fixed by the proposal. Fourier transform based operators struggle with localized features due to lack of spatial localization in spectral kernels. Convolutional neural operators do not adaptively enlarge receptive fields and dilated convolutional operators are limited to periodic inputs due to the use of the FFT.\n\nThe method used is dilated convolution (DC) blocks with squeeze-and-excitation (SE) modules that jointly capture e wide receptive fields and dynamics in addition to channel-wise attention. Dilation rates allow the receptive field to focus on critical regions to model long range dependencies.\nThe dilations allow the receptive field to grow without downsampling or increasing the number of parameters.\nThe SE block adjusts the importance of each feature channel. It uses global average pooling to compute channel-wise statistics which are transformed into scaling weights and applied with element-wise multiplication.\nThe blocks are combined into a DS layer which begins with a dilated conv block followed by a SE block. The full model follows a lifting, processing and project setup and if fully convolutional.\n\nExperiments are performed on four benchmarks datasets (Airfoil, Pipe, Darcy and Navier Stokes) also used by previous operator approaches, spanning structured and regular grids. From Table 2 test errors are on-par or better than the transformer baseline except for the Navier-stokes dataset. For the other datasets performance appears to be on-par with the transformer model with faster inference time. The performance in inference time appears to be on-par with FNO and is worse in some cases. Accuracy is better than FNO also except for the Navier-Stokes dataset where it is worse on both counts (accuracy and time).\n\nIn the discussion (second last paragraph page 8) the paper states “... higher accuracy than the original FNO on all three datasets”. I wonder why this does not count the Navier-Stokes dataset in table 2?", "claims_and_evidence_yn": "Yes", "claims_explanation": "The main claim of the paper is a method that is lightweight, fast and accurate for PDE surrogate modeling tasks. Speed and accuracy are shown by experiments on four datasets where the method is shown to be significantly faster than the transformer models with similar accuracy on three out of the four datasets. The performance on the fourth dataset is somewhat worse in accuracy but better in time.\n\nThe model is lightweight by being fully convolutional and has significantly fewer parameters, shown in Table 13.", "audience_yn": "Yes", "audience_explanation": "The model presents a simple PDE surrogate model which combines architectural features (Dilated convolutions, squeeze-and-excitation block) to develop a lightweight operator model. This should be of interest to researchers in efficient PDE surrogate modeling.", "requested_changes": "Section 4.2 says “these layers reduce and restore the number of channels…”. Explanation as to how and where this is done should be included at this point.\n\nCould you add an explanation of dilation rates and adaptive receptive fields in the main paper itself.? In what sense are these adaptive if they have to be chosen per dataset?\n\nCan you explain why you don’t consider all four datasets when summarizing the performance comparison with FNO in section 6 paragraph 1?\n\nTypos\n\nSection 2. DNCO should not be bold.\nIn section 4.1 G(a) = u should be written as u = G(a)\nHow is the inference time in Table 2 computed? Is this per example? per batch?", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)