The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operators
Authors: Mansi Sakarvadia
Kareem Hegazy
Amin Totounferoush
Kyle Chard
Yaoqing Yang
Ian Foster
Michael W. Mahoney
Venue: ICLR 2026 Poster (2026)
Keywords: Partial differential equations
Neural operators
super-resolution
zero-shot super-resolution
multi-resolution training
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: Machine-learned operators fail to perform zero-shot super-resolution and are susceptible to aliasing; we propose a multi-resolution training scheme to enable accurate cross-resolution inference.

Abstract: A core challenge in scientific machine learning, and scientific computing more generally, is modeling continuous phenomena which (in practice) are represented discretely. Machine-learned operators (MLO) have been introduced as a means to achieve this modeling goal, as this class of architecture can perform inference at arbitrary resolution. In this work, we evaluate whether this architectural innovation is sufficient to perform “zero-shot super-resolution,” namely to enable a model to serve inference on higher-resolution data than that on which it was originally trained. We comprehensively evaluate both zero-shot sub-resolution and super-resolution (i.e., multi-resolution) inference in MLOs. We decouple multi-resolution inference into two key behaviors: 1) extrapolation to varying frequency information; and 2) interpolating across varying resolutions. We empirically demonstrate that MLOs fail to do both of these tasks in a zero-shot manner.
Consequently, we find MLOs are not able to perform accurate inference at resolutions different from those on which they were trained, and instead they are brittle and susceptible to aliasing. To address these failure modes, we propose a simple, computationally-efficient, and data-driven multi-resolution training protocol that overcomes aliasing and that provides robust multi-resolution generalization.

Published as a conference paper at ICLR 2026
THE FALSE PROMISE OF ZERO-SHOT SUPER- RESOLUTION IN MACHINE-LEARRNE OPERATORS
Mansi Sakarvadia : Kareem Hegazy Amin Totounferoush", Kyle Chard', Yaoqing Yang'. Ian Foster', Michael W. MaMonee 'University of Chicago, 'Lawrence Berkeley National Laboratory. 'University of Stuttgart, DDattoouth College, "International Computer Science Institute, Univeerstyy of California, Berkeley
ABSTRACT
A core challenge in scientific machine learning, and scientific computing more generally, is modeling continuous phenomena which (in practice) are represented discretely. Machine-learned operators (MLO) have been introduced as a means achieve this modeling goal, this class of architecture can perform inference at arbitrary resolution. In this work, we evaluate whether this architectural in- novation is sufficient to perform "zero-shot super-resolution." namely to enable a model to serve inference on higher-resolution data than that on which it was originally trained. We comprehensively evaluate both zero-shot sub-resolution and super-resolution (i.c., multi-rcsolution) inference in MLOs. We decouple multi-resolution inference into two key behaviors: 1) extrapolation varying fre- quency information; and 2) interpolating across varying resolutions. We empiri- cally demonstrate that MLOs fail to do both of these tasks in a zero-shot manner. Consequently, we find MLOs are not able perform accurate inference at resolu- tions different from those on which they were trained, and instead they are brittle and susceptible to aliasing. To address these failure modes, we propose simple, computationally-cffiiiii and data-driven multi-resolutionn training protocol that overcomes aliasing and that provides robust multi-resolution generalization.
INTRODUCTION
Modeling physical systems governed by partial differential equations (PDEs) is critical to many computational science workflows:
= (Si),
(I)
where M is an approximation of the PDE's solution operator, S1 is the input state of the system, and S2 is the predicted state. Central to this problem formulation is that continuous physical systems must be sampled and, therefore, modeled discretely. For discrete model, M, to be useful in repre- senting phenomena of different scales, scientists require the ability to use it at different resolutions accurately. For example, when modeling fluid flow, scientists often use adaptive mesh refinement (Berger & Oliger, 1984), technique that increases simulation resolution in areas that require high accuracy (c.g., regions turbulence), and coarsens it in less critical regions.
Traditionally, the approximation . is constructed by numerical methods which, by design, can be employed at arbitrary discretization (Forrester et al., 2008; Cozad et al., 2014; Asher et al., 2015; Su- dret et al., 2017; Alizadeh et al., 2020; Kudela & Matousek, 2022). However, numerical methods are computationally expensive. Alternatively, machine-learned operators (MLOs), a class of data-driven machine learning (ML) models which parameterize the solution operator families of PDEs, have been proposed (Raissi et al., 2019; Li et al., 2020a; Lu et al., 2021b; Kovachki et al., 2023; Raonic al., 2023). Although querying MLOs at arbitrary discretization is computationally inexpensive, it is not obvious that this can be done accurately. The Fourier Neural Operator (FNO) (Li et al., 2020a), a specific MLO, claimed to address the discretization challenge in zero-shot manner (Li et al., 2020a; Tran et al., 2021; George et al., 2024; Li et al., 2024b; Azizzadenesheli et al., 2024). The claim is that FNO can be trained at resolution m and then serve accurate inference at resolution
Correspondence da aguchi oago.edu, Code: httpa: h/tgt///ttttt oom hwm.///aaaavad....mmmmmmm h/wpeaator...ooooooooooooo lastnn

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Figure 1: Aliasing in zero-shot super-resolution. Model trained resolution 16 data, and eval- uated at varying resolutions: 16, 32, 64, 128. Top Row: Sample prediction for Darcy flow; notice striation artifacts resolution 128. Middle Row: Average test set 2D energy spectrum of label and model prediction. Bottom Row: Average residual spectrum normalized by label spectrum.
> without training on additional high resolution data e.g., zero-shot super-resolution. This claim of zero-shot super-resolution, true, is especially attractive in settings where generating, and training on, high-resolution data is computationally expensive.
In this paper, we evaluate the claim of zero-shot super- (and sub- uub))resolution inference in MLOs. We document substantial disparity in model performance across data of different discretizations, suggesting that MLOs are generally incapable of accurate inference at resolutions greater or less than their training resolution (i.e., zero-shot multi-resolution inference). Instead, we find that MLOss often misrepresent unseen frequencies and incorrectly infer their behavior in data whose discretization differs from its training discretization, i.e., they exhibit form of aliasing (Fig. 1). In addition, informed optimization constraints (Li et al., 2024b) and (ii) band-limited learning (Raonic al., 2023; Gao al., 2025). We find that neither enables zero-shot multi-resolution, as they do not address the central issue: MI.Os, like all machine- learned models, cannot typically generalize beyond their training data (Yang et al., 2023; Liu al., 2020; Krueger et al., 2021). We establish that the discretization at which MI. ML.s are trained impacts the discretization which they accurately model the system.
study two previously proposed solutions: (i)
To enable multi-resolution inference, we propose multi-resolution training, simple, intuitive, and principled data-driven approach which trains models on data multiple resolutions. We profile the impact of different multi-resolution training approaches, finding that optimal multi-resolution performance can often be achieved via training data sets that contain mostly low resolution (less expensive) data and very little high resolution (more expensive) data. This permits to achieve low computational overhead, while also increasing the utility single MLO.
To summarize, the main contributions of our work are the following:
1. We assess the ability of trained MLOs generalize beyond their training resolution. We demonstrate that MLOs struggle to perform accurate inference at resolutions higher or lower than which are they trained on, and instead they exhibit aliasing. Based these sults, we conclude that accurate zero-shot multi-resolltion inference is unreliable (Sec. 3).
2. We evaluate two intuitive pamnachhs--incopprratinn physics-informed constraints during training, as well as performing band-limited learning- and we find that neither approach enables reliable multi-resolution generalization. (Sec.
3. We propose and test multi-resolution training, where we include training data of varying resolutions (in particular, small amount of expensive higher-resolution data and larger amount of cheaper lower-resolution data), and we show that multi-resolution inference improves substantially, without significant increase in training cost (Sec. 5).
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Figure Accurate multi-resolution inference requires both interpolation extrapolation. Original:
Signal is sampled at a rate greater than its Nyquist frequency. Interpolation: Adapting to new sampling rates of a given signal. Extrapolation: Adapting to new frequency information under constant sampling rate. Super- Resolutis: Sampling system at higher rate which enables the capture of higher frequency information (interpolation extrapolation). Aliasing: High-frequene information misrepresented as freeueegycccy information due insufficient sampling.
2 BACKGROUND ON SIGNAL PROCESSING AND ALIASING
We start by discussing the practice of training ML models to represent continuous systems via dis- crete data. Next, we outline the implications of aliasing in ML as it relates to multi-resolution inference. Finally, we formally define "zero shot multi-resolution" inference in discrete context.
Discrete Representations of Continuous Systems. The fundamental challenge in discretely rep- resenting continuous systems lies in the choice sampling rate. The Whitaker-Nyyqiit-SShee -hannoo sampling theorem established that given sampling rate r, the largest resolved frequency is r/2 (Unser, 2002; Shannon, 1949; Whittaker, 1928). Thus, ML models will be trained on discrete rep- resentations where only some of the frequencies are fully resolved. Resolving higher-order frequen- cies greater than r/2, consequently, becomes an out-of-distribution task. Aligning these discrete models' predictions with the underlying continuous system open problem (Krishnapriyan al., 2021: Queiruga et al., 2020; Ott et al., 2021; Ren et al., 2023; Takamoto et al., 2022; Subel et al., 2022; Chattopadhyay et al., 2024).
Aliasing, When sampling a continuous signal at rate r, aliasing occurs when frequency components greater than r/2 projected onto lower frequency basis functions (Fig. 2) (Gruver et al., 2022). Thus, content with frequency n > r/2 is observed as a lower frequency contribution:
n mod r
if (n mod r) < r/2
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In an MI. context, when inferring at different discretizations of given signal, aliasing can manifest as the divergence between the energy spectrum of the model prediction and the expected output. Aliasing indicates a model's failure to the underlying continuous system.
Zero-shot multi-resollutinn inference. We define multi- resoiiireooution inference the ability to do in- ference at multiple resolutions (e.g., sub- and super-resolution). The zero-shot nulti-resolltion task employs an ML model, which is trained data with some resolution and tested on data with dif- ferent resolution. Zero-shot multi-resolution inference raises two important questions with respect to the generalization abilities of trained models (see Fig. 2):
1. Resolution Interpolation. How do models behave when the frequency information in the data remains fixed, but its sampling rate changes from training to inference? Can the model interpolate the fully resolved signal to varying resolutions?
2. Information Extrapolation. How do models behave when the resolution remains fixed, but the number of fully resolved frequency components changes from training to inference? For super-resolution, this means can the model extrapolate beyond the frequencies in its training data and model higher frequency information?
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Figure 3: Resolution Interpolation. Four FNOs are trained on Darcy data at resolutions {16, 32, 64, 128) from left right with constant frequency information (low-pass limit of 8f), and are tested on varying resolutions. We assess if each model can generalize to data with varying sampling rate. We visualize spectra of the normalized residuals across test data. Notice, residual spectra (error) increases substantially in the low frequencies. Lower residual energy at all frequencies is better.
3 ASSESSING MULTI-RESOLUTTI GENERALIZATION
We examine the zero-shot multi-resolution abilities of FNO, an architecture for which this claim has been previously made (Li et al., 2020a). We study the multi-resolution inference task from an out-of-distribution data perspective by decoupling what it means for a model to perform inference resolution different from that used during training. Specifically, in Sec. 3.1, we assess whether models trained system sampled rate are capable of both interpolating accurately new sampling rates (Fig. 2(b) and extrapolating accurately to additional/fewer frequencies present in data (Fig. 2(c)). We systematically test an FNO's ability to do both objectives and show neither are achieved. In light of these failure modes, in Sec. 3.2, we then assess the spatial zero-shot sub- and super-resolutonn capabilities of FNOs and show the claim does not hold (Fig. 2(d)).
We evaluate FNO on three standard scientific datasets: Darcy, Burgers, and Navier Stokes. For each dataset we optimize FNO hyperparameters via grid-search described in Appendix B.
3.1 BREAKING DOWN MULTI- CAPABILITIES
Resolution Interpolation. We assess an FNO trained on data of a specific resolution can general- ize to data of both lower and higher resolution under fixed frequency information. Specifically, we keep the set of populated frequencies constant the train and test data, while varying the resolution of the test data. We do this by applying low-pass filter all data the highest resolution, and then down-sampling as needed. The sampling rates of all data are large enough to resolve all remaining frequencies.
We begin with a simple experiment: we train an FNO on a Darcy flow dataset at resolution N 16 and assess the trained model's performance across test datasets at varying resolutions {16, 32, 64, 128}, all low-pass filtered with limit 8f where f is the frequency unit 2 Z/N. In Fig. 3(a), we visualize the average spectral energy of the model predictions normalized by the spec- tral energy of the unfiltered ground truth for each test dataset. For the test data with resolutions that are different from the training data, we observe sharp increases in their residual's energy spectrum in frequencies greater than 8/. This is especially concerning since the model was never trained on nor shown inference data containing frequencies greater than 8/. other words, FNOs, trained in zero-shot manner, fail to reliably interpolate to varying resolutions.
In Figs. 3(b-),, we repeat the same experiment at varying training resolutions (e.g. 32, 64, 128) with low-pass limit 8f. We notice that at each training resolution, the model consistently and in- correctly assigns high energy in frequencies greater than 8f across all test resolutions. We perform corresponding experiments for the Burgers dataset with low-pass limit 64f and resolutions {128, 256, 512, 1024) and Navier Stokes dataset with low- low-pass limit 32f and resolutions E (64, 128, 255, 510} and observe the same failure mode (Appendix C: Figs. 1 11-12). We conclude that changing
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Figure 4: Information Extrapolation. Four FNOs trained on Darcy data of resolution 128 (constant sampling rate) and low-pass filtered with limits (8, 16, 32, 64) (varying frequency information) from left to right. Test if each model can generalize to data with varying frequency information. Visualizing spectrum of the normalized residuals across test data. Notice, residual spectra (error) increases substantially in the high frequencies. Lower residual energy at all frequencies is better.
resolution at test time is akin to out-of-distribution inference: the model never trained on data with broad range of sampling rates and consequently fails to generalize.
Information Extrapolation. We assess FNO trained on data containing a fixed of frequen- cies can generalize to data containing both fewer and additional frequencies under fixed resolution. Specifically, we keep the resolution constant but vary the number of populated frequencies in the train and test datasets by applying varying low-pass filters to data fixed resolution.
We begin with a simple experiment: an FNO is trained on a Darcy flow dataset at resolution 128 which is low-pass filtered at limit 8f. In Fig. 4(a), we assess the trained model's 's performance across four versions test dataset, all of which have the same sampling resolution (128) but are filtered with low-pass limits {8, 16, 32, f (e.g., increasing amounts of frequency information). There is a sharp increase the residual spectra in higher frequencies across all test sets; the residual error increases the test and training filters diverge. In other words, FNOs, trained in zero-shot manner, fail to extrapolate on data with previously unseen frequency information.
In Figs. 4(b-d), we repeat the same experiment varying training data low-pass filter limits (e.g, 16, 32, 64.)). Each model consistently and incorrectly assigns high energy in the high frequencies regardless of whether the test data contained any high-frequency information. This is concerning failure mode indicating FNOs do not generalize both in the presence of frequencies greater than, and the absence of frequencies less than, what was present in their training data. We perform corresponding experiments for the Burgers dataset with resolution 1024 and low-pass limits E (64, 128, 256, 512). and Navier Stokes dataset with resolution 510 and low-pass limits E (8, 16, 32, 255}. f and observe the same failure mode (Appendix C: Figs. 13-14). We conclude that varying the frequency information at test time effectively out-of-diittibbution inference as the model was not trained data with such variation in frequency information and, therefore, fails to generalize.
3.2 ZERO-SHOT
INFERENCE
We study whether FNOs are capable of spatial multi-resolution inference: simultaneously changing the sampling rate and frequency information. For each dataset in (Darcy. Burgers, Navier Stokes), ). we train a model on data at resolutions (16, 32, 64, 128). (1024, 512, 256, 128.). (510, 255, 128, 64), respectively.
Fig. I, we see that models trained at low resolutions do not generalize to high resolutions. Similarly, in Fig. 5, we again see a failure to generalize and instead observe high- frequency artifacts in model predictions in multi-resolution settings. Further, for time-varying PDEs, such Navier Stokes, we observe that these high frequency aliasing artifacts compound across time steps (Fig. 15). In Fig. 16, we show that models trained at a given resolution do not achieve low loss across all test resolutions. In fact, losses vary by 1 1x, 2x, and 10x across test resolutions for the Darcy, Burgers, and Navier Stokes datasets, respectively. Therefore, we conclude that Fourier neural operators not capable of consistent zero-shot super- sub-resolution.
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Figure 5: FNOs do not generalize to higher or lower resolutions. Model trained on Navier Stokes dataset resolution 255 (indicated by *), evaluated resolutions 510, 255, 128. Top Row: Ground truth, prediction at resolutions 510 (super-resolution), resolution 255 (same as train resolution), and resolution 128 (sub-resolution). Bottom Row: Average energy spectra over test data.
4 EVALUATING POTENTIAL CORRECTIVE METHODS
Here we assess two previously proposed strategies for accurate zero-shot multi-resolution inference: physics-informed optimization objectives (Li et al., 2024b) and band-limited learning (Raonic et al., 2023; Gao et al., 2025). We find that neither enables accurate nulti-resolution inference.
4.1 PHYSICAL OPTIMIZATION CONSTRAINTS
Physics-informed optimization constraints have been proposed means of achieving accurate inference in the zero-shot super-resolution setting (Li et al., 2024b). For each dataset (Darcy, Burgers, Navier Stokes), we train FNOs at all avaliable resolutions. We optimize each set of model parameters with a dual optimization objective C(0) = (1 w) * Ldata (0) + w * physs (0), where (dma is the original data-driven loss (mean squared error) with additional physics-ifformed loss Ephys. which explicitly enforces that the governing partial differential equation is satisfied. We use the physics losses of Li al. (2024b) and detail the implementation in Appendix E.
We find that the data data-driven loss always outperforms any training objective that includes a physics constraint (Fig. 17). We determine this by evaluating E (0, 0.1, 0.25, 0.5) for (Darcy. Burg- ers, Navier Stokes) } at resolutions 64, 512, 255, respectively. Among the objectives that contain a physics constraint, we observe clear trend: the lower the physics constraint is weighted, the better test performance the model achieves. This indicates that physics constraints make it more challeng- ing for the model be trained optimally despite extensive hyperparameter tuning (perhaps due to practical reasons such as being difficult to optimize (Krishnapriyan et al., 2021; Subramanian et al., 2022: Gao & Wang. 2023; Wang et al., 2023)).
To further illustrate, we use the smallest w 0.1 and compare the physics-informet optimization with solely data-driven optimization. mized with physics loss generally diverge more substantially across test resolutions than models optimized with only a data loss. Models optimized with physics constraints even fail to accurately fit their training distributions (Fig. 6(c)), and fail to generalize to both super- and sub-resolution data (Fig. 6(a.b,d)). See Appendix E for results on all datasets. We conclude that physics informed constraints do not reliably enable multi-rect resolutoon generalization.
Fig. 6, the predicted spectra of data from models opti-
4.2 BAND-LIMITED LEARNING
We study two approaches which propose learning band-limited representations of data: convolu- tional neural operators (CNO) (Raonic et al., 2023) and the Cross-Resolutia Operator-Learning (CROP) pipeline (Gao et al., 2025). Both CNO and CROP have been proposed as alias-free so- lutions to enable multi-rese nulti-resolution inference (Bartolucci et al., 2023). CNOs are fixed fxee-resoluttion models; to use them, one must first interpolate their input to the model's training resolution, do a forward pass, and then interpolate back to the original dimension. The CROP pipeline is gen-

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Figure 6: (Physies-Ifformed) Optimization Evaluation. FNO trained on Burgers data at resolu- tion 256 (indicated by *) with and without physics optimization constrains. Average test spectra visualized. Spectra generated by the models trained with physics+data loss do not match the ground truth. Physics Loss term weighting w = 10%. Full results in Appendix E, Figs. 19-21.
eral and extends to any class of model by interpolating inputs (both at training and inference) to and from a -iee-diieniional representation before and after forward pass.
We train and evaluate a CNO and CROP+FNO model on all (Darcy, Burgers, Navier Stokes) datasets. Models are optimized using optimal hyper-pane hyper-parameters which were found via a grid search detailed in Appendix B. In Fig. 7, we vi- sualize the predicted spectra of model's trained resolution 16 data across test data of res- olution {16, 32, 64, 128]. We observe that the CNO does accurately learn the band-limited representation of its training data: the spectra matches that of the ground truth until frequency 8f after which, by design, it drops sharply. This means that while CNO does not alias, it cannot predict frequencies higher than what was seen during training. Similarly, we observe that the CROP model, accurately fits lower frequencies, but struggles to fit high frequencies accurately across resolutions.
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Figure Bandlimited Learning Evaluation. Models trained on Darcy dataset at resolution 16 (indicated by *), evaluated resolutions 16, 32, 64, 128. Average test set 2D energy spectrum of model predictions and ground truth. Notice both model prediction spectrums diverge from ground truth after frequency 8.
We note more broadly that the design band- hond-limiting model's training data and predictive capacity counter-intuitive to the goal of multi-resolltion inference, in which, broad range of frequencies must be modeled accurately. Band-limiting a model's predictive capacity may enable accurate fixed-resouttion representations, but ensures that high-frequency information is not predicted accurately (or at all). We conclude that band-limited learning limits a model's utility for multi- resolution inference (full results in Appendix F).
5 MULTI-RESOLUTION TRAINING
We hypothesize that the reason models struggle to do zero-shot multi-resolution inference because data representing a physical system at varying resolutions is sufficiently out-of- diitr--dittiibttion to model's fxee-resolution training data. To remedy this, we propose data-driven solution: multi- resolution training (i.e., training on more than one resolution) (Li et al., 2024a; Rowbottom et al., 2025; Berner al., 2025; George et al., 2024; Tang et al., 2024; Lyu al., 2023; Ma et al., 2024; Lanthaler et al., 2024).
We compose multi-resolution datasets by randomly sampling different proportions of training data at varying resolutions (ri,..., ) where ra is the proportion of resolution training data and 4. We begin by evaluating dual-resolution training. et al. (2024a) have previously shown that dual- resolution active learning enables more accurate resou-rsoouution inference. We extend this finding and evaluate if resoluisnontion training can enable accurate resolutenooltion inference. For each dataset in (Darcy, Burgers, Navier Stokes), combine data across resolutions in pairwise manner

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Figure 8: Multi-resolution training. FNO trained on multi-resolution Darey data. Top row: Av- erage number of pixels in data sample in the training set. Lower number of pixels enables faster data generation and model training. Middle row: The ratios of data within each resolution bucket. Bottom row: The average test loss across different resolutions. Lower loss is better. Notice: mixed resolution datasets achieve both low average data size and low loss (ideal scenario).
creating n(n-1) dual-resolution sets; the ratio of data samples between the two resolutions varied over p E 1...S....0225...,,,,......0
In Fig. 8(a-f), we observe for pair-wise training, the test performance for data that corresponds to the two training resolutions is generally better, but there not consistent gains for the two non-training resolutions. This indicates that models perform best on the data resolutions on which they are trained.
To improve multi-resolution capabilities, investigate the impact of including data from all res- olutions. We first assess an equal number of samples across resolutions. In Fig. 8(g), the test performance across all resolutions improves which confirms that multi-resolution training benefits multi-resolution inference. Next, we ask: Can we improve the computationally efficiency of multi- resolution training? To do this, the training dataset must be composed of primarily low resolution data as it is both the cheapest to generate and train over (Fig. 32). We compose two additional multi-resolution datasets: {0..7, 0. 0.1, 0.1), (0.9, 0.5, 0.3, 0.2)). In Fig. 8(h, i), models remain competitive across test resolutions, even we decrease the amount of high-resolution data. In Fig. we observe the consistent trend for all datasets: models are able achieve balance between dataset size and multi-resolution test loss via multi-resolution training. Optimizing the ratios across all resolutions remains an exciting future direction. Full results Appendix G.
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Figure 9: Data Cost vs. Loss Tradeoff. Lower data size and test loss are desirable (bottom left corner). We generally notice the "All Res." datasets form the Pareto front of achieving optimal data size vs. test loss. "m" and "m+n"" indicate the one or two resolutions that were included in the training set. "All Res." indicates that dataset contains points from all resolutions available. "Avg. Data Size" is the normalized average number of pixels in data point.
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6 RELATED WORK
Modeling PDEs via Deep Learning. Three prominent approaches exist to discretely model PDEs with deep learning. 1) Mesh-Free models learn the solution operator to specific instance of a PDE (Yu et al., 2018; Raissi et al., 2019; Bar & Sochen, 2019; Smith et al., 2020; Pan & Duraisamy, 2020). Mesh-free models can be queried to return measurement at any time and space coordinate. While, this approach means that a single model can resolve its solution at arbitrary discretization, it has two shortcomings: (i) Inference costs increases with number points queried. (ii) Models cannot generalize beyond the specific PDE instance it was trained on. 2) Fixed-Mesh models remedy both issues by learning a solution operator for a PDE family over a fixed-reslutee
mesh (Guo et al., 2016; Zhu & Zabaras, 2018; Adler & Öktem, 2017; Bhatnagar et al., 2019; Khoo et al., 2021). The inference costs of fixed-mesh models are lower than traditional numerical methods at corresponding mesh resolutions. However, fixed-mesh models fall short when one is interested in modeling scale phenomena that cannot be resolved via the fixed-mesh resolution (e.g.. high-frequency information in turbulent systems ). 3) Mesh-invariant models, unlike fixed-mesh models, are capable of doing inference at arbitrary mesh resolutions (Li et al., 2020b;a; Lu et al., 202 lb; Bhattacharya et al., 2021; Nelsen Stuart, 2021; Patel al., 2021; Rahman et al., 2022; Fanaskov & Oseledets, 2023). They have been proposed as means to learn mesh-invariant solution operators to entire PDE families cheaply: train on low-resolution data, and use in a zero-shot fashion on arbitrary resolution data (e.g., 7ero-shot multi-resolution). In this work, we examine the zero-shot multi-resolution utility of mesh-invariant models.
Aliasing (and corrective measures) in Deep Learning. Sources of aliasing in deep learning include both artifacts of a pixel grid which models learn and amplify and the application of point-wise non-nonlinearities to intermediate model representations (Karras et al., 2021; Gruver et al., 2022; Wilson, 2025). A straightforward approach, first introduced in generative adversarial networks, to aliasing is to up sample signal before applying a non- non-linearity followed by down sampling the signal (Karras et al., 2021). Bartolucci et al. (2023) and Raonic et al. (2023) extend the application of anti-alias activation function design to scientific machine learning: while this does prevent aliasing, it does not enable models trained at a specific resolution to resolve higher frequencies in higher resolution data. Gao al. (2025) proposed a framework that "lifts" arbitrarily land-limited space, to both train and do inference in. We
stem nonlincarity-caute
discretization data to fixed-resolur investigate the sources of aliasing the context of zero-shot multi-resolution inference and show that proposed solutions fall short in remedying the core issue:
generalization.
7 CONCLUSION AND FUTURE WORK
For machine-learned operators to be as versatile numerical methods-based approaches for mod- cling PDE's they must perform accurate multi-resolution inference. To better understand an MLO's abilities, we break down the task of multi-resolution inference and assess trained model's ability to both extrapolate to higher/lower frequency information in data and interpolate across varying data resolutions. We find that models trained on low resolution data and used for inference on high-resolution data can neither extrapolate nor interpolate, and therefore, more generally fail to do accurate multi-resolution inference. Changing the resolution of data inference time is akin to out- of-distribution inference: models have not learned how to generalize in such settings. We document that models used in a zero-shot multi-resolution setting are prone to aliasing. We study the utility of two existing polutions-physicee -informdd constrains and learning band-limited learning-and find that neither enable accurate multi-resolution inference.
We introduce a simple and principled approach to enable accurate multi-resolution inference: multi- resolution training. We first show that models perform best at resolutions they have been trained on. We then extend this finding and demonstrate that one can computationally efficiently achieve the benefits of multi-resolution training via datasets composed with mostly low-resolution data and small amounts of high-resolution data. This enables accurate multi multi-resolution learning with the added benefit of low data geairrateeenionnnn and model training cost. A promising future direction remains the automated selection of nulti-resolution training data using strategies like active learning.

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ACKNOWLEDGEMENT
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Department of Energy Computational Science Graduate Fellowship under Award Number DE-SC00231 12. YY acknowledges support from DOE Award DE-SC0025584 and Dartmouth College. AT was funded by Deutsche Forschungsgemein- schaft (DFG, German Research Foundation) under Germany's ' Excellence Strategy EXC 2075 390740016 and acknowledges the support by the Stuttgart Center for Simulation Science (Sim'Tech). We would also like to acknowledge the DOE Competitive Portfolios grant and the DOE SeiGPT grant.
REPRODUCIBILITY STATEMENT
To enable reproduction of our results we will link to the code base upon acceptance. We addition- ally include details about hyperparameter tuning in Appendix B, models (and their configuration details) in Appendix H, datasets (and their configuration details) Appendix A, data extrapola- tion/interpolation implementation details in Appendix C.I. To ensure reliability of results include an additional sensitivity analysis in Appendix L. Unless otherwise specified, all experiments were NVIDIA A100-PCIE with driver version 163.01 CUDA Toolkit version was 121,.131; Py Torch version was 2.9. 1cc the "neuraloperator" library (Kos- saifi et al., 2025) version was 2.0.0. We report results from replicating subset of experiments across multiple computing architectures to ensure reproducibility in Appendix N.
using Python 3.10
REFERENCES
Jonas Adler and Ozan Öktem. Solving ill-posed inverse problems using iterative deep neural net- works. Inverse Problems, 33(12):124007, 2017.
Reza Alizadeh, Janet K Allen, and Farrokh Mistree. Managing computational complexity using surrogate models: A critical review. Research in Engineering Design, 31(3):275-298, 2020.
Michael Asher, Barry FW Croke, Anthony J Jakeman, and Luk JM Peeters. review of surrogate models and their application to groundwater modeling. Water Resources Research, 51(8):5957- 5973, 2015.
Kamyar Azizzadenesheli, Nikola Kovachki, Zongyi Li, Miguel Liu-Schiaffini, Jean Kossaifi, and Anima Anandkumar. Neural operators for accelerating scientific simulations and design. Nature Reviews Physics, 6(5):320-325 2024.
Leah Bar and Nir Sochen. Unsupervised deep learning algorithm for PDE-based forward and inverse problems, arXiv preprint arXiv:1904.055: 05417, 2019.
Francesca Bartolucci, Emmanuel de Bezenac, Bogdan Raonic, Roberto Molinaro, Siddhartha Mishra, and Rima Alaifari. Representation equivalent neural operators: framework for alias- free operator learning. Advances in Neurul Information Processing Systems, 36:69661-69672. 2023.
Marsha J Berger and Joseph Oliger. Adaptive mesh refinement for hyperbolic partial differential equations. Journal of computational Physics, 53(3):484-512 1984.
Julius Berner, Miguel Liu-Schiaffini, Jean Kossaifi, Valentin Duruisseaux, Boris Bonev, Kam- yar Azizzadenesheli, and Anima Anandkumar. Principled approaches for extending neural architectures to function spaces for operator learning. ArXiv, (is//500610973,, 2025. URL https:/ /apii...... semanticscho
Saakaar Bhatnagar, Yaser Afshar, Shaowu Pan, Karthik Duraisamy. and Shailendra Kaushik. Predic- tion of aerodynamic flow fields using convolutional neural networks. Computational Mechanics, 64(2):525-555, 2019.
10

---

Published as a conference paper at ICLR 2026
Kaushik Bhattacharya, Bamdad Hosseini, Nikola B Kovachki, and Andrew M Stuart. Model reduc- tion and neural networks for parametric PDEs. The SMAI journal of computational mathematics, 7:121-157, 2021.
Ashesh Chattopadhyay, Y. Qiang Sun, and Pedram Hassanzadeh. Challenges of learning multi- scale dynamics with Al weather models: Implications for stability and one solution, 2024. URL https:/ /ttpps://axivv org/ abs 2304. 07029.
Alison Cozad, Nikolaos V Sahinidis, and David C Miller, Learning surrogate models for simulation- based optimization. AICHE Journal, (000)2211-2221) 2014.
Vladimir Sergeevich Fanaskov and Ivan V Oseledets. Spectral neural operators. In Doklady Math- ematics, volume 108, pp. S226-S232. Springer, 2023.
Alexander Forrester, Andras Sobester, and Andy Keane. Engineering design via surrogate mod- elling: A practical guide. John Wiley & Sons, 2008
Wenhan Gao and Chunmei Wang. Active learning based sampling for high-dimensional nonlinear partial differential equations. Journal of Computational Physics, 475: 1848, 2023.
Wenhan Gao, Ruichen Xu, Yuefan Deng, and Yi Liu. iiscreiizaion-invarianeee? On the discretiza- tion mismatch errors in neural operators. In The Thirteenth International Conference on Learning Representations, 2025.
Robert Joseph George, Jiawei Zhao, Jean Kossaifi, Zongyi Li, and Anima Anandkumar. Incremental spatial and spectral learning of neural operators for solving large-scale PDEs. Transactions on Machine Learning Research, 2024. ISSN 2835-8856. URL
n
forum?id- 6cPQ0bp0.
Nate Gruver, Marc Finzi, Micah Goldblum, and Andrew Gordon Wilson. The Lie derivative for measuring learned equivariance. arXiv preprint arXiv:2210.029844 2022.
Xiaoxiao Guo, Wei Li, and Francesco lorio. Convolutional neural networks for steady flow ap- proximation. In Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining, pp. 481-490, 2016.
Hendrycks. Gaussian error linear units (Gelus), arXiv preprint arXiv:1606.08415, 2016.
Tero Karras, Miika Aittala, Samuli Laine, Erik Härkönen, Janne Hellsten, Jaakko Lehtinen, and Timo Aila, Alias-free generative adversarial networks. Advances in neural information processing systems, 34:852-863, 2021.
Yuehaw Khoo, Jianfeng Lu, and Lexing Ying. Solving parametric PDE problems with artificial neural networks. European Journal of Applied Mathematics, 32(3) 2021.
Jean Kossaifi, Nikola Kovachki, Zongyi Li, David Pitt, Miguel .iu-Schiaffini, Valentin Duruisseaux, Robert Joseph George, Boris Bonev, Kamyar Azizzadenesheli, Julius Berner, al. A library for learning neural operators. arXiv preprint 2rXiv:2412.10354 2024.
Jean Kossaifi, Nikola Kovachki, Zongyi Li, David Pitt, Miguel Liu-Schiaffin,, Valentin Duruisscaux, Robert Joseph George, Boris Bonev, Kamyar Azizzadenesheli, Julius Berner, and Anima Anand- kumar. A library for learning neural operators. arxiv preprint arXiv:2412. 10354, 2025.
Nikola Kovachki, Zongyi Li. Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Neural operator: Learning maps between function spaces with applications to PDEs. Journal of Machine Learning Research, 24(89):1-97, 2023.
Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Char- acterizing possible failure modes in physics-ifformed neural networks. Advances in neural infor- mation processing systems, 34:26548-200 2666600 2021.
David Krueger, Ethan Caballero, Heem-- Jacobsen, Amy Zhang, Jonathan Binas, Dinghuai Zhang, Remi Le Priol, and Aaron Courville. Out-of-distribttion generalization via risk extrapo- lation (rex). In International conference machine learning, pp. 5815-5826. PMLR, 2021.

---

Published as a conference paper at ICLR 2026
Jakub Kudela and Radomil Matousek. Recent advances and applications of surrogate models for finite element method computations: A review. Soft Computing, (6(24):3370-137333 2022.
Samuel Lanthaler, Andrew M Stuart, and Margaret Trautner. Discretization error of fourier neural operators. arXiv preprint arXiv:2405.0222: 2024.
Shibo Li, Xin Yu, Wei Xing, Robert Kirby, Akil Narayan, and Shandian Zhe. Multi-resolution active learning of Fourier neural operators. In Sanjoy Dasgupta, Stephan Mandt, and Yingzhen Li (eds.), Proceedings The 27th International Conference on Artificial Intelligence and Statistics, volume 238 of Proceedings of Machine Learning Research, pp. 2440-2448, PMLR, 02-04 May 2024a. URL htwphttt///ccm. http://pooceeeeigneeeeeee www.mmr..m... pwwss/v23ss..eeee wtw.://ss........22eee.e.
244 html.
Zongyi Li, Nikola Kovachki, Kamyar Arizzadenesheli, Burigede Liu, Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Fourier neural operator for parametric partial differential equations. arXiv preprint arXiv:2010.08892 2020a.
Zongyi Li, Nikola Kovachki, Kamyar Arizzadenesheli, Burigede Liu, Kaushik Bhattacharya, An- drew Stuart, and Anima Anandkumar. Neural operator: Graph kernel network for partial differ- ential equations. arXiv preprint 0rXiv:2003.03483 2020b.
Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadeneshelli and Anima Anandkumar. Physics-informed neural operator for learning partial differential equations. ACM/ / IMS J. Data Sci., 1(3), May 2024b. doi: 10.1 10.13446688056.. URL
https: //oii. wrw.orrggooooo.. 10. 10111114344555
Weitang Liu, Xiaoyun Wang, John Owens, and Yixuan Li. Energy-based ofistrriitiiiiutio detec- tion. Advances in neural information processing systems, 33:2 53:21164-21145, 2020.
Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nature machine intelligence, 3(3):218-22) 2021a.
Lu Lu, Pengzhan Jin, Guofei Pang, Zhongqiang Zhang, and George Em Karniadakis. Learning nonlinear operators via DeepONet based on the universal approximation theorem of operators. Nature machine intelligence, (()):118-229 2021b.
Yanfang Lyu, Xiaoyu Zhao, Zhiqiang Gong, Xiao Kang, and Wen Yao. Multi-fidelity prediction of fluid flow based on transfer learning using fourier neural operator. Physics of Fluids, 35(7), July 2023. ISSN 7666. doi: 10.1063/5.015555. URL http://dx: do1. org.i..orr/io 10.1//1/.1111//// 5.555555
Xianlin Ma, Rong Zhong, Jie Zhan, and Desheng Zhou. Enhancing subsurface multiphase flow simulation with fourier neural operator. Heliyon, 10(18), 2024.
Nicholas H Nelsen and Andrew M Stuart. The random feature model for input-output maps between banach spaces. SIAM Journal on Scientific Computing, 43(5):33222-33343, 2021.
Katharina Ott, Prateek Katiyar, Philipp Hennig, and Michael Tiemann. ResNet after all: Neural ODEs and their numerical solution. In International Conference on Learning Representations, 2021.
Shaowu Pan and Karthik Duraisamy. Physics-informed probabilistic learning of linear embeddings of nonlinear dynamics with guaranteed stability. SIAM Journal on Applied Dynamical Systems, 19(1):400-509 2020.
Ravi G Patel, Nathaniel A Trask, Mitchell A Wood, and Eric C Cyr. A physics-informed operator gression framework for extracting data-driven continuum models. Computer Methods in Applied Mechanics and Engineering, 373: 13500, 2021.
Alejandro F Queiruga, N Benjamin Erichson, Dane Taylor, and Michael W Mahoney. Continuous- in-depth neural networks. arXiv preprint arXiv:2008.02388, 2020.
Md Ashiqur Rahman, Zachary E Ross, and Kamyar Azizzadenesheli. U-NO: U-shaped neural operators. arXiv preprint 122 2022.
12

---

Published as a conference paper at ICLR 2026
Maziar Raissi, Paris Perdikaris, and George E Karniadakis. Physics- innssce-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational physics, 378:686-707, 2019.
Bogdan Raonic, Roberto Molinaro, Tim De Ryck, Tobias Rohner, Francesca Bartolucci, Rima Alai- fari, Siddhartha Mishra, and Emmanuel de Bézenac. Convolutional neural operators for robust and accurate learning of PDEs. Advances in Neural Information Processing Systems, 36:77187- 77200, 2023.
Pu Ren, N Benjamin Erichson, Shashank Subramanian, Omer San, Zarija Lukic, and Michael W Mahoney. Superbench: super-resolution benchmark dataset for scientific machine learning. arXiv preprini 2rXiv:2306.140000 2023.
James Rowbottom, Stefania Fresca, Pietro Lio, Carola-Bibiane Schönlieb, and Nicolas Boullé. Multi-level monte carlo training of neural operators. arXiv preprint arXiv:2505.12940, 2025.
C.E. Shannon. Communication in the presence of noise. Proceedings the doi: 10.1
the IRE, 1949.
Jonathan D Smith, Kamyar Azizzadenesheli, and Zachary E Ross. Eikonet: Solving the eikonal equation with deep neural networks. IEEE Transactions on Geoscience and Remote Sensing, 59 (12):16655-1066, 2020.
Subel, Guan, A Chattopadhyay, and P Hassanzadeh. Explaining the physics of transfer learning data-driven subgrid-scale closure to different turbulent flow. arXiv preprint 2rxXi:2206.0319: 724, 2022.
Shashank Subramanian, Robert M Kirby, Michael W Mahoney, and Amir Gholami. Adaptive self- supervision algorithms for physics-informer neural networks. arXiv preprint arXiv:2207.04084 2022.
Bruno Sudret, Stefano Marelli, and Joe Wiart. Surrogate models for uncertainty quantification: An overview. 2017 11th European conference on antennas and propagation (EUCAP). pp- 793-797. IEEE, 2017.
Makoto Takamoto, Timothy Praditia, Raphael Leiteritz, Daniel MacKinlay, Francesco Alesiani, Dirk Pflüger, and Mathias Niepert. PDEBench: An extensive benchmark for scientific machine learning. Advances in Neural Information Processing Systems, 35:1596-1611 2022.
Hewei Tang, Qingkai Kong, and Joseph P. Morris. Multi-fidelity fourier neural operator for fast modeling of large-scale geological carbon storage. Journal of Hydrology, 629:130641, 2024. ISSN 0022-1694. doi: http:://di.org/I00
URL https: :/www.
sciencedirect com/ scienceeeeeece artiaaattiee httpc/////r.r.ceea//////2211111/
Alasdair Tran, Alexander Mathews, Lexing Xie, and Cheng Soon Ong. Factorized fourier neural operators. arXiv preprint 13X122211..33802, 2021.
Michael Unser. Sampling-50 years after Shannon. Proceedings of the IEEE, 88(4):569-587, 2002.
Sifan Wang, Shyam Sankaran, Hanwen Wang, and Paris Perdikaris. An expert's guide to training physics-informdd neural networks. arXiv preprint arXiv:2308.080 08466888.888888.. 2023.
John M Whittaker. The "Fourier" theory of the cardinal function. Proceedings of the Edinburgh Mathematical Society, 1(3):169-176, 1928.
Andrew Gordon Wilson. Deep learning is not so mysterious or different. 12X10:2503.02113. 2025.
arXiv preprint
Liu Yang, Siting Liu, Tingwei Meng, and Stanley Osher. In-context operator learning with data prompts for differential equation problems. Proceedings of the National Academy of Sciences, (20039))2230144222,, 2023.
Bing Yu et al. The deep Ritz method: a deep learning-bast based numerical algorithm for solving varia- tional problems. Communications in Mathematics and Statisties, 6(1):1-12 2018.
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Yinhao Zhu and Nicholas Zabaras. Bayesian deep convolutional encoder-decoder networks for surrogate modeling and uncertainty quantification. Journal of Computational Physics, 366:415- 447, 2018.
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A DATA
We study three standard scientific datasets: Darcy, Burgers, and turbulent incompressible Navier Stokes, as released in PDEBench (Takamoto al., 2022). We summarize each dataset here; for full details, see Takamoto et al. (2022):
Darey. We study the steady-state solution of 2D Darcy Flow the unit square with viscosity term a(x) as input of the system. We learn the mapping from a(x) to the steady-state solution described by:
-V(((=)V(<))= f(x), E (0, 1) u(z) 0, 1)0,1)²
The force term is a constant value f
Burgers. We study Burgers' equation which is used model the non-linear behavior and diffusion process in fluid dynamics:
@e((,,)) + Oz (x.x)) v/T0,, x E (0. 1),4 E (0,2)
(3)
u(0, .) = uo(2), x E (0, 1).
(4)
The diffusion coefficient is constant value 0.001.
(Turbulent) Inhomogencous, Incompressible Navier Stokes. We study a popular variant of the Navier Stokes equation: the incompressible version. This equation is used model dynamics far lower than the speed of propagation of waves in the medium:
V u = 0, p(0.v + v Vv) - Vp + nA..
(5)
Takamoto et (2022) employ an augmented form of (5) which includes vector field forcing term
p + V. Vv) = Vp + nAv + u.
The viscosity is a constant value v=0.01. We convert the incompressible Navier Stokes dataset to vorticity form to enable direct comparison with Li et al. (2020a).
Darcy (2D)
Burgers (1D)
Navier Stokes (2D)
10a
Energy
10a
10-1
10-1
10-1
10a
100
10:
10a
Freqeuncy
Freqeuncy
Fregeuncy
Figure 10: Dataset Energy Spectrum. Average energy spectra over test datasets. Notice that Navier Stokes is in the turbulent regime. K = Kolmogorov coefficient.
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B HYPER-PARAMETER SEARCH
FNO Hyper-parameter Tuning. For each dataset E {Darcy, Burgers, Navier Stokes), we do grid search for optimal training hyper-parameetrs:: learning rate E {1e-2, le-3, lc-4, le-5]; and weight decay & {le-5, lc-6, le-7]; for both the data driven loss (i.e., mean squared error) and the respective data+physics driven loss. Each model was trained for 150 epochs. For the models optimized with the data+physics loss, we optimized the physics loss term's weighting coefficient w {0.1, 0.25, 0.5). For Darey, Burgers, and Navier Stokes, we do this hyper- papammee search for data at resolution 64, 512, 255, respectively, and we then use the optimized parameter values for each dataset to train models the remaining resolutions. See Tab. for the optimal hyper-parameter each dataset/loss combination.
CROP/CNO Hyper-parameter Tuning. For each dataset E (Darcy, Burgers, Navier Stokes), we do grid search for optimal training hyper-parameters: learning rate E {1e-3, Ie-4, le-5), for the data driven loss (i.e., mean squared error). Each model was trained for 150 epochs. For Darcy, do this hyper-paranen
Burgers, and Navier Stokes, respectively; and we then use the optimize parameter values for each dataset to train models at remaining resolutions. See Tab. 2 for the optimal hyper-parameter for each dataset/loss combination.
search for data resolution 64, 512, 255,
Table I: Optimal FNO peree-parameters from hyper-parameee search outlined in Appendix B. *NS batch size had to be reduced to for multi-resolution training experiments (see Sec. 5), there- fore used lower learning rate in that setting. w=Physic Loss Coefficient (see Sec. 4.1).
Data
Loss
Learning Rate
Weight Decay
Batch Size
Darey
Data
le-3
le-5
128
Darcy
Data+Physics
0.1
le-2
le-5
128
Burgers
Data
le-3
le-5
Burgers
Data+Physics
0.1
le-3
le-5
64
Navier Stokes
Data
le-2
le-6
Navier Stokes
Data+Physics
0.1
lc-4
le-5
4
Navier Stokes
Data
1e-5
le-6
Optimal CNO/CROP/DeepONe hyper-paramele parameters from hyper-parameter search out- lined in Appendix B. *The original CROP implementation did not include a ID version, so we omit CROP for the ID Burgers dataset.
Table 2:
Data
Loss
Model
Learning Rate
Weight Decay
Batch Size
Darcy
Data
CNO
0.0001
le-5
128
Darey
Data
CROP
0.001
le-5
128
Darcy
Data
DeepONet
0,001
le-5
128
Burgers*
Data
CNO
0.001
le-5
64
Navier Stokes
Data
CNO
0,001
le-6
1
Navier Stokes
Data
CROP
0.001
le-6
1
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C INFORMATION EXTRAPOLATION AND RESOLUTION INTERPOLATION
C.I EXTRAPOLATION AND INTERPOLATION IMPLEMENTATION DETAILS
Down sampling Procedure: For the resolution interpolation experiments, data was sampled different resolution by employing standard down sampling of the data (keeping every Nth ele- ment) from the maximal data resolution. This is the same down sampling procedure used by PDEBench (Takamoto et al., 2022).
For the information extrapolation experiments, data was filtered with a standard stop-band filter which restricts any frequency components above the desired filter limit N. Concretely, to low pass filter data, we transform the data to the frequency domain via an FFT and shift it such that the lowest frequency component is the center of the spectrum. We determine which frequency components are greater than our limit N by assessing which components fall outside radius of N from the center, we then set the energy of any frequency components greater than N to zero. Finally, the (filtered) data is transformed back to the spatial domain via an inverse FFT.
Low-pass Filtering Procedure:
C.2 FULL RESULTS
Here, we include the full experimental results of studying FNOs' abilities to do both information extrapolation and resolution interpolations, as described See. 3.
Train Data
Train Data
Train Data Resolution (512)
Train Data Resolution (1024)
Resolution (128)
Resolution (256)
10a
Norallzze Rosieuals
10a
10a
10-3.
(a)
(b)
(c)
(d)
400
200
400
200
400
200
400
Frequency
Frequency
Frequency
Frequency
Resolution (128)
Resolution (256)
Resolution (512)
Resolution (1024)
Train Lmmi
Figure 11: Interpolation. Four FNOs trained on Burgers data low-pass limit 64 (constant fre- quency information) and down sampled to resolutions from left to right. We test if each model can generalize to data with varying sampling rate. We visu- alize the normalize residual spectra across test data. Notice that the residual spectra (error) increases substantially in the low frequencies. Lower energy at all wave numbers is better.
27..2566512,10244 (varying sampling rate)
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Train Dala Resolution (64)
Train Data Resolution (128)
Train Data Resolution (255)
Train Data
Resolution (510)
1016
Normalizes Residuall
107
10*
(a)
(b)
(d)
100
200
100
200
100
200
100
200
Frequency
Frequency
Frequency
Frequency
Resolution (64)
Resolution (128)
Resolution (255)
Resolation (510)
Train Limit
Figure 12: Interpolation. Four FNOs trained on Navier Stokes data low-pass limit 32f (constant
frequency information) and down sampled to resolutions {64.128,255,5 (varying sampling rate) from left to right. We test if each model can generalize to data with varying sampling rate. We visualize the normalize residual spectra across test data. Notice that the residual spectra (error) increases substantially in the low frequencies. Lower energy at all wave numbers is better.
(64)
(128)
Train (256)
(512)
Normalize zeed dusiduals
10
(a)
(b)
(c)
(d)
200
400
200
400
200
400
200
400
Frequency
Frequency
Frequency
Frequency
Filter Limit (64)
Filter Limit (128)
Filter Limit (256)
Filter Limit (512)
Train Liiit
Figure 13: Extrapolation. Four FNOs trained on Burgers data of resolution 1024 (constant sam-
pling rate) and low-pass filtered with limits [64, (64,128,28) 126,256, (varying frequency information) from left to right. We test each model can generalize data with varying frequency information. We visualize the normalize residual spectra across test data. Notice that the residual spectra (error) increases substantially the high frequencies. Lower residual energy at all wave numbers better.
Train Data Filter (8)
Train Data Filter (16)
Train Data Filter (32)
Train Data Filter (255)
Norma lird Rcsddals Energy
10
(a)
(b)
(c)
(d)
200
100
200
100
Frequency
Frequency
Frequency
Frequency
Limit
Filter Limit
(32)
Filter Limit
Train Limit
Figure 14: Extrapolation. Four FNOs trained Navier Stokes data of resolution 510 (constant sampling rate) and low-pass filtered with limits {8.16.32,255), (varying frequency information) from left to right. Test if each model can generalize to data with varying frequeney information. We visualize the normalize residual spectra across test data. Notice that the residual spectra (error) increases substantially in the high frequencies. Lower residual energy at all numbers better.
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D EVALUATING SUB- AND SUPER-RESOLUTION
Here we included the full experiment results of studying FNOs' abilities to do zero-shot multi- resolution inference as described in Sec. 3.
Ground Truth Time 10
Ground Truth Time
Ground Truth Time 16
Predicted Time 10
Predicted Time 13
Predicted Time
Figure 15: Aliasing artifacts compound over time. FNO trained on resolution 255 Navier-Stokes data, evaluated resolution 510. Left: Ground truth evolution of NS fluid flow. Right: Corre- sponding FNO predictions at resolution 510. Notice that the high frequency artifacts become more prevalent over time.
Darcy
Burgers
Navier-Stokes
10
4E-04
7E-04
1E-03
2E-03
128
6E-04
6E-04
6E-04
1E-02
2E+00
2E+00
1E+
20+00
Train Resolution
3E-04
256
4E-04
4E-04 04
1E-02
128
2E+00
2E+00
2E+
2E+00
128 64
1E:03
4E-04
3E-04
9E-04
512
4E-04
4E 04
4E-04
1E-02
255
3E+00
2E+03
2E+00
2E+03
2E-03
5E-04
3E-04
3E-04
1024
1E-03
1E-03
1E-03
5E-04
510
2E+10
2E+11
6E+ 04
2E+00
16
32
64
128
128
256
512
1024
64
128
255
510
Test Resolution
Test Resolution
Test Resolution
Figure 16: FNOs do not reliably generalize to higher or lower resolutions. Heatmaps losses incurred by FNO trained and tested at varying resolutions (lower is better). When the test resolution varies from the training resolution, the models often incur substantial increase in loss.
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E PHYSICS-INFORMED OPTIMIZATION
Here, we use the physics losses of Li et al. (2024b) which explicitly enforce that the governing PDE satisfied. The governing PDEs are detailed in Appendix A.
Below we include the results of tuning the physics loss weighting coefficient w in Figs. 18 and 17. We then include the full comparisons of training each dataset (Darey, Burgers, Navier Stokes) each resolution with both w E {0, 0.1} in Figs. 19-20.
Darcy
Burgers
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Figure 17: (Physics-Informel Optimization. Increasing the proportion of physics-informnt loss the optimization objective corresponds to increased test loss. Lower MSE is better. Darcy trained resolution 64, Burgers trained at resolution 512, and Navier Stokes trained at resolution 255.
Darcy
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Figure 18: (Physies-Informed) Optimization Objective. Physics informed constraints never achieves better performance than using pure data-driven constraints. Darcy trained at resolution 64, Burgers trained at resolution 512, and Navier Stokes trained at resolution 255.
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Test 128
Test 64
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Data Physics Loss
Figure 19: Darcy. Energy spectra for models trained at a specific resolution (y-axis) and tested at multiple resolution (x-axis), with and without physics optimization constraint, The spectra generated by the models trained with physics+data loss do not match ground truth. We set w = 10%.
Test 1024
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Figure 20: Burgers. Energy spectra for models trained at a specific resolution (y-axis) and tested at multiple resolution (x-axis), with and without physies optimization constraint. The spectra generated by the models trained with physics+data loss do not match ground truth. We set w = 10%.
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Test 510
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1036
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10a
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Data+ Physics Loss
Figure 21: Navier Stokes. Energy spectra for models trained at a specific resolution (y-axis) and tested at multiple resolution (x-axis), with and without physics optimization constraint. The spectra generated by the models trained with physics+data loss do not match ground truth. We set w = 10%.
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F BAND-LIMITED LEARNING
Here, we include the full experimental results of training models with mixed resolution datasets, as described in Sec. 4.2. Note that Gao et al. (2025) did not release a ID CROP pipeline; therefore, were unable to test CROP with Burgers.
First, we observe that band-limited learning, in which models able to both infer and learn over band-limited representations of data, accurately learns the frequency region of the data included in the band-limit. However, these models struggle to learn do not learn anything outside this range (see Figs. 7, 26, 24). The implication of this is that hand-iimieddapppracchee suffice for modeling data within prespecified range, as long as the band-limit range wide enough, and the model will never need to be used to infer on data containing additional frequency information. However, we observe the band limited approach under performs multi-resolution training at fitting the full spectrum in the multi-resolution inference setting; this is because the resolution of data, and consequently the resolved frequencies, are changing (Fig. 9). For both Darey (Fig. 24) and Burgers (Fig. 25), notice that muultiressolutoon training out out-performs band limited approaches. In Fig. 7, we observe that the predetermined band-limit leads to error in the high frequency range.
A scenario in which CNO and CROP appear to perform well is datasets in which the majority of the energy concentrated in the predetermined band-limit (e.g.. Navier Stokes, see Fig. 10). In this setting, see that band-limited fit the lower frequencies in the spectrum very well (Fig. 26). However, we also see in Fig. 26, that multi-resolution training is the only method that consistently predicts both the correct amount of energy across the full spectrum. Band limited approaches fail to fit the high frequency range of the spectrum.
We note that band-limited approaches do not accurately fit frequencies outside of their predeter- mined limit. This makes them effective for fixed-resolution inference, but they are ineffective for multi-resolution inference. Alternatively, we demonstrated in Sec. that multi-resolution inference can be scaled to new data resolutions (and therefore new parts of the spectral energy spectrum) via scaling up representative samples the training dataset. We conclude that multi-resolution training is a more flexible and scalable approach to enabling accurate multi-resolution training at all parts of the energy spectrum.
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(a) Test Res. 510
(b) Test Res. 255*
(c) Test Res. 128
(d) Test Res. 64
10a
10-2
10-3
10-1
Frequency
Frequency
Frequency
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16a
MSE
10a
Physics
Figure 22: Spectral Comparison Navier Stokes. Top Row: Average predicted spectra for test data at varying resolutions across all methods. Bottom row: Average mean squared error loss over test data at varying resolutions across all methods. Zero-shot methods: CNO, FNO, Physics Informed and CROP are all zero-shot methods, meaning there are trained at specific resolution (255, indicated by "), and evaluated resolutions 510, 255, 128, 64. Data-driven method: Multi- resolution training; notice that multi-resolution training is the only method that consistently fits both the high and low parts of spectra. Band-limited methods: CNO and CROP are both band- limited methods which are trained in zero-shot manner at fixed resolution; observe that they only accurately fit the low frequencies.
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(a) Test Res. 128
(b) Test Res. 64
(c) Test Res. 32
(d) Test Res. 16*
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Physics Informed
Figure 23: Spectral Comparison Darcy. Top Row: Average predicted spectra for test data at varying resolutions across all methods. Bottom row: Average mean squared error loss over test data at varying resolutions across all methods. Zero-shot methods: CNO. FNO, Physics Informed and CROP are all zero-shot methods, meaning there are trained specific resolution (16, indicated by *), and evaluated at resolutions 128, 64, 32, 16. Data-driven method: Multi-resolution training; notice that multi-resolution training the only method that consistently fits both the high and low parts of the spectra. Band-limited methods: CNO and CROP are both linitedmited methods which are trained zero-shot manner fixed resolution; we observe that they only accurately fit low frequencies.
CROP2D
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Test Resalution
Figure 24: CROP+ENO and CNO trained Darey On average, both CROP+FNO and CNO incur higher losses across resolutions compared to both FNO (Fig. 16) and multi-resolution training (Fig. 9).
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CNO1D
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1024
Test Resolution
Figure 25: CNO trained on Burgers. On average, CNO incur higher losses across resolutions compared to both FNO (Fig. 16) and multircesolution training (Fig. 9). We note that despite our hyperparameter search (Tab. 2) the CNO model trained on resolution 128 failed to converge.
CROP2D
CNO2D
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64
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Figure 26: CROP+ENO and CNO trained on Navier Stokes. On average, both CROP+FNO and CNO incur lower losses across resolutions compared to both FNO (Fig. 16) and multi-resolution training (Fig. 9).
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G MULTI-RESOLUTION TRAINING
Here, we include the full experimental results of training models with mixed resolution datasets, as described in Sec. 5. We provide an overview comparison across methods Fig. 27. See Figs.28-33 28-30 for our main results. We plot the association between increased dataset size and increased training time in Fig. 32.
Ground Futh
Multi Res.
CND
FNO
Physics Informed
CROP
Tist LosE
Ras
10
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Rox 32
Test Loos
10
Ress, 64
Tist LOGS
10
Rex 128
Toxt LOGE
Ground
Informed
CROP
Figure 27: Assessing multi-resolution inference. Column 1: Expected prediction for Darcy flow varying resolutions. Columns 2-6: Sample prediction for Darcy flow varying test resolutions. Column 7: Average mean squared error test loss at each resolution (lower is better). Zero-shot methods: CNO, FNO, Physics Informed and CROP are zero-shot methods, meaning the model was trained at specific resolution (16) and evaluated at resolutions 16, 32, 64, 128. Data-driver method: Multi-resolution training; notice that multi-resolution training consistently outperforms zero-shot methods.
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Ratio
(590)
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34 1-10"
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Figure 28: Darcy Multi-Resolution Training. FNO trained on multi-resolution Darcy data. Each row include two sub-rows; each row is delineated the the dual-resolution training ratio (indicated in y-axis label). The top sub-row illustrates the ratios of data within each resolution bucket. The bottom sub-row indicates the average test loss across different resolutions. Lower loss is better. Notice in the mixed resolution datasets achieve both low average data size and low loss (ideal scenario).
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Rati
(5%)
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Textt
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Rabio (95%)
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Figure 29; Burgers Multi-Resolution Training. FNO trained on multi-resolltion Burgers data.
Each row include two sub-rows; each row is delineated the the dual-resolution training ratio (indi- cated in y-axis label). The top sub-row illustrates the ratios of data within each resolution bucket. The bottom sub-row indicates the average test loss across different resolutions. Lower loss better. Notice in the mixed resolution datasets achieve both low average data size and low loss (ideal see- nario).
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Ratio
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Figure 30: Navier Stokes Multi-Resolutian Training. FNO trained on multi-resolution Navier Stokes data. Each row include two sub-rows; each row is delineated the the dual-resolution training ratio (indicated in y-axis label). The top sub-row illustrates the ratios of data within each resolution bucket. The bottom sub-row indicates the average test loss across different resolutions. Lower loss is better. Notice in the mixed resolution datasets achieve both low average data size and low loss (ideal scenario).
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Darcy
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Figure 31: Data size corresponds with training time. We notice a clear trend with Darcy and Navier Stokes datasets: as data size increases, so does average training time per epoch. For Burgers, this trend is less clear. However, we note that the Burgers dataset is several orders of magnitude smaller and therefore can be used with high batch size (see Tab. 1), which reduces the computa- tional gains achieved from smaller sized dataset.
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G.1 MULTI-RESOLUTE TRAINING PERFORMANCE GAINS
Here we profile both the memory savings (see Tab. 3) and wall-clock training time (see Tab. 4) saving due the reduction in dataset size from multi-resolution training compared with the single- resolution training counterpart. We empirically observe the smaller dataset size is associated with reduction in training time in Fig. 32.
Table 3: Impact of Multi-Resolution Training on Memory. Reporting the size of datasets in Gigabytes (GB) for multi-resolution dataset with ratios (0.9, 0.5, 0.3, 0.2) vs. single-resolution maximum resolution dataset along with the percent decrease dataset size. The smaller the dataset, the better.
Dataset
Single-Res. Size (GB)
Multi-Res. Size (GB)
% (4)
Darcy
0.54
0.02
96%
Burgers
0.34
0.005
98%
Navier Stokes
0.96
0.04
96%
Table 4: Impact of Multi-Resolution Training on Wall-clock Training Time. Reporting the training time (over 150 epochs) for multi-resolution dataset with ratios (0.9, 0.5, 0.3, 0.2) vs. single- resolution maximum resolution dataset along with the percent decrease in dataset size. The lower the training time, the better.
Dataset
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% (t)
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0.33
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Figure 32: Data size corresponds with training time. We notice clear trend with Darcy and Navier Stokes datasets: as data size increases, so does average training time per epoch. For Burgers, this trend is less clear; however, we note that the Burgers dataset several orders of magnitude smaller and therefore can be used with high batch size (see Tab. 1) which reduces the computational gains achieved from a smaller sized dataset.
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H MODEL IMPLEMENTATIONS AND CONFIGURATIONS
The Fourier Neural Operator is described in detail in (Li et al., 2020a); we closely follow their implementation which can be found at https: /neurauuaal operator: githui.thhh... io/dev/ index. html. We detail key model configuration parameter choices here:
Max Modes:
When training and evaluating models to do zero-shot super-resolution or sub-resolution and multi-resolution training/inference,
train with the max modes pa- rameter set to half the maximum training resolution. Further, we do an ablation study of the modes parameter in Appendix I assess impact of FNO's ability do information extrapolation and resolution interpolation.
2. Layer count: (standard implementation)
3. Hidden channels: 32 (standard implementation)
4. Activation Function: All main paper experiments utilize the Gelu nonlinear- ity (Hendrycks, 2016). Further, we assess the impact of utilizing anti-aliasing activation function on zero-shot super and sub-resolution, information extrapolation, and resolution interpolation in Appendix K (Karras et al., 2021).
5. All other parameters: All other FNO parameters were initialized to default values detailed in the standard neura neuraloperator Python package (Kossaifi et al., 2024).
The CNO is described in detail in (Raonic al., 2023); implementation which can be found at https:/ /ttp:://githbb.. com/ camlab-ethh Convol ut i ona NNeura Neeral parameter choices here:
closely follow their
wow.eettretrreeeeeeeeeeeeeeeeeee /
mmain. We detail key model configuration
1. Residual block count: 6 (standard implementation)
2. Latent size: 64
3. Input Resolution: Varries based on training data resolution.
4. All other parameters: All other CNO parameters were initialized default values de- tailed in the official CNO implementation: https; htg.hhb..ccomm com/ hamlab-ethe/ Convolutioo ina Mtt.aaaaooineeeeeooooorrrtttrrra
maain.
The CROP pipeline is described in detail in (Gao al., 2025); closely follow their imple- mentation for CROP+FNO which can be found at https: :ops:// ///ggttti thub. com wwwhangaoh111111 TCLR25-CROP / reee//ee///eeee ma i.. We note that they did not include 1D CROP implementation, so we exclude evaluation of CROP on the ID Burgers dataset. We detail key model configuration parameter choices here:
1. Hidden channels: 32 (standard implementation)
2. Latent size: 64
3. Max Mode: 32 (half latent size)
4. All other parameters: All other CROP parameters were initialized to default values de-
tailed in the official CROP implementation: https: /ttptttttttttt/t///// thub. com/ wenhangao21/ ICIR25-CROEE ttte/eeee/////eeee m
The DeepONet model described in detail (Lu et al., 2021a); we closely follow the implemen- tation of (Gao et al., 2025) which can be found at https /i thub. com cwmmowonhtet ht///whhange /ww/hwggghh111111 ICLR25- CRRSP tttee/ree///eeee sinee//minn mainn. We detail key model configuration parameter choices here:
1. Branch layer count: 4 (standard implementation)
2. Trunk layer count: (standard implementation)
3. Branch layer width: 128 (standard implementation)
4. Trunk layer width: 128 (standard implementation)
5. All other parameters: All other DeepONet parameters were initialized to default values detailed the official implementation of (Gao et al., 2025); https: /ttt:////ttbb//// com/ frnhangaa21/IcIee.
tree main.
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I
MAX MODES
key design decision in the FNO architecture is the parameter m that indicates maximum frequen- cies keep along each dimension the Fourier layer during the forward pass. This has implications both during training (which frequencies are learned in the Fourier layers) and at inference (which frequencies are predicted over in the Fourier layers). that it can make use all frequencies in its imput: this is especially critical in the multi-resolution set- ting where data of varying discretization will have varying frequency information. Here, we study the impact of varying m. In the zero-shot multi-resolutio inference setting, in Figs. 34 and 33, we find that that across all variation in m, the models assign high energy the high-frequencies comment that in the context of multi-resolution inference, it does
Sec. 3, the FNO is always initialized such
(e.g., aliasing). More broadly, not make sense to set m to a value less than the largest populated frequency in a model input, as it ensures that the model cannot make that frequency information greater than in the Fourier layers. the event frequency information above m not useful to prediction (e.g., noise), we ad- vocate low-pass filtering and down sampling the data to more compressed representation of data prior to inference to remove unwanted frequencies and ensure faster inference.
Train data
Train data
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25 Frequency
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Figure 33: Resolution Interpolation. Four FNOs trained on Darcy data low-pass limit = 8 (constant frequency information) and down sampled to resolutions [16,32,64,12)) 128)) (varying sampling rate) from left to right, and top bottom with max modes m 18,16,32,64) We test if each model can generalize to data with varying sampling rate. Visualizing the spectrum of the normalized residuals across test data. Notice that the residual spectra (error) inereases substantially in the low frequencies. Lower energy at all wave numbers better. We see that across all variation in m. the models assign high energy in the high-frequencies.
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Train data Filter (8)
Train data Filter (16) Train data Filter (32) Train data Filter (64)
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Filter limit (8)
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Filter limit (64)
Figure 34: Information Extrapolation. Sixteen FNOs trained on Darcy data of resolution 128 (constant sampling rate) and low-pass filtered with limits (8,16,32,64) (varying frequency infor- mation) from left to right, and top bottom with max modes (8,16.32,44) We test if each model can generalize to data with varying frequency information. Visualizing the spectrum of the normalized residuals across test data. Notice that the residual spectra (error) increases substantially in the high frequencies. Lower residual energy at all wave numbers is better. We observe that across all variation in m, the models assign high energy in the high-frequencies.
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J ASSESSING DEEPONET
We extend our investigation include DeepONet el., 2021a). DeepONet is prominent MLO, however, it notably does not claim to perform accurate zero-shot super- and sub-resolution. Despite this, we still investigate the zero-shot super-resolution, sub-resolution, information extrapolation, and resolution interpolation abilities of DeepONet. We then assess if multi-resolution training can enable more accurate multi-resolution inference. Prior to the investigation, we note that DeepONet operates at fixed resolution on the input and is only resolution independent on the output.
search over learning rates E {le 2, 1e-3,, 1,,1e,4, 4, le- 5): we set static weight decay wd = le 5 as we do not notice significant performance changes w.r.t. weight decay for FNO/CNO/CROP (See Tables and 2). We find the optimal learning rate le 3 which report in Tab. 2.
We begin by performing a
Next, we assess DeepONet's ability perform information extrapolation and resolution interpolation via experiments detailed in Sec. 3.1 and find that it cannot do either accurately (sce Figures 35 and 36). There is little variability in Deponent's 's information extrapolation performance across test datasets; there is similar level of aliasing across all test datasets regardless of frequency content.
Train Data
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Figure 35: Resolution Interpolation w/ DeepONet. Four DeepONets trained Darcy data resolutions {16. 32, 64, 128} from left to right with constant frequency information (low-pass limit of 8f), and are tested on varying resolutions. We assess each model can generalize to data with varying sampling rate. We visualize spectra of the normalized residuals across test data. Notice, residual spectra (error) increases substantially in the low frequencies. Lower residual energy at all frequencies is better.
Train Data Filter (8)
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Train Limit
Figure 36: Information Extrapolation w/ DeepONet. Four DeepONets trained on Darcy data of resolution 128 (constant sampling rate) and low-pass filtered with limits {8, 16, 32, 64) (varying frequency information) from left right. Test if each model can generalize to data with varying frequency information. Visualizing spectrum of the normalized residuals across test data. Notice, residual spectra (error) increases substantially in the high frequencies. Lower residual energy at all frequencies is better.
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We then assess if DeepONet perform zero-shot super-reso reperrresolutoon and sub-resolution As found with other MLOs, we find that cannot accurately perform zero-shot super-resolution and sub-resolution (See Figures 37 and 38).
DeepONet2D
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Figure 37: FNO and DeepONet trained on Darey. On average both models incur higher losses across test resolutions that differ from the train resolution. This means both FNO and DeepONet cannot accurately do zero-shot super-resolution or sub-resolution.
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Figure 38: DeepONet for Darcy. Energy spectra for models trained at specific resolution (y-axis) and tested at multiple resolution (x-axis). The spectrums generated by the models not match ground truth for test resolutions that diverge from the training resolution.
Finally, we assess if multi-resolution training can improve DeepONet's data-driven multi-resolution performance. We employ the same experimental methodology outlined in Sec. 5. We find that multi-resolution training improves DeepONet's performance across test resolutions (see Fig. 39). We observe that DeepONet is more sensitive than FNO w.r.t. the ratios of data across resolutions: DeepONet benefits from having more data samples evenly spread out across resolutions. We hypoth- the output, meaning
esize this is due to the fact that DeepONet is only resolution independent that during training the model must learning mappings from exclusively low resolution inputs to varying resolution outputs (low through high). Because low resolution inputs contain less frequency
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components than high high-resolution inputs, it is harder learning task than learning between the same resolution input output mappings (as FNO does).
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Figure 39: Multi-resolution training for DeepONet. DeepONet trained on multi-resolution Darcy data. Top row: Average number of pixels data sample in the training Lower number of pixels enables faster data generation and model training. Middle row: The ratios of data within each resolution bucket. Bottom row: The average test loss across different resolutions. Lower loss is better. Notice: mixed resolution datasets achieve both low average data size and low loss (ideal scenario).
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K ASSESSING ANTI-ALIAS NONLINEARITIES
We extend our investigation FNO to assess the impact nonlinearities play on model aliasing in the multi-resolution inference setting. As discuss in Sec. 6. Karras et al. (2021) observed that the application of point-wise nonlinearities to intermediate model representations introduces high- frequency components that cannot be resolved via the given resolution. To mitigate this source of aliasing, they propose anti-aliasing nonlincarities which simply interpolate data to greater reso- lution by scalar factor, apply the desired mon-ooniinearty the higher resolution representation, and then interpolate the signal back down to the original resolution (thus removing many of the introduced high-frequency components). The CNO architecture utilizes these anti-alias nonlinear- ities. We now investigate the impact of anti-alias nonlinearities to disentangle architecture- vs. data-pipeline-driven aliasing for FNO.
We begin by assessing the FNO with anti-aliasing nonlinearities' ability to perform information extrapolation and resolution interpolation via experiments detailed Sec. 3.1 and find that it cannot do either accurately (see Figures 40 and 41). This confirms that the out-of-distribution nature of data plays a significant role aliasing for FNO.
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Figure 40: Resolution Interpolation w/ FNO+Anti-Aliasing Activation. Four FNOs w/ anti-
aliasing activation functions are trained on Darcy data at resolutions {16, 32, 64, 128} from left to right with constant frequency information (low-pass limit of 8f), and are tested on varying reso- lutions. We assess if each model can generalize to data with varying sampling rate. We visualize spectra of the normalized residuals across test data. Notice, residual spectra (error) increases sub- stantially in the low frequencies. Lower residual energy at all frequencies is better.
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Figure 41: Information Extrapolation w/ FNO+Anti-Aliasingg Activation. Four FNOs w/ anti- aliasing activation functions are trained on Darcy data of resolution 128 (constant sampling rate) and low-pass filtered with limits {8. 16, 32, 64) (varying frequency information) from left right. Test if each model can generalize data with varying frequency information. Visualizing spectrum of the normalized residuals across test data. Notice, residual spectra (error) increases substantially in the high frequencies. Lower residual energy all frequencies is better.
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We then assess if FNO with anti-aliasing nonlinearities can perform vero-shot super-resolution and sub-resolution. As we found with other MLOs, we find that it cannot accurately perform zero-shot super-resolution and sub-resolution (See Figures 42 and 43). Therefore, conclude that while anti- aliasing nonlinearities stem architecture- aliasing artifacts in the model forward pass, they do not address the data-driven aliasing artifacts introduced during inference data of resolutions that differ from a model's 's training resolution.
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Figure 42: FNO w/ Anti-Aliasing vs. GeLu activation functions trained on Darcy. Both models incur higher losses across test resolutions that differ from the train resolution. This means that acti- vation functions alone are not the source of FNO aliasing in the zero-shot multi-resolution setting; this confirms that data resolutions different than the model's training resolution sufficiently out- of-distribution. Zero-shot multi-resolution inference is unreliable regardless of activation function choice.
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Figure 43: FNO w/ Anti-Aliasing activation function for Darcy. Energy spectra for models trained at a specific resolution (y-axis) and tested at multiple resolution (X-axis). The spectrums generated by the models do not match ground truth for test resolutions that diverge from the training resolution.
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L SENSITIVITY ANALYSIS
We include an extended sensitivity analysis our resolution interpolation, information extraction and zero-shot superr/sub-resolution experiments for the Darcy dataset. We report the mean and stan- dard deviation over these experiments over three random seeds and show consistent performance across all seeds. In the main paper, we ensured the robustness of our results by replicating observed trends across wide array of datasets (Darcy, Burgers, Navier-Stokes) as a proxy for generaliz- ability of results. Doing multi-seeded experiments and Navier-Stokes) due to computational cost. By including both multi-seeded experiments and experiments over multiple datasets, we ensure that results are generalizable.
infeasible for the larger datasets (Burgers
We begin by assessing the FNO's ability to perform information extrapolation and resolution in- terpolation via experiments detailed in Sec. 3.1 over three random seeds and find that it cannot do either accurately (see Figures 44 and 45). Results are consistent over multiple seeds.
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Figure 44: Resolution Interpolation over 3 Random Seeds. Four FNOs are trained on Darcy data resolutions {16, 32, 64, 128) from left right with constant frequency information (low-pass limit of 8f), and are tested on varying resolutions. We assess if each model can generalize to data with varying sampling rate. We visualize spectra of the normalized residuals across test data. Notice, residual spectra (error) increases substantially in the low frequencies. Lower residual energy at all frequencies better. Results averaged over three random seeds; shaded region depicts standard deviation.
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Figure 45: Information Extrapolation over 3 Random Seeds. Four FNOs trained on Darey data of resolution 128 (constant sampling rate) and low-pass filtered with limits {8, 16, 32, 64) f (varying frequency information) from left to right. Test if each model generalize to data with varying frequency information. Visualizing spectrum Notice, residual spectra (error) increases substantially in the high frequencies. Lower residual energy at all frequencies is better. Results averaged over three random seeds; shaded region depicts standard deviation.
the normalized residuals across test data.
We then assess if FNO can perform zero-shot super-resolution and sub-resolution over three random seeds (see Fig. 46). We observe very similar performance across all seeds: FNO consistently fails
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to perform accurate zero-shot super-/sub gapper/sub-resouttinn resolution. We report mean and standard deviation of this experiment across all three seeds in Tab. 5.
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Figure 46: FNO on Darcy over three random seeds. On average FNO incurs higher losses across test resolutions that differ from the train resolution across all three random seeds.
Table 5: Assessing FNO zero-shot super-/sub-resolutie over multiple seeds. Reporting mean and standard deviation of MSE over random seeds.
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M ASSESSING DATA DOWN-SAMPLING METHOD
Unless otherwise states we follow the standard PDEBench convention for changing the resolution of our data in our experiments: strided downsampling Takamoto et al. (2022). Put simply, strided downsampling reduced the resolution of a signal by sampling every Nth point; therefore, to reduce the resolution of signal by 50%, every other point the signal would be kept (and the rest dis- carded). We choose strided downsampling as it simulates realistic experimental settings in which you have measurements of system at fixed intervals and no method resolve intermediate points between intervals (e.g., sensors for monitoring real-world system).
We now investigate the impact of sampling strategy by assessing the zero-shot super-/sub-resolution capability of models trained on data downsampled via data striding vs. average pooling vs. bilinear interpolation. In all cases, we consistently observe a failure to accurately fit the high-frequency regime when conducting zero-shot super-resolution (see Figures47-50).
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Figure 47: FNO on Darcy with varied data downsampling strategies. FNO trained and tested on data that downsampled via from the max resolution of 128 via three strategies: strided down- sampling, average pooling, and bilinear interpolation. On average FNO incurs higher losses across resolutions that differ from the train resolution across all three downsampling strategies.
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Figure 48: FNO trained on Darcy (strided downsampling). Energy spectra for models trained at specific resolution (y-axis) and tested at multiple resolution (x-axis). The spectrums generated by the models do not match ground truth for test resolutions that diverge from the training resolution. The train and test data that are less than 128 are downsampled via strided downsampling. The gray line indicates the maximum frequency present in the model's 's training data.
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Figure 49: FNO trained on Darcy (average pooling). Energy spectra for models trained
specific resolution (y-axis) and tested multiple resolution (x-axis). The spectrums generated by the models do not match ground truth for test resolutions that diverge from the training resolution. The train and test data that are less than 128 are downsampled via average pooling. The gray line indicates the maximum frequency present in the model's training data.
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Figure 50: FNO trained on Darcy (bilinear interpolation). Energy spectra for models trained at specific resolution (y-axis) and tested at multiple resolution (x-axis). The spectrums generated by the models do not match ground truth for test resolutions that diverge from the training resolution. The train and test data that are less than 128 are downsampled via bilinear interpolation. The gray line indicates the maximum frequency present in the model's 's training data.
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N BENCHMARKING PERFORMANCE ACROSS COMPUTING ARCHITECTURES
We profile the performance of FNO on the super-resolution inference task in both the zero-shot and multi-resolution training setting across multiple computing architectures. In the zero-shot super- resolution task, we train an FNO on resolution 16 Darcy data. In the multi-resolution training-based super-resolution task, we train an FNO on resolution, 25% 32 resolution, 25% 64 resolution, 25% 128 resolution data. We evaluate both models on test data of varying resolutions: 16, 32, 64, 128.
Darey dataset comprised of 25% 16
In all experiments, we use Python 3.10 and "neuralopertor" library (Kossaifi et al., 2025) version 2.0.0. Notably, we were not able to acquire access to AMD GPU hardware and older NVIDIA hardware (e.g., GPUs in the NVIDIA RTX/PI00 lines which were used in the previous work (Li et al., 2020a) were not available to us). Further, while we did have access to the Intel Data Center GPU Max Series (128GB HBM), the hardware-compatible PyTorch version 2.8. 1.8.000+gitba56110 did not support the full stack of operations used by FNO layers so were were unable to profile performance on that hardware. We tested the follow computing architectures:
1. AMD EPYC 7763 (Milan) CPU; Py Torch version 2.9.1
2. NVIDIA A A100-PCIE-40GB with driver version 550.163.01; CUDA Toolkit version 12,4.131; PyTorch version 2c
3. NVIDIA A40 with driver verion 590.44.01; CUDA Toolkit version 13.1,80; PyTorch ver- sion 2.10.0tcu130
4. NVIDIA H100 NVL with driver verion 590.44.01; CUDA Toolkit version 13.1.80; PyTorch version 2.1000ccu130
5. NVIDIA H200 NVL with driver verion 590.44.01; CUDA Toolkit version 13.1,80; PyTorch version 2.10.0+cu100
6. Apple M2 Pro; Metal 4; PyTTrch version 2.10.0
We observe that there is wide variance in error across different hardware (see y-axes Fig. 51). Despite this, we confirm the core trend of multi-resolution training outperforming zero-shot super resolution is consistent across computing architectures. In Figures 53-57, we note that the predicted spectra of the zero-shot model consistently diverged from the label spectra after frequency (the maximum observable frequency in the training data) across all hardware; this was not the case for the model trained on multi-resolution data. This suggests that variance in hardware not the reason for poor zero-shot super-resolution, but rather it is the -iitt---iittibttionn nature of the zero-shot inference task. While implementation differences in hardware-software stacks give rise to different levels of error in the model's forward pass it does so consistently across different types of model training regimes (e.g., single-resolution vs. multi-resolution). Therefore, we conclude that these hardware performance differences are not key drivers of data-driven aliasing artifacts in FNO.
Finally, we caution users to be aware of the wide-range of FNO performance across different com- puting architectures and invite the community to further replicate our experiments on additional architectures. To enable this, in addition to releasing our full set of experimental code', we release both these hardware tests and more comprehensive set of tests to reproduce representative subset of the experiments from this paper'.
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Figure 51: Zero-Shot v.s. Multi- Training for
X-axis: under the curve (AUC) of the normalized spectrum of the residuals the model predictions (lower bet- ter). Y-axis: Test resolution. Across all computing architectures, notice that the multi-resolution model consistently out- out-performs the zero-shot model across all test resolutions.
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Example Prediction
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AMD CPU. Notice that the model predicted spectra diverge the zero-shot super- resolution setting after frequeney 8 (the maximum frequency present in the training data). Overall, the multi-resolution model predicted spectra are more accurate.
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Figure 53: NVIDIA A100. Notice that the model predicted spectra diverge the zero-shot super- resolution setting after frequency 8 (the maximum frequency present in the training data). Overall, the multi-resolution model predicted spectra are more accurate.
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resolutions: 16, 32, 64, 128. Top Row: Sample prediction Darcy flow. Middle Average test set 2D energy spectrum of label and model prediction. Bottom Row:
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Figure 54: NVIDIA A40. Notice that the model predicted spectra diverge in zero-shot super- resolution setting after frequency 8 (the maximum frequency present in the training data). Overall, the multi-resolution model predicted spectra are more accurate.
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Figure 55: NVIDIA H100. Notice that the model predicted spectra diverge the zero-shot super- resolution setting after frequency 8 (the maximum frequency present in the training data). Overall, the multi-resolution model predicted spectra are more accurate.
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Example Prediction
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Figure 56: NVIDIA H200. Notice that the model predicted spectra diverge the zero-shot super- resolution setting after frequency 8 (the maximum frequency present in the training data). Overall, the multi-resolution model predicted spectra are more accurate.
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Train Limit
Normalized Recidual
(b) Multi-resolution Training. Model trained on dataset with (25% resolution 16, 25% resolution 32, 25% resolution 64, 25% resolution 128} data, and evaluated at varying resolutions: 16, 32, 64, 128. Top Row: Sample prediction for Darcy flow. Middle Average test 2D energy spectrum of label and model prediction. Bottom Row: Average residual spectrum normalized by label spectrum.
Row:
Figure 57: Apple M2 Pro. Notice that the model predicted spectra diverge in the zero-shot super- resolution setting after frequency 8 (the maximum frequency present the training data). Overall, the multi-reso/ution model predicted spectra are accurate. Note: While the FNO model is capable inference on this computing architecture, we observe significant distortions the model predictions: therefore, we suspect the hardware-compatible PyTorch version 2.10.0 for the Apple M2 Pro may not be fully compatible with the "neuraloperator" V2.0 library.
54


Reviews:
[{"review_id": "93TTDz79XK", "reviewer": "Reviewer_am35", "summary": "This paper empirically investigates whether Machine-Learned Operators (MLOs) can reliably generalize across different resolutions when modeling Partial Differential Equations (PDEs), a capability often referred to as zero-shot super-resolution or sub-resolution. Through well-structured experiments covering interpolation, extrapolation, and super-resolution scenarios, the authors demonstrate that Fourier Neural Operators (FNOs) consistently fail when evaluated at unseen resolutions, largely due to aliasing and out-of-distribution effects. The study further shows that physics-constrained optimization and band-limited training, do not address these issues. As a mitigation strategy, the authors propose a simple multi-resolution training protocol, which substantially improves accurate results but still falls short under severe resolution mismatches.", "questions": "- A single consolidated table summarizing final model configurations (e.g., number of modes, hidden dimensions, etc.) for each dataset would improve clarity and ease of implementation.\n\n- Since aliasing is highly sensitive to data resampling, the paper should more clearly specify the interpolation and filtering operations used in generating different resolutions.\n\n- While the empirical evidence is compelling, the paper would benefit from a brief theoretical discussion.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "It is clearly written, well-structured, and addresses an important yet often overlooked question in the MLOs. The authors effectively decompose the problem into resolution interpolation, information extrapolation, and both, providing a rigorous framework for analysis. The work’s key contribution lies in extensive empirical evaluation under different settings.", "weaknesses": "While the title refers broadly to “Machine-Learned Operators (MLOs),” the experiments primarily focus on the FNOs. Given that FNOs represent only one class of MLO architectures, extending the analysis or discussion to other operator-learning models would help clarify how general these findings truly are.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "acay5zsmcQ", "reviewer": "Reviewer_MdwW", "summary": "The paper interrogates the widely circulated claim that mesh‑invariant machine‑learned operators (MLOs)—notably Fourier Neural Operators (FNOs)—can do zero‑shot multi‑resolution inference: train on one grid and generalize to higher (or lower) resolutions without additional data. It decomposes the problem into two behaviors: resolution interpolation (same frequency content, new sampling rate) and information extrapolation (same sampling rate, new frequency content). Carefully controlled experiments on Darcy (2D), Burgers (1D), and incompressible Navier–Stokes (2D) from PDEBench reveal that FNOs fail at both: residual energy spectra spike exactly in bands that should remain quiet under the experimental controls, evidencing aliasing; artifacts grow through time‑rollouts for Navier–Stokes. “Zero‑shot fixes”—adding physics‑informed losses or adopting band‑limited approaches (CNO, CROP)—do not rescue multi‑resolution generalization: physics terms usually hurt, and band‑limited models fit low frequencies but drop high‑frequency content by design. The authors then propose a simple multi‑resolution training recipe (mix mostly low‑res with a small fraction of high‑res), documenting a favorable cost–accuracy Pareto front across tasks. Figures and appendices provide spectral diagnostics, max‑modes ablations, and resolution×resolution heatmaps that collectively undermine the zero‑shot super‑resolution narrative and replace it with an actionable practice.", "questions": "1. Evaluation is limited primarily to Fourier-based operator architectures. Including at least one non-Fourier, mesh-invariant operator (e.g., DeepONet, U-NO, or multiwavelet neural operator) would strengthen the generality of the findings and demonstrate whether the observed zero-shot failure modes persist across different operator families.\n\n2.  For the Navier–Stokes experiments, additional quantitative diagnostics—such as errors on enstrophy, energy spectra, or divergence and conservation measures—are needed to clarify the apparent discrepancy between low mean-squared error and degraded spectral fidelity in band-limited models.\n\n3. An explicit investigation of anti-alias filtering around nonlinearities, or of antialiased up-/down-sampling within the operator pipeline, can determine whether the reported interpolation and extrapolation failures are due to architectural aliasing rather than inherent data-distribution shifts.\n\n4. The study shows that physics-informed losses underperform across weighting choices. Testing alternative formulations—such as residual-only constraints, adaptive weighting or curriculum schemes, or discretization-matched residuals—could be informative to rule out optimization artifacts and isolate the underlying cause of this degradation.\n\n5.  Reporting mean ± standard deviation across multiple random seeds for key figures and tables, together with an ablation of the low-/high-resolution data ratios in the multi-resolution training protocol, are needed for more robust evidence and practical guidance for determining data-budget trade-offs.\n\n6. It remains unclear whether the proposed multi-resolution training improves robustness to out-of-distribution conditions beyond discretization changes, such as shifts in physical parameters (e.g., viscosity, forcing, boundary conditions). \n\n7.  Public release of scripts for generating controlled low-pass and resampled datasets, computing normalized residual spectra, and reproducing the reported Pareto-front trade-off plots are required for areproducibility and transparency.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Good conceptual framing: Separating resolution interpolation from information extrapolation clarifies why “train low, test high” fails and grounds the evaluation in sampling theory.\n\nRigorous diagnostics: Normalized residual energy spectra consistently reveal aliasing when resolution or frequency content shift, time‑rollout visuals show error compounding in Navier–Stokes.\n\nBreadth of evaluation: Three canonical PDEs, systematic train/test grids, and resolution×resolution loss heatmaps; max‑modes ablations show failures persist across spectral truncation choices.\n\nActionable guidance: Multi‑resolution training (mostly low‑res + some high‑res) improves robustness with modest cost, forming a clean Pareto front between average data size and error.\n\nThe study complements claims of discretization invariance in operator learning (FNO, DeepONet, mesh‑invariant variants) and recent aliasing/anti‑aliasing discussions by providing decisive, controlled counter‑evidence and a practical alternative.", "weaknesses": "1. Most conclusions rest on FNO (plus CNO/CROP wrappers). Including non‑Fourier operator learners (e.g., DeepONet/U‑NO/multiwavelet or graph‑kernel operators) would better support claims that span “MLOs” broadly.\n\n2. Results appear single‑seed without confidence intervals or standard deviation reporting, negative conclusions look stable but would benefit from variance reporting across seeds and data resamplings.\n\n3. In turbulence, band‑limited methods can achieve competitive MSE while failing spectrally; adding physics‑aware diagnostics (e.g., enstrophy/energy spectra errors, divergence) would clarify practical significance.\n\n4. While max‑modes are ablated, there is no explicit test of antialiased nonlinearity/resize operations (well‑known in vision/GANs) inside FNO blocks; this could disentangle architecture‑ vs. data‑pipeline‑driven aliasing.\n\n5. The study focuses on resolution/frequency shifts; it would be useful to probe parameter OOD (e.g., viscosity/forcing) to test whether the proposed multi‑resolution recipe transfers to broader distribution shifts.\n\n6. There is a qualitative data‑size vs. epoch‑time trend, but a more explicit accounting (wall‑clock, memory, FLOPs) for the zero‑shot baselines vs. multi‑resolution training would sharpen the practical trade‑offs.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "MWbzVHFaLv", "reviewer": "Reviewer_2rDL", "summary": "The paper validates the ability of trained MLOs to generalize beyond their training resolution and shows that accurate zero-shot multi-resolution inference is unreliable.\n\nThey find that neither approach, incorporating physics-informed constraints during\ntraining and performing band-limited learning, enables reliable multi-resolution generalization.\n\nAlso, the authors propose and test multi-resolution training, where they include training data of varying\nresolutions (in particular, a small amount of expensive higher-resolution data and a larger\namount of cheaper lower-resolution data).", "questions": "1. How the findings in the paper can be applied to the real-world problem?", "limitations": "", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. They systematically analyze the zero-shot multi-resolution abilities of FNO (Fourier Neural Operator) with several methods including resolution interpolation and information extrapolation\n\n2. The paper visualizes the results and their findings clearly which leads to better readability of the paper.", "weaknesses": "1. The paper does not show how the findings in the paper can be applied to real applications including deep learning-based training/inference.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "sAeulsy8PW", "reviewer": "Reviewer_eXr2", "summary": "The paper evaluates the hypothesis that Fourier Neural Operators (FNOs) and similar Machine-Learned Operators (MLOs) can generalize across resolutions, i.e., perform \"zero-shot super-resolution\", when trained on low-resolution data. They test the hypothesis on Darcy flow, Burgers', and Navier-Stokes equations and show that FNOs fail to generalize to unseen resolutions or frequency bands. They contribute these failurs as aliasing phenomena and propose a simple fix: multi-resolution training, mixing low- and high-resolution samples, which improves generalization without major cost increase.", "questions": "The main claims appear to echo the insights of the following work:\n[1] Neural Operator: Learning Maps Between Function Spaces With Applications to PDEs, Kovachki et al., JMLR 2023\n[2] Discretization-invariance? On the Discretization Mismatch Errors in Neural Operators, Gao et al., ICLR 2025\n\nCould you clearly explain how your analysis provides new insight beyond these works? In particular, is the aliasing perspective quantitatively novel, or primarily a reinterpretation of previously reported resolution-mismatch failures?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The work has clear, well-motivated experiments\n- The systematic decomposition of the problem (frequency extrapolation vs resolution interpolation) is nice", "weaknesses": "- \"zero-shot super-resolution\" phrasing misleads readers from computer vision/image SR domains; the task is not SR in the standard sense.\n- The insight that trained models don't extrapolate beyond training distribution is general ML knowledge and the work contributes only limited conceptual novelty.\n- The proposed solution (multi-resolution training) is expected and straightforward with limited novelty for the ML community.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)