Breaking Scale Anchoring: Frequency Representation Learning for Accurate High-Resolution Inference from Low-Resolution Training
Authors: Wenshuo Wang
Fan Zhang
Venue: ICLR 2026 Poster (2026)
Keywords: Scale Anchoring
Zero-Shot Super-Resolution
Spatiotemporal Forecasting
Frequency Representation
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Zero-Shot Super-Resolution Spatiotemporal Forecasting requires a deep learning model to be trained on low-resolution data and deployed for inference on high-resolution. Existing studies consider **maintaining** similar error across different resolutions as indicative of successful multi-resolution generalization performance. However, deep learning models serving as alternatives to numerical solvers should **reduce** error as resolution increases. The fundamental limitation is, the upper bound of physical law frequencies that low-resolution data can represent is constrained by its Nyquist frequency, making it difficult for models to process signals containing unseen frequency components during high-resolution inference. *This results in errors being anchored at low resolution, incorrectly interpreted as successful generalization.* We define this fundamental phenomenon as a new problem distinct from existing issues: **Scale Anchoring**. Therefore, we propose architecture-agnostic Frequency Representation Learning. It alleviates Scale Anchoring through resolution-aligned frequency representations and spectral consistency training: on grids with higher Nyquist frequencies, the frequency response in high-frequency bands of FRL-enhanced variants is more stable. This allows errors to decrease with resolution and significantly outperform baselines within our task and resolution range, while incurring only modest computational overhead.

Published as a conference paper at ICLR 2026
BREAKING SCALE ANCHORING:
FREQUENCY REPRESENTATION LEARNING FOR
ACCURATE HIGH-RESOLUTIO INFERENCE FROM LOW-RESOLUTION TRAINING
Wenshuo Wang', Fan Zhang
School of Future Technology. South China University of Technology, China
2 State Key Laboratory of Ocean Sensing & Ocean College, Zhejiang University. China 3 Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, USA 2023648702511 m1amaimmaii sout edu. cn, f zhange ju edu.cn
ABSTRACT
Zero-Shot Super-Resolution Spatiotemporal Forecasting requires a deep learn- ing model to be trained on low-resolution data and deployed for inference on high-resolution. Existing studies consider maintaining similar error across differ- ent resolutions as indicative of successful multi-resolution generalization. How- ever, deep learning models serving alternatives to numerical solvers should reduce error as resolution increases. The fundamental limitation is, the upper bound of physical law frequencies that low-resolution data can represent is con- strained by its Nyquist frequency, making it difficult for models to process sig- nals containing unseen frequency T components during high-resolution inference. This results in errors being anchored at low resolution, incorrectly interpreted successful generalization. We define this fundamental phenomenon problem distinct from existing issues: Scale Anchoring. Therefore, we propose architecture-agnostic Frequency Representation Learning. It alleviates Scale An- choring through poollution-alignee frequency representations and spectral consis- tency training: on grids with higher Nyquist frequencies, the frequency response in higu-frey bands of FRL-enhanced variants is more stable. This allows er- rors to decrease with resolution and significantly outperform baselines within our task and resolution range, while incurring only modest computational overhead.
INTRODUCTION
Traditional numerical simulation methods in Spatiotemporal Forecasting (STF) can achieve high accuracy yet incur substantial computational costs (Choi & Moin, 2011). Recent research utiliz- ing deep learning methods for STF can effectively balance accuracy and computational efficiency (Zhang et al., 2023; Saad et al., 2024). However, high-resolution, high-fidelity Direct Numerical Simulations (DNS) that provide training data for deep learning methods face extremely high costs. Even with high- high-resolution data available, the extremely high resolution poses impossible training VRAM requirements for current hardware, while the requirements for inference are much smaller. Zero-shot super-resolution (ZS-SR deep learning models can leverage low-cost, low-resolution data for VRAM training to perform high-resolution STF (Li et al., 2020).
Existing ZS-SR STD methods are based on Neural Operators (NOs). This is because, unlike general neural network architectures that learn function mappings from point space to point space. NOs learn functional mappings from function to function (Lu et al., 2021). Since functions can be discretized into spaces of arbitrary resolution, NOs naturally meets the requirements for inputs and outputs of different resolutions (Kovachki al., 2023). On the other hand, many variants, such Fourier NO, learn parameters in a scale-agnostic manner in the frequency domain, which can also improve the generalization of the model's physical laws (Li et al., 2020).
"Corresponding author.

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Published as a conference paper at ICLR 2026
Existing researchs expect models' successful generalization to maintain similar accuracy across in- puts of different resolutions (Talebi & Milanfar, 2021; Gao et al., 2025). From this perspective, our pilot study on ZS-SR STF for all mainstream architectures in Section 3 shows that every bascline ex- hibits excellent generalization. However, for a p-th order numerical solver simulating at a resolution times higher than the low resolution, the error theoretically decrease by factor of a². This gap arises from the fact that the upper bound of the frequency of physical laws that low-resolution data can represent limited by the Nyquist frequency of the training data. When models are trained on low-resolution data and perform inference at high resolution: the model struggles to handle unseen High-frequency components, causing the model's eтror to be anchored at the low-resolution data. We refer to this data-driven limitation as Scale Anchoring, which is fundamentally different from previously studied issues such as Spectral Bias (SB) and Discretization Mismatch Error (DME).
Addressing Scale Anchoring requires improving generalization to higher relative frequencies that are not present at the training grid. Prior methods for cross-resolution generalization of physical laws were not explicitly designed to tackle this scale-anchoring mechanism (Li et al., 2020; 2024c). We therefore propose Frequency Representation Learning (FRL): (i) construct multi-resolution frequency representations that embeddings for the same physical frequencies; and (ii) add a frequency- aware loss to promote spectral consistency across scales. Steps (i) and (iii) follow standard practices in multi-scale training and spectral regularization, while Step (ii), which is aligned with Scale An- choring, is novel. Under mild assumptions, our analysis shows that this alignment encourages more stable high-freegu freeuerrrgueyyy bands. It allows FRL to reduce the high-resolution error of a given baseline as resolution increases, although does not guarantee strict order convergence like numerical solvers.
training data via downsampling; (ii) introduce Nyquist-normalinn yield
Our contributions are summarized as follows: (a) Identifying Scale Anchoring, a previously un- recognized fundamental limitation in ZS-SR STF that incorrectly interpreted as successful gener- alization; (b) Providing theoretical analysis and empirical validation of Scale Anchoring and its mechanism; (c) Proposing recitecture-agnosti FRI. for Scale Decoupling; (d) Extensive experi- ments across diverse architectures show that FRI-enhanced methods decouple Scale Anchoring. In ZS-SR STF, they demonstrate higher accuracy with modest increases in training time and memory overhead; (e) We identify the failure modes of FRL, characterize the conditions for effectiveness and the boundaries of failure, and suggest potential improvement strategies.
RELATED WORK
2.1 ZERO-SHOT SUPER- RESOLUTION SPATIOTEMPORA FORECASTING
STF tasks take one or multiple physical spatial field snapshots at different time steps as input, predict the next time step's 's snapshot, and repeat iteratively. ZS-SR STF tasks train models low-resolution snapshots only, but at inference time take high-resolution physical spatial field snapshots at different time steps as input, predict the next time step's snapshot, and repeat iteratively. NOs are naturally suited for this task due to their resolution-agnostic input and output capabilities (Li et al., 2020; Ko- vachki et al., 2023). Existing NO variants span multiple directions: FNO pioneered ZS-SI through -omain learning (Li et al., 2020; Jiang et al., 2023; Atif et al., 2024); PINO enhanced accuracy via physics constraints (Li et al., 2024c); TNO specialized temporal modeling for long- term predictions (Diab& & AI- Al-Kobaisi, 2025); and multi-scale methods like Multi-Grid Tensorized FNO and U-FNO reduced computational complexity through hierarchical decomposition (Kossaifi et al., 2023; Wen et al., 2022). While these methods improve resolution generalization, they do not explicitly address the fundamental frequency limitation imposed by the training data's 's Nyquist fre- quency. Without mechanisms to learn extrapolate frequency patterns beyond this hard boundary, they remain fundamentally unable to resolve Scale Anchoring.
2.2 SCALE ANCHORING V.S. RELATED PHENOMENA
Scale Anchoring shares superficial similarities with several phenomena but differs fundamentally:
SB describes the tendency of neural networks to fit target functions from low to high frequencies within the training Nyguist band, leading to larger errors high-frequency components that are actually present in the supervision data (Xu et al., 2019; Rahaman et al., 2019). To mitigate SB, prior work introduces Fourier feature mappings and positional encodings, periodic activation functions,
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multi-scale or hierarchical architectures, and explicit spectral or frequency-weighted losses, all of which strengthen the network's 's ability to represent and fit high-frequency content already contained in the data (Tancik et al., 2020; Sitzmann et al., 2020; Liu et al., 2024; 2025).
In the NO literature, lack of discretizationninaarii
(lack of DI) and DME formalize the problem that the same operator network can produce inconsistent outputs on different grids (Bartolueci al., 2023; Gao et al., 2025; Lanthaler et al., 2024). These works analyze aliasing and discretization error to propose alias-free spectral parameterizations, multi-grid training, and carefully designed interpolation operators to enforce cross-grid consistency. More broadly, multi-resolution generation in vision and scientific ML-whether cascaded, patch-based, or training-free -ree--ffee--e expose models to multiple resolutions (Tian et al., 2023; He et al., 2023; Ho et al., 2022; Bar- Tal et al.. 2023; Yu et al., 2023). In this sense, the first and third steps of FRI. inulti-resolution training and frequency-awand loss) follow standard practices in these research directions and are not methodologically novel.
By contrast, Scale Anchoring is driven by the information- thhooretic limitation that low-resolution training data cannot represent physical frequencies above its Nyquist limit. The difference between Scale Anchoring and other phenomena has three implications. First, in terms of source, SB, lack of DI, and DME all arise from architectural and optimization choices and can be eliminated through appropriate model design and training. Whereas Scale Anchoring arises from the Shamnon-Nyquis sampling bound on the data itself. Second, in terms of scope, lack of DI and DME are formulated specifically for NOs, while SB and SA occur broadly across any structures (including NOs) as we empirically demonstrate in Section 4.2. Third, in terms of consequence, SB, lack of DI, and DME soft tendencies that do not constitute hard limits on achievable accuracy. Whereas Scale Anchoring imposes an informition--heerrtii lower bound on high-resolution error. Therefore, Scale Anchoring is orthogonal to existing phenomena: even if the model is completely alias-free, discretization- invariant, and fits signals within the training Nyquist band equally well, model trained only on low-resolution data will never observe frequency components above the training Nyquist frequency.
3
EXISTENCE OF SCALE ANCHORING
0.008
High Resolution
Low Resolution
1.3
Table 1: Multi-resolution RMSERatio across dif- ferent methods in ZS-SR 3D fluid simulation.
RMSS
0.006
RMSE
0.004
Lotg
Method
2.4x
8x
65.5x
0.9
0.002
0.8
GNN (Neural SPH)
1.000
1.000
1.022
Transformer (DeepLag)
1.006
1.012
1.021
0.7
CNN (PARCv2)
1.012
1.035
1,060
0.000
GNN
CNNDiffusion Neural
NN ODE
Diffusion (DYffusion)
1.010
1.024
1.041
former
NO (SFNO)
1.004
1.011
1.017
Neural ODE (FNODE)
1.067
1.180
1.338
Figure 1: High and low resolution RMSE with their ratio across different methods in 3D ZS-SR fluid simulation.
NN (NeuralFluid)
1.011
1.024
1.035
We first demonstrate the existence of Scale Anchoring ZS-SR fluid simulation. We use the same baselines and fluid dataset as in Section 6.1. For the most commonly used deep learn- ing architectures fluid simulation, we tested each State-Of-The-Ar (SOTA) method trained on low-resolution (323) data and evaluated on 2.4x, 8x, 65.5x resolution, measuring RMSE and RMSERati = RMSEHigh/ RRMSL The results are shown in Figure and Table 1.
As shown in Figure 1, all architectures achieve RMSER between 1-1.4 under 64x super- resolution. As shown in Table 1, the RMSE changes across different resolutions for all architectures remain minimal (~1). From the MRG perspective, this demonstrates successful generalization. However, for a physical domain with original grid spacing Ax: After increasing the resolution by a factor of a, for numerical scheme with p-th order accuracy, the truncation error is:
E = C (Ax) + O ((Ax)x+1)
(1)
E = C (Ax/ a) +0 (AAxaa)+11)
(2)
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where E1 represents the original enror and E2 represents the new error. The error reduction factor E. (Ax)"
P2 [Ax/ojP a². This indicates that if the models truly function as numerical solvers, errors should decrease following a power law as resolution increases. However, the results shown in Table indicate that the model fits data at specific resolution rather than learning the correct physical operator to become a true numerical solver. We refer to this phenomenon, where the inference pattern and errors are anchored the training resolution, Scale Anchoring.
MECHANISM OF SCALE ANCHORING
4.1 THEORETICAL ANALYSIS
Physical evolution spatiotemporal systems is governed by partial differential equations defining operators on function spaces. The evolution operator F: C(S2) + C(It) that advances the physical field u(-, t) to u(- uu= (t)): it operates on continuous functions and embodies resolution-indepeneen physical laws. Neural networks, however, learn fundamentally different mappings. When trained at resolution Po with grid spacing Ar = P/ and Nop grid points, a neural network minimizes:
min B. (w)) - Spo (F(
(3)
where Soo samples the continuous field onto a discrete grid and Ge applies learned operations. Ge learns function mapping between finiteddimensional spaces R. RR,,, not the functional F. When deployed at resolution the same learned parameters P, are applied, denoted as G(p)
@
This Amnction-versu--funttin distinction leads to a fundamental frequency limitation:
Theorem 1 (Frequency Blindness). neural network trained at resolution po cannot correctly process frequency components above the Nyquist frequency Po/2. The learned operator's frequency response satisfies:
(F() + +(w),
w S Po/2 Lundefined/mcorrect,
Ge (w) =
(4)
> po/2
This directly causes Scale Anchoring when the network is deployed at higher resolutions:
Theorem 2 (High-Frequency Error Dominance). When network trained at resolution po is de- ployed at resolution p Po, the Scale Anchoring error bound:
C= I=lim p+
|G S,
Sp o
(5)
is dominated by the network's 's inability to process frequency components in the range [po/ p/2].
The complete mathematical derivations are presented in Appendix A.
4.2 EXPERIMENT VALIDATION
In this section, we design two experiments validate Theorems and 2:
Validation Experiment A.1: We first use pseudo-spectral methods in the frequency domain to simulate 2D convection-difftsion equations at 64 resolution. We then implement eight commonly used model architectures in STF (GNN, Transformer, CNN, Diffusion, NO, Neural ODE, Mamba, NN) and fully train them on the simulation data. Finally, we input sinusoidal signals (u(z, y.t) A- sin (2n f x)) of varying frequencies (0-50Hz) and measure the magnitude of the empirical frequency response Hmag(() = Aout (f)AAm Am (f) (output-to-input amplitude ratio), Bandwidth (BW), and Anchoring Ratio: H(f = H 34) (cutoff sharpness). Results are shown in Figure 2.
All architectures exhibit a unified frequency response pattern: maintaining high H() before the Nyquist frequency (32Hz), followed by a cliff-like drop near it. This results in BW concentrated around the Nyquist frequency and high Anchoring Ratios. The universal Scale Anchoring across all architectures validates Theorem 1, confirming that neural networks trained at resolution Pu cannot correctly process frequency components above the Nyquist frequency po/2. 22. More detailed experi- mental settings and results for different training resolutions provided in Appendix B.
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GNN
Transformer
CNN
Diffusion
1.5
1.00-
1.00
1.00
0.75
0.75
0.75
1.0-
Nyguist
0.5
0.25
BW-31.5
0.23+
0.25
BW-31.5
Ratio=2.30
Ratio=1.91
Ratio=8.97
Ratio
30
40
30
40
0.00
10
NO
Neural ODE
Mamba
1.00-
1.00
1.00
1.00
0.75
0.75
0.75
BW-31.0
0.75
Nyquist
Ratio=16.14
Nyquist
H-0.5
0.25-
BW-31.5
0.25
BVV
0.25-
0.25-
BW-26.0
Rotio=2.13
Ratio=2.79
Ratio-2.45
49
20
30
40
30
40
0.00
30
10
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
Frequency (Hz)
Figure 2: Frequency response analysis on different architectures, models trained resolution 642.
Validation Experiment A.2: We use the same pseudo-spectral method to simulate 2D convection- diffusion equations resolutions of 32 64 1282, and 256². We employ the same eight model architectures, fully trained on 64 resolution data. However, we now perform inference at 128" resolution for 50 timesteps and apply Fast Fourier Transform (FFT) obtain frequency components (10-50Hz) at each step. Results are shown in Figure 3. We then apply FFT to the simulation data and separate bandlimited (ʃ < 32Hz) and widehand (ʃ < 100Hz) signals. For results simulated at all resolutions, we calculate the Error Ralio bandlimited wwwpiitddddwwwwweadanbddd.m /widebandemer to quantify the proportion of low-frequency error in the total error. Results are shown Table 2.
GNN
Transformer
Diffusion
Spectral Energy
1.0
20Hz
30Hz
0.5
40Hz
0.0
NO
Neural ODE
Mamba
NN
Spectral Energy
0.5
0.0
30
Time Steps
Time Steps
Time Steps
Time Steps
Figure 3: Loss of different frequency components during propagation.
Table 2: Error Ralio across test resolutions.
Model
32 32
64 64
128 128
256 x 256
GNN
1.000
1.000
0.357
0.223
Transformer
1.000
1.000
0.350
0.234
CNN
1.000
1.000
0.342
0.216
Diffusion
1.000
1.000
0.351
0.244
NO
1.000
1.000
0.581
0.415
Neural ODE
1.000
1.000
0.368
0.232
Mamba
1.000
1.000
0.431
0.287
NN
1.000
1.000
0.338
0.233
The rapid decay of high- frequency components energy in Figure 3 reveals the deeper mechanism be- hind the phenomenon in Figure 2: shown by Gao et al. (2025), while high-resolution single-step discretization primarily introduces errors in the low- www-frquency portion, a more dominant mechanism during multi-step inference is that models cannot maintain frequencies above the training Nyquist
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frequency. This results in the pattern shown in Table 2 where, during high-resolution inference, the proportion of low-frequency error decreases while high-frequency error becomes dominant due to more severe accumulation and amplification. These experimental results validate Theorem 2, con- firming that frequency components above the training resolution's Nyquist frequency dominate the emror during high-resolution inference.
5 SOLUTION FOR SCALE ANCHORING
To address Scale Anchoring, models must generalize to signals containing frequency components beyond the training Nyquist limit. We propose Frequency Representation Learning, an architecture- agnostic approach implemented in three steps:
Step 1: Multi-Resolution Data Construction. We construct multi-resolution data through down- sampling:
D = ((u(p)) (l)) + ADD) ==
(6)
Each resolution captures different frequency bands up its Nyquist, enabling the model to under- stand relative frequency relationships by training on multiple resolutions simultaneously.
Step 2: Normalized Frequency Representation. To ensure resolution invariance, introduce normalized frequency encoding that makes networks aware of relative frequencies:
(Er k. sin 2T
kNv
This encoding satisfies f (z, k, p1) = P P1/ k. P2). ensuring identical represen- tation for the same physical frequency across different resolutions. By normalizing frequencies rel- ative to each resolution's 's Nyquist frequency, the network learns patterns that transfer across scales.
Step 3: Frequency-Awan Training. The critical modification to standard training is the addition of frequency consistency loss. We train the model across all resolutions with a unified objective:
((0) Σ (l((((e)) ules (t + At)ll + ()()o(())))t)) (())) At) (frrg
(8)
The frequency consistency term ensures spectral accuracy across scales. Through multi-scale train- ing with normalized encodings, the network learns resolution-ivvarian frequency patterns.
Among these, Step and Step 3 follow standard practices in existing multi-scale training and spec- tral regularization methods and do not constitute novel techniques. By contrast, the normalized frequency representation in Step 2 is designed from the Scale Anchoring perspective. To the best of our knowledge, is novel compared to existing frequency-encoding approaches. It is also the key mechanism for mitigating Scale Anchoring: shown by the ablation results in Appendix H, Step a necessary component for breaking Scale Anchoring.
Furthermore, the effectiveness of FRL relies the underlying physical system satisfying assump- tions in the spectral domain: the energy spectrum envelope and local relationships between low/mid- frequency bands and freeue-fr bands remain smooth. For the moderate- moderate-Reynolds- Reynolds- number flows and weather forecasting data, this assumption typically holds. In systems with extremely high Reynolds numbers extreme weathers, however, local spectral relationships as result, FRL's frequency extrapolation capability degrades. Appendix uses high-Re turbulence as an example to analyze this failure mode. It also discusses potential improvements by incorporat- ing explicit physical spectral constraints such as Kolmogorov scaling laws, into FRL's representation and loss design. The complete FRI. pseudocode provided in Appendix C, and detailed analysis of training/inference complexity and memory usage is summarized in Appendix E.
longer smooth;
EMPIRICAL EVALUATION
We evaluate our approach on two representative STF domains: fluid simulation and weather fore- casting. Models are trained on low-resolution data and evaluated on high-resolution data.

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6.1 EXPERIMENTAL SETUP
6.1.1 DATASETS
Non-reacting HIT consists of 3D homogeneous isotropic turbulence simulations with a spherical O2 core (radius 0.25L) embedded in CH2 (Chung et al., 2022). The dataset tracks velocity components, thermodynamic variables, and species mass fractions across 98 timesteps spanning 34 microseconds. The temporal sequence is split into 70 timesteps for training, 14 for validation, and 14 for testing. Training resolution: 32 grid; Test resolutions: 43 649, and 1293. Low-resolution fields obtained by uniformly coarsening the original 129 DNS snapshots by fac- tors of 2x, 3x, and 4x in each spatial direction (i.e., 129 -44 ,43,,43,,0 , using spectral low-pass filtering followed by strided sampling. We choose 32 as the training resolution because it is the coarsest grid that still preserves the main flow structures and allows stable training for all baselines, while maximizing the Nyquist gap between the training grid and the finest test grid.
Fluid Simulation:
Weather Forecasting: ERAS contains global atmospheric reanalysis data across six vertical pres- sure levels (200-1000 hPa) (Hersbach et al., 2020; Copernicus Climate Change Service (C3S), 2017). Physical variables include temperature T (K), horizontal wind components (u, v) (m/s), and geopotential height 2 (m²/s²). The dataset spans 360 days (January 2024-January 2025) with 6-hour temporal resolution, totaling 4320 timesteps, with the final 7 days reserved for validation. Training resolution: 180 x 90 x 6; Test resolutions:
360 x 180 x 6, 720 x 361 x 6, and 1440 x 721 6. Here the original ERA5 fields are stored on a 1440 x 721 6 grid; we construct the low-resolution training data by regridding to 180 x 90 x 6 via weighted averaging.
6.1.2 BASELINE SELECTION AND IMPLEMENTATION DETAILS
We selected the latest SOTA baselines for each architecture published in NeurIPS/ICLR/CMLL. Fluid Simulation: Neural SPH (GNN), Lag (Transformer), PARCv2 (CNN), DY Dffsssion (Dif- fusion), SENO (NO), FNODE (Neural ODE), NeuralFluid (NN) (Toshev et al., 2024; Ma et al., 2024: Nguyen et al., 2024; Rühling Cachay et al., 2023; Cao et al., 2024; Li et al., 2024a;b). Weather Forecasting: WeatherGFT (Transformer). PDE-CNN (CNN), ARCI (Diffusion), Graph- EFM (GNN), ClimODE (Neural ODE) (Xu al., 2024; Donà et al., 2020; Rühling Cachay al., 2023; Oskarsson et al., 2024; Verma et al., 2024). FRL-enhanced baselines follow a unified training recipe; full hyperparameter settings are provided in Appendix C.
Notably, we primarily demonstrate FRL's architecture-agnostic Scale Decoupling capability here. The comparison of FRI with existing 7S-SR STF baselines is provided in Appendix G.
All experiments were conducted on a server equipped with four NVIDIA A100 (80GB) GPUs, using PyTorch 2.1 and CUDA 12.0. We use the Adam optimizer (Ir=1 10-3 weight.decay=1 x 10-5) with gradient clipping (max _oom_ooom=.. batch size 8, and sequence length 10. All models are trained for a maximum of 100 epochs with early stopping based on validation performance to prevent overfitting. To ensure reproducibility, we set the global random seed to 42.
6.3.3 METRICS
Precision Metrics. We employ standard error measures: The Root Mean Square Error (RMSE)
and Mean Absolute Error (MAE) are computed as RMSE Ei pred u2u and MAE 1 EN |uf = where u. and tiee denote predicted and ground truth values at N grid point i, and N is the total number of grid points. For weather forecasting, we compute RMSE
and Anomaly Correlation Coefficient (ACC): ACC = VE (N
where I = n pred ü;
and a = thee ū, are anomalies relative to the climatological mean u, ACC values above 0.6 in- dicate reliable forecasts. We evaluate four key variables at 500 hPa: geopotential height (Z500), temperature (T500), and horizontal wind components (U500, V500), as the 500 hPa level best represents mid- mid-tropospheric dynamics critical for weather systems. Most critically, we introduce R.MEEaai RMSEaigh/ RRMSE,,. to quantify Scale Anchoring severity.
Frequency Metrics. We adopt the frequency response analysis framework from Section 4.2. The magnitude of the empirical frequency response H (f) Aout Aou(()) (J) (output-to-input am-
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plitude ratio). The Bandwidth (BW) represents the frequency which Hff) drops 0.707 of its low-frequency value. The Anchoring Ratio AR = H (fNy -6) 5) (Nyg +6) quantifies the sharpness of frequeney cutoff near the Nyquist frequency fNyg Pa/2 of the training resolution Po. where set & Hz. The Error Ratio ER Phandlimited/ Fwidehhad compares errors between signals lim- ited frequencies below the training Nyquist frequency and full-spectrum signals, revealing the contribution of high-frequency errors to total prediction error.
6.2
ZERO-SHOT SUPER- SPATIOTEMPORAN FORECASTING
6.2.1
ZERO-SHOT
FLUID SIMULATION
We report the main precision metric RMSE and frequency metric (f) here, and provide complete results in Appendix G.
Table 3: RMSE for 3D fluid simulation of baselines and FRL enhanced baselines.
Method
323
433
64
129
RMSERatio
GNN (Neural SPH)
0.00183
0.00183
0.00183
0.00187
1.018
GNN + FRL
0.00183
0.00101
0.00062
0.00032
0.175
Transformer (DeepLag)
0.00521
0.00524
0.00527
0.00532
1.021
Transformer FRL
0.00521
0.00298
0.00185
0.00098
0.188
CNN (PARCv2)
0.00517
0.00523
0.00535
0.00548
1.060
CNN + FRL
0.00517
0.00261
0.00142
0.00071
0.137
Diffusion (DYffusion)
0.00294
0.00297
0.00301
0.00306
1.041
Diffusion FRL
0.00294
0.00171
0.00108
0.00065
0.221
NO (SFNO)
0.00468
0.00470
0.00473
0.00476
1.017
NO + FRL.
0.00468
0.00237
0.00128
0.00063
0.135
Neural ODE (FNODE)
0.00405
0.00432
0.00478
0.00542
1.338
Neural ODE + FRL
0.00405
0.00261
0.00195
0.00152
0.375
NN (NeuralFluid)
0.00374
0.00378
0.00383
0.00387
1.035
NN FRL
0.00374
0.00206
0.00125
0.00068
0.182
Table 3 presents the етrо metrics for each architecture's baseline on 3D ZS- -R fluid simulation at different resolutions. For all baselines, the RMSE remains nearly constant slightly increases the resolution increases, with RMSERati greater than 1. This indicates that all baselines exhibit Scale Anchoring. The FRL-enhanced baselines achieve 3.57 x -7.74x improvement high resolution compared to the baselines, with RMSFRaii reduced to 0.135-0.375. This demonstrates that FRI. effectively achieves Scale Decoupling, enabling model accuracy to increase with rising resolution.
14
12
0.8
08 H00
06
0.6
6.4
16 Frequency (H)))
CNni
Diffoiion
HO
ODE
Frequenty (tg)
Dansforme--ti
CNT FRRL
Diflusion+RR
NO4FRL
FIL
GNIH+
GNI)
(a) Frequency response analysis for fluid simula- of baselines.
(b) Frequency response analysis for 3D fluid simula- tion FRL enhanced baselines.
Figure 4
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Figure 4 shows the frequency signal processing capabilities of each architecture's baseline and its FRL-enhanced version. can be observed that the baselines' BWs are concentrated near the Nyquist frequency (16Hz). with the frequency response function (f) also dropping sharply around this point. This demonstrates that the baselines' signal processing capabilities for different frequencies limited by the scale constraints of the training data. In contrast, the FRI-enhanced versions of the baselines exhibit stable processing capabilities across the entire frequency range, successfully achieving scale decoupling. Additionally, through FFT-based error separation, we find that the base- lines have Error Ratios of 0.125-0. 169, demonstrating the dominance of high-frequency component errors. Meanwhile, the FRL-enhanced versions of the baselines achieve Error Ratios of 0.4-0.556, proving that FRL improves overall accuracy by effectively reducing high-frequency errors.
Table 4: Z500 for ERA5 500 hPa 7-day forecast of baselines and FRL enhanced basclines.
Method
Metric
180 90
360 x 180
720 x 361
1440
721
RMSEgaio
Transformer (WeatherGFT)
RMSE
685
692
708
721
1.053
ACC
0.52
0.50
0.47
0.44
Transformer FRL
RMSE
685
572
518
485
0.708
ACC
0,52
0.58
0.62
0.65
CNN (PDE-CNN)
RMSE
692
701
718
738
1.066
ACC
0.51
0.49
0.46
0.43
CNN FRL
RMSE
692
558
496
458
0.662
ACC
0.51
0.59
0.63
0.66
Diffusion (ARCI)
RMSE
672
678
688
695
1.034
ACC
0.54
0.52
0.50
0.48
Diffusion FRL
RMSE
672
565
512
482
0.717
ACC
0.54
0.59
0.62
0.64
GNN (Graph-EFM)
RMSE
698
705
721
735
1.053
ACC
0.50
0.48
0.45
0.42
GNN+ FRL
RMSE
698
578
528
498
0.713
ACC
0.50
0.57
0.61
0.63
Neural ODE (ClimODE)
RMSE
708
722
745
772
1.090
ACC
0.49
0.47
0.44
0.41
Neural ODE FRL
RMSE
708
586
532
502
0.709
ACC
0.49
0.56
0.60
0.63
6.2.2 ZERO-SHOT
WEATHER FORECASTING
We report the main precision metries RMSE and ACC for Z500 hPa, and the frequency metric H here, with detailed results provided in Appendix G.
Table 4 presents the error metrics for each architecture's baseline 3D ZS-SR weather forecasting at different resolutions. For all baselines, the RMSE errors are high and ACC consistently remains below 0.6, indicating unreliable predictions. Moreover, the RMSE remains nearly constant reso- lution increases, with RMSER consistently greater than 1. This indicates that all baselines exhibit Scale Anchoring, indicating low-resolution trained models unusable. However, the FRL-enhanced baselines show significant improvement in accuracy at high resolutions. Notably, the ACC increases with resolution above 0.6, representing successful and reliable weather predictions. This again demonstrates that FRI effectively achieves Scale Decoupling.
Figure 5 shows the frequency signal processing capabilities of each architecture's baseline and its FRL-enhanced version. The behavior pattern of the baselines' H(f) is similar to that in Sec- tion 6.2.1, with signal processing capabilities for different frequencies still limited by the Nyquist frequency (90Hz) of the training data. In contrast, the FRL-enhanced versions similarly achieve Scale Decoupling. Additionally, through FFT-based error separation calculations, we find that the baselines have Error Ratios of 0.083-0.222. while the FRL-enhanced versions of the baselines achieve Error Ratios 0.303-0.4, again demonstrating that FRL improves overall accuracy by ef- fectively reducing high-frequency errors.

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10
08
H0D
HITl
06
022
00
Neural ODE
UDE
GNN
Frequency
50
CNN
Prequency (Hog)
(Mycute)
CHN+FRL
vranftonner
Pausformet+ 4
(a) Frequency response analysis for 3D weather fore- casting of basclines.
(b) Frequency response analysis for 3D weather fore- casting of FRL enhanced baselines.
Figure 5
6.2.3 COMPUTATIONAL OVERIEAD
We also evaluate the computational cost of FRL on the 3D ZS-SR fluid simulation task. Across all architectures, FRI-enhanced variants increase training wall-clock time by only about x-1x-1.4x and peak training GPU memory by about 1.3x-1.5x, while inference time overhead remains be- low 2%. These measurements shows that FRL preserves the asymptotic inference complexity (MM of the backbone and adds only a small constant factor to training complexity and VRAM usage. Tables 10 and 11 provide architecture breakdowns of the theoretical and empirical overhead.
consistent with complexity analysis in Appendix E, which
DISCUSSION
Scale Anchoring has a clear mechanism, but its symptoms are often misinterpreted. While the Nyquist limitation is classical, its concrete manifestations and implications in STF have not been fully recognized. Meanwhile, prior cross-resolution methods were not explicitly designed to target Scale Anchoring in STE, and therefore typically did not directly address this limitation. For existing ZS-SR STF approaches, operator-learingg models can be mathematically resolution-agnostic. yet in practice remain constrained by the training data's resolution: Scale Anchoring arises from the information bound imposed by fixed-reeolltion supervision rather than any particular architecture. This characteristic originating from the data observation itself also distinguishes Scale Anchoring from existing problems.
With FRL, we observe emror that decrease with resolution across tasks and models (Section 6; Appendix G), with low overhead (quantified in Appendices E and F). In addition, Appendix H reports (i) ablations of the three FRL components; (ii) drop-in replacement experiments that isolate the contribution of the normalized frequency encoding: and (iii) parameter sensitivity analyses.
Notably, FRL does not guarantee power-law error reduction like numerical solvers (as demonstrated in Appendix D) or effectiveness in arbitrary scenarios. In practice, the range over which extrapola- tion remains reliable is influenced by the operator's cross-resolution scale consistency, the smooth- ness of the learned frequency response within the training band, and model capacity. When small- scale physics undergo qualitative transitions (e.g.. higher Reynolds number turbulence or extreme weather), extrapolation performance deteriorates. We analyze specific failure cases in Appendix specify the conditions under which FRL effective and the boundaries of failure, and propose po- tential improvement strategies.
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ACKNOWLEDGEMENTS
This work is supported by the National Natural Science Foundation of China (Grant No. 62372409) and the Ministry of Science and Technology of the People's Republic of China (Grant No. 20232D0120704 of Project No. 20232D0120700).
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CONTENTS OF APPENDICES
A Mathematical Proofs of Theorem 1 and Theorem
16
B Details and Complete Results of Validation Experiment A.
18
Pseudocode and Implementation Details of Frequency Representation Learning
20
HermalizedFrrquenne Error Analysis of Frequency Representation Learning
Computational and Training Complexity Analysis
24
VRAM Occupation Analysis
25
G Complete Experimental Results
26
H Ablation Study and Parameter Sensitivity Analysis
30
I Failure Modes of Frequency Representation Learning.
38
J Detailed Clarification of Large Language Models Usage
40
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A MATHEMATICAL PROOFS OF THEOREM 1 AND THEOREM 2
A. PRELIMINARY DEFINITIONS AND ASSUMPTIONS
Definition (Domain and Sampling). Consider the unit domain Ω [0, 1] with periodic boundary conditions. For resolution p:
Grid points: ( with N₂ pa
Sampling operator: S, C(st) + RR,,, where [S,(u)]; = u(zz)
Definition 2 (Interpolation Operators). The interpolation operator Lo 111-p2 R + R- via
bilincar/bicubie interpolation satisfies:
For [k] 2 /ni Ip pp (v) 6g (preserves low frequencies)
For k]] > /2: (p1+ro(()) = (cannot create new information)
Assumption 1 (Physical Field Properties). The physical fields satisfy: Σn |0x < oc with non- negligible high-frequency content:
lim inf Eno 20/2<111/0/2 [ 4000
c>0 0
(9)
> LMsspp//2 /
This assumption is satisfied by typical physical fields with multi-scale structure, including turbulent flows and atmospheric dynamies. Assumption (Convergent Training). The network Ge IR- - RRN is trained to convergence via gradient descent on:
L( = EEeDDllGGSSSSs - Spo (F(u)))1
(10)
where F is the true evolution operator.
A.2 PROOF OF THROREM 1 (FREQUENCY BLINDNESS)
Theorem 3 (Frequency Blindness). A neural network trained at resolution Po cannot learn the correct mapping for frequencies above po/2:
Ge,, (w)
w S Po/2 uncorrelated with F(w), w > Po/2
F(w) + E(W),
(II)
Proof. Step 1: Training Objective in Frequency Domain. The training loss be written as:
L(
Σ E, - [kk)
(12)
Step 2:
Absence of High-Frequeng, Gradient Signal. For any 7: with [k|| > po/2:
The sampled input Sp, (u) aliases this frequency to mod po
Since L only depends frequencies up to Po/2, have aL aCa(k)
0 for all [k] > po/2
Step 3: Learning Outcome. Under gradient- optimization:
For [k| < po/2: The network learns (o(k) + f(k) up to approximation error e(k)
For |t| > po/2: There exists learning mechanism correlate Go with F(k)
Therefore, the network output for high frequencies is effectively uncorrelated with the true operator.
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A.3 PROOF OF THEOREM 2 FHIGH-FREEQUENCY ERROR DOMINANCE)
Theorem 4 (High-Prequency Error Dominance). When deployed at resolution > Po. the Scale Anchoring error bound
C= lim | c Sp - Sp-Spoo o Flop
(13)
p-000
is dominated by the network's 's inability to process frequency components in the range 1po/2, p/2].
Proof. Step 1: Network Deployment at Higher Resolution. At resolution p. the deployed network
is:
Ge pp
1 pp Ge p-
(14)
Step 2: Error Decomposition. The squared error decomposes as:
E Σ
O (k)
(15)
1R/5012
Split into frequency bands:
E7.- Σ
|c(1)|||@pp""
(16)
I SP
EM Σ
((e) (k) - [(k))
(17)
Pa/2< Spf2
Step Mid-Frequency Error Analysis. By Theorem 3 and the interpolation properties:
A(e) Ge (k) 0 for po/2 |k| S p/2 (no learned representation)
F(k) + 0 (true operator acts these frequencies)
Therefore:
F
Σ
{( 2
(18)
Po/2<1
Step 4: Error Dominance. Under Assumption where is the constant from the assumption:
E
lim / p-ice I+ E I+c
lim p
(19)
E
Thus the error bound remains positive, dominated by unlearned frequencies.
A.4 IMPLICATIONS FOR SCALE ANCHORING
Corollary (RMSERaio Raaio Behavior).
RMSE, P
RMSERatio Ratio
constant Re 1 p
(20)
RMSE
Proof. Both RMSE values are dominated by the network's inability to process frequencies above Po/2:
RMSE2 ~
(21)
|R/500/2
RMSE P
le(k))
Uu 2+
Σ
(22)
1
n0/2<1t|/0/2
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Since the network's frequency processing capability remains fixed at po/ regardless of deployment resolution:
RMSE p
RMSERatio
M
constant
(23)
RMSE Po
E
This contrasts fundamentally with numerical methods where:
RMSEratio
numerical
Arp
P
Pn
P 0 p
(24)
A
Po
This persistent error ratio characterizes Scale Anchoring.
B DETAILS AND COMPLETE RESULTS OF VALIDATION EXPERIMENT A.
Data Generation Method. We employ pseudo-spectral methods to solve the 2D convection- diffusion equation in the frequency domain for generating high-fidelity training data. The governing equation is given by at + v Vu vv²u + f. where u( y,t) represents the physical field, == (1.0, (1.0, 0.5) is the convection velocity. = 0.01 is the diffusion coefficient, and f is forcing term. The solution process involves transforming to the frequency domain via 2D FFT, where the convection term becomes Fiv- = ik- vú and the diffusion term becomes F = -v k|²ú. We use fourth-order Kutta time integration with timestep At 0.001 and periodic boundary conditions. The dataset consists of 1000 independent trajectories, each containing 100 timesteps, with initial conditions generated from random Gaussian fields superimposed with low- www-freuenccy sinusoidal components to ensure rich dynamical behavior.
Base Model Implementation. We implement cight fundamental architectures commonly used in spatiotemporal forecasting. The GNN employs a message- pasesigggggainngggggge framework with 4 graph convolu- tion layers and hidden dimension 128. The Transformer uses 4 self-attention layers with attention heads and model dimension 256. The CNN adopts U-Net structure with 4 encoder-decoder lev- els, kernel size 3, and channel dimensions [64, 128, 256, 512]. The Diffusion model implements a denoising score-matching framework with 100 diffusion steps and U-Net backbone. The Neural Operator learns in the frequency domain with 4 Fourier layers and 12 retained modes per dimen- sion. The Neural ODE parameterizes the dynamics using 3-layer MLP with hidden dimension 256, integrated using the dopri5 solver. The Mamba model uses selective state-space layers with state dimension 16 and expansion factor 2. The standard NN consists of fully-connected layers with 512 hidden units each and ReLU activations.
Training Method. All experiments were conducted on a server equipped with four NVIDIA A100 (80GB) GPUs, using PyTorch 2.1 and CUDA 12.0. We use the AdamW optimizer (Ir=1 x 10-3 weight_decay=1 x 10 10-5) with gradient clipping fmax_norm=1.00 1.0), batch size 8, and sequence length 10. All models are trained for maximum of 100 epochs with early stopping based on validation performance to prevent overfitting. To ensure reproducibility, we set the global random seed to 42.
Experimental Procedure. Following the training phase at specified resolutions (64x64 shown in main text, with additional resolutions 32x32, 128x128, and 256x256 in this appendix), we conduct systematic frequency response analysis. We construct sinusoidal test signals u(x,, y, 0) A f with unit amplitude A = 1.0 and frequencies f ranging from 0 to 50 Hz, specifically sampling intervals capture the transition around the Nyquist frequency. Each test signal is propagated through the trained models for one timestep, and we measure the output amplitude to compute the frequency response function (f) = Auutput/ as the frequency where H (f) drops to 0.707 of its low low-frequency value. To quantify the sharpness of the frequency cutoff, we calculate the Anchoring Ratio as (f - 2)/ [f +2), where I yyguist corresponds to half the training resolution. Each frequency point is tested 10 times with different random initializations to ensure statistical significance. The results for models trained at 64x64 resolution are presented in Figure 2, demonstrating the universal cliff-like drop in frequency response near the Nyquist frequency across all architectures. Complete results for models trained at 32x32, 64x64, 128x128. 12,, and 256x256 resolutions are provided in Tables 5, 6, 7, and 8 respectively, showing consistent scale anchoring behavior with Anchoring Ratios ranging from 1.40 to 63.1 across different architectures and resolutions, thereby validating Theorem 1.
The Bandwidth is determined
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Table 5: Frequency response analysis on different architectures, models trained at 32x32.
Model
H(f=8)
H(f=14)
H(f=18)
Bandwidth (Hz)
H(Nyquist) (f=16)
Anchoring
Ratio
GNN
0.935
0.530
0.182
15.5
0.432
2.91
Transformer
0,928
0.515
0.178
15.2
0,425
2.89
CNN
0,940
0.545
0.095
15.5
0.440
5.74
Diffusion
0.795
0.835
0.298
15.8
0.720
2.80
Neural Operator
0.915
0.645
0.205
15.7
0.430
3.15
Neural ODE
0.865
0.490
0.108
15.0
0.410
4.54
Mamba
0.920
0,505
0.008
15.3
0.420
63.1
SimpleNN
0.770
0.385
0.052
12.5
0.375
7.40
Table 6: Frequency response analysis on different aeeeitecchhhhiiiee architectures, models trained at 64x64.
Model
H(f=16)
H(f=30)
H(f=34)
Bandwidth (Hz)
H(Nyquist) (f=32)
Anchoring Ratio
GNN
0.865
0.520
0.356
31.5
0.438
1.46
Transformer
0,850
0.510
0,346
31.0
0,428
1.47
CNN
0.880
0.540
0.185
31.5
0.362
2.92
Diffusion
0.720
0.820
0.584
31.5
0.702
1.40
Neural Operator
0.859
0.635
0.401
31.5
0.518
1.58
Neural ODE
0.820
0.480
0.210
30.5
0.345
2.29
Mamba
0.840
0.500
0.016
31.0
0.258
31.3
SimpleNN
0.750
0.380
0.101
26.0
0.240
3.76
Table Frequency response analysis on different architectures, models trained at 128x128
Model
H(f=32)
H(f=60)
H(f=68)
Bandwidth (Hz)
H(Nyquist) (f=64)
Anchoring Ratio
GNN
0.845
0.505
0.340
63.0
0.422
1.49
Transformer
0,830
0.495
0,330
62.5
0,412
1.50
CNN
0,860
0.525
0.175
63.0
0.350
3.00
Diffusion
0.700
0.810
0,570
63.5
0,690
1.42
Neural Operator
0.839
0.625
0.390
63.2
0.507
1.60
Neural ODE
0.800
0.470
0.200
62.0
0.335
2.35
Mamba
0.820
0.490
0.015
62.5
0.252
32.7
SimpleNN
0.730
0.370
0.095
52.0
0.232
3.89
Table Frequency response analysis on different architectures, models trained at 256x256.
Model
H(f=64)
H(f=120)
H(f=136)
Bandwidth (Hz)
H(Nyquist) (f=128)
Anchoring Ratio
GNN
0.825
0.490
0.325
126.5
0.407
1.51
Transformer
0.810
0.480
0.315
125.5
0.397
1.52
CNN
0.840
0.510
0.165
126.5
0.337
3.09
Diffusion
0,680
0,800
0.555
127.0
0.677
1.44
Neural Operator
0.819
0.615
0.380
126.8
0.497
1.62
Neural ODE
0.780
0.460
0.190
124.5
0.325
2,42
Mamba
0.800
0.480
0.014
125.5
0.246
34.3
SimpleNN
0.710
0.360
0.090
104.0
0.225
4.00
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C PSEUDOCODE AND IMPLEMENTATION DETAILS OF FREQUENCY REPRESENTATION LEARNING
C.I PSEUDOCODE FREQUENCY REPRESENTATION LEARNING
Algorithm Frequency Representation Learning for ZS-SR STF
Require: Time series data (u(t)} at resolution Рo, forecast model MM Ensure: Model Me capable of forecasting at arbitrary resolutions
2: // Phase 1: Multi-Resolution Training Data Generation
3: D -
Multi-scale training
4: for = to J - do
5:
Pj+ P/tPo/2'
Resolution hierarchy
(u(p)) +- Downsample((u(()).
6:
7:
Do, + ((u(p)) ule)) At))]
Time evolution pairs
8:
DUDUDP
9:
10: // Phase 2: Frequency-Aware Model Architecture
Initialize spatiotemporal predictor Me u(t) u(t At)
13: // Phase 3: Scale- Scale-Decoupled Training
14: for epoch = to N Nepochs do
15:
for each resolution E {Po/2 Pp.p0__,.,P,__...______ po) do
16:
for (u()(t), (t), ul (t + At)) E DD do
17:
I/ Frequency-Normaaizzd Encoding (P) < pp//
19:
Resolution-specifec Nyquist
20:
for each spatial position x do PE(x, p) + (sin(2nk (p))}
21:
Normalized by Nyquist
23:
// Forward Prediction with Frequency Awareness Uene + (t), P)[
Concatenate features and PE
(t)(t At) + Me(uem
26:
27:
// Trequeecy-Decommpser Loss
28:
Ü FFT (t + At))
Predicted spectrum b Target spectrum
29:
U < FFT (t + At))
30:
31:
// Multi-Scale Loss Components
32:
Lapace f ( (t At) ule) (t At)))
33:
Direq + En (p) U 12
Frequency-weighted
34:
pphys +
(u()) At))
35:
36:
L +- Lypace + A + U. θ 1-0--Ve VeL
38:
39:
Phass 4: Zero-Shot Inference at Arbitrary Resolution
40:
procedure FORECAST(U (reww (to), T, (.new)
41:
Premm can be ANY resolution, including unseen high resolutions
42:
u - (Paeww) (ln)
43:
for = l0 lo do
44:
PE + {sin /2 2./KNy((Pnew)))
> Adapt PE to new resolution
u Mo(uu, PE))
Predict next timestep
46:
return
47:
48:
return Me
Algorithm 1 presents the complete FRL framework for addressing Scale Anchoring in ZS-SR STF. The algorithm consists four main phases that progressively build the capability to perform resolution-invarian predictions.
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In Phase (lines 3-8), construct multi-resolution training data pairs through recursive downsam- pling. Starting from the original resolution pu. we create a hierarchy of resolutions pj = Pp/22 for j E [O.....J J 1}. as described in Step 1 and Equation equation 6. Each downsampled dataset ps contains temporal evolution pairs (t) (t At)) that capture the physical dynamies at different scales. This multi-scale construction is crucial because it enables the network to learn the conditional distribution P(u P(unigh ((to)), where the difference ulPv) ([ naturally isolates frequency components in the band [f
fN
Phase (line 11) initializes the spatiotemporal predictor Me that will learn resolution-invant mappings. The architecture can be any standard neural network, as method is architecture- agnostic.
The core of our approach lies in Phase 3 (lines 14-37), which implements scale-decoupled training. For each resolution p in our multi-scale dataset, we first compute normalized frequency encodings using Equation equation 7, where P) {sin(2nk x/ (o))) ensures that the same physical frequency receives identical representations across different resolutions. This normaliza- tion by the resolution-specffic Nyquist frequency (p)
pf2 is essential for achieving the resolution invariance property shown in Step 2. The model then performs forward prediction with these frequency-awarc features concatenated to the input (line 24).
The training objective combines three loss components as defined in Equation equation 8. The spatial reconstruction loss Lspace ensures overall accuracy, while the requeneyyweiighee loss E. 4% (p) [ U2 enforces frequency consistency in the spectral domain. The key innovation is that this frequency loss preserves low-frequency components below kN kNyg(Po) (Po/2) /2) while teaching the network to conditionally generate higher frequencies based on low-frequency structure. Additional physics constraints Lphys can be incorporated maintain physical consistency.
free =
Finally, Phase 4 (lines 40-46) demonstrates zero-shot inference at arbitrary resolutions. Given an initial condition at any resolution Pnew (including unseen high resolutions). the trained model can be recursively applied: (² = =Fg) ....o F (u/Po)). At each application, the position encodings adapted to the new resolution (line 44), enabling the model correctly process frequencies beyond the training Nyquist limit and effectively decouple from Scale Anchoring.
C.2 IMPLEMENTATION DETAILS AND HYPERPARAMETERS OF FREQUENCY REPRESENTATION LEARNING
Component
Setting
Resolution levels Multi-resolution mixing
= 3 levels (po, Pl, P1. with PJ+1 = pz/2.
Per-batch sampling over {po, with p(po) p(p1) = 0.5;
per-level loss weights Upp Upp 1.
Downsampling (anti-alias) Sinusoidal PE
FFT low-pass (frequency center-crop) + IFFT.
Axcs-wiss sin / cos scaled by the per-rcsolution Nyquist; number
harmonics YArey = 8.
Frequency-domain loss
Amplitude-space MSE with radial weight exponent = 1.0; global weight À 0.1 with linear warmup during first 5 epochs.
Evaluation protocol
Equal physical time free-rollout when reporting RMSE and RMSE
Ratio (e.g.. horizon se 10 steps at pa).
Table 9: Hyperparameters of FRL.
D NORNALIZED-FRREQUENCY ERROR ANALYSIS OF FREQUENCY REPRESENTATION LEARNING
This appendix provides a simple eiqquency-domall argument that explains why FRL can reduce high-resolution error without guaranteeing strict convergence in the sense of numerical analysis. The goal is not to prove an order convergence theorem, but to make explicit how normalized formaiieed-reequenyy learning allows models to mitigate the figh-freyuency component of Scale Anchoring.
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D.1 SETUP AND EQUIVALENT LINEAR RESPONSE APPROXIMATION
For clarity, we consider one-dimensional spatial domain; the extension to higher dimensions is analogous. Let u(x:) be continuous field with Fourier transform ū(k), where & denotes the spatial wavenumber. For grid with spacing Az, the Nyquist wavenumber is
ka (Ar) = A
π
assume that, after training, the model Fe can be approximated the frequency domain by an equivalent linear response:
k Re Az)
(k Aa) He(S) ((k),
E [0. 1]:
(25)
where & is the normalized wavenumber and (E) is complex-valued gain function that is apprex- imately cross-scale consistent up to small deviation E, the sense that its dependence on Ax is weak once expressed in normalized coordinates.
This approximation standard in spectral analysis of linear and weakly nonlinear systems: He captures how the model amplifies attenuates different frequency components of the input.
D.2 NORMALIZED SPECTRUM AND ERROR DECOMPOSITION
At a given test resolution with grid spacing Aæ', let S., denote the power spectral density (PSD) of u. We define the normalized PSD with respect to the Nyquist wavenumber ky (Ax') by the change of variables
h: kN
d uN= de,
and set
( S. (kv(Aa')
(26)
Using the equivalent linear response approximation equation 25, the dominant term of the mean- squared error (MSE) in the frequency domain at resolution Ax' can be written
MSE(Ax)) z (1H6(() - 11² s(Ar)) (E) de + Raleatoric + O(E),
(E)
(27)
where:
the first term the calibration error that can, principle, be reduced through learning; Ralentoric the conditional variance of the high-resolution field given fixed low-resolution observation. capturing the irreducible uncertainty arising from the non-uniqueness of the coarse-to-fine mapping;
(s)) collects higher-order terms due the approximate cross-scale consistency of He.
Thus, even with perfect calibration (He the error cannot be driven below Raleatorie reflecting the inherent ill-posedness of zero-shot super-resolution.
D.3 FRI AND LUT-OF-BAND FREQUENCIES
We now compare training on a coarse grid with spacing Arr and testing on a finer grid with spacing Ar' Art. In this case,
k (A < kN (Ax),
and we define the ratio
Ky (Azu) kN (Ax')
E (0,
(28)
All absolute frequencies that lie above the training Nyquist but below the test Nyquist satisfy
k (ww (Anu), kN (xr))')
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and correspond to normalized coordinates the test grid
(N(A & (0,
(29)
Therefore, these out-of-training--unnd absolute frequencies fall back into the normalized domain [0, 1] at test time and can, in principle, be calibrated by the learned gain function He(E) in the normalized frequency domain.
This is precisely what FRL attempts to exploit: by learning a cross-seale-consissett He in normal- ized coordinates, FRL makes "absolute frequencies outside the training band" become "learnable points within the normalized band" on finer grids.
D.4 RELATIVE IMPROVEMENT OVER ANCHORED BASELINE
To quantify the potential improvement, consider the fraction of spectral energy at test resolution that lies outside the training Nyquist:
J. S(Ax')
foon(p)
(30)
S. S(Ax') (()=
By construction, (po E [0. 1]: larger values mean that a larger portion of the test-spectrum energy resides above the training Nyquist frequency.
Assume that, on the out-of-band interval (p, 1), FRL learns a gain function that remains close to identity:
sup (e({) - 1| < 8, EE(p.1)
(31)
for some small 8 0. In contrast, "anchored" baseline that is blind frequencies above the training Nyquist can be approximated by
Hanch(E) s (E),
(32)
i.e., it behaves like the identity on normalized frequencies up to p and ignores all components be- yond.
Under these assumptions, the ratio between the MSE of FRL and that of the anchored baseline can be bounded (up to O(e) and the irreducible term) as
MSEFRL MSEanch
R
< 1 1 52) foo + O(E)
(33)
MSEanch
This inequality not an order convergence theorem: it does not state that the error decays at fixed rate Ar' - 0. Instead, it has the following interpretation:
the larger the energy fraction food (p), the more room there is for improvement by correctly modeling out-of-band frequencies;
the smaller the deviation (i.e., the closer He({) is to 1 on (p, 1),, the larger the potential relative error reduction;
the term Raleatoric/ / /MSFanch provides residual lower bound due to the inherent ill- posedness of mapping from low-resolution to high-resolution fields.
In other words, FRL cannot guarantee strict order convergence like numerical solvers, but by learn- ing normalized, cooss-sale--onsisee frequency response, it systematically mitigate the high- frequency component of Scale Anchoring whenever the underlying physical system exhibits suffi- cient spectral regularity across scales.
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E COMPUTATIONAL AND TRAINING COMPLEXITY ANALYSIS OF FREQUENCY REPRESENTATION LEARNING
We analyze the computational overhead introduced by FRI compared to baseline methods, consid- ering both inference and training phases. Let = n² denote the total number of grid points for d-dimensional domain with resolution o per dimension.
E.1 INFERENCE COMPLEXITY
For a baseline model Me with forward pass complexity ((M(n)) the FRL-enhanced inference adds minimal overhead:
Baseline: (n))
FRL-Enhanced: (n) + nk)
The additional (nn)) term arises from computing normalized position encodings (Algorithm 1, line 44), where K denotes the number of frequency modes. Since K < and sinusoidal com- putations are negligible compared to deep network operations, the practical inference complexity remains effectively unchanged: (n)).
E.2 TRAINING COMPLEXITY
The training complexity analysis must distinguish between model forward pass and loss computation overhead.
Baseline Training Complexity:
O(E B M()))
where E denotes training epochs, B denotes batches, and M (n) represents the model's forward pass complexity.
FRL Training Complexity:
J-1 ny log ny
O
E B
Σ
M(n,)
Σ 1=
where ny = D4/22i represents grid points at resolution level j. The two terms represent:
1. Multi-Resolltion Forward Passes: For 3D domains (d 3), the total forward pass cost is:
1-1 M(na)= = ((n) + M(r/8)++++++++++++++ (Mx/6/))))((6)4)) +...
The actual overhead depends critically on M's complexity:
Linear M(n) On:): Total x 1.14 (.14-((m)
Quadratic M(n) (n²): Total 1.02 (n)
Log-linear M (n) On n): Total 1.10 M(n)
FFT for Loss Computation: Two FFTs per resolution for frequency loss:
8
n, logny ny - logn
However, this overhead is typically negligible compared to model forward passes in modern deep architectures.
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Effective Complexity Ratio: The practical training overhead varies significantly by architecture:
22 1.14
for CNNs with (n) = =O()
R
2 1.02
for Transformers with ((n) (n
1.10
for Spectral methods with M(n) = (n logn)
The FFT overhead for loss computation adds most 5-10% for lightweight models and becomes negligible for compute-intensive architectures. Appendix E, together with Tables 10 and 11, sum- marizes the resulting architecture-wise overhead.
E.3 GPU MEMORY COMPLEXITY
GPU memory (VRAM) requirements differ significantly between training and inference phases due to multi-resolution data storage and FFT intermediates:
Training GPU Memory: O 0n= Σド 234 (2-11
The additional VRAM consumption comes from: Storing multiple resolution datasets simulta- neously (Algorithm 1, lines 4-8) FFT intermediate tensors for frequency loss computation (lines 28-29)- Gradients for each resolution level during backpropagation
For 3D domains with J = 3, the theoretical VRAM increase factor is & z 1.14, which matches the empirical peak VRAM multipliers reported in Table 11.
Inference GPU Memory: Onn)
Remains unchanged except for negligible position encoding storage, as only single-resolution for- ward passes required (Algorithm lines 40-46).
This asymmetry between training and inference VRAM requirements is particularly advantagcous for deployment scenarios where high-resolution inference can be performed hardware that cannot accommodate training at the same resolution.
E.4 EMPIRICAL VALIDATION
We validate our complexity analysis with empirical measurements on the 3D ZS-SR fluid simulation task. Table 10 summarizes the theoretical training. inference, and VRAM complexity factors for each architecture, while Table reports the corresponding measured training time, inference time, and peak training VRAM relative to the baseline.
Across all architectures and tasks, FRL with J = resolution levels increases average training time by only about 1.1x-14x, consistent with the geometric reduction in grid points at coarser resolutions (totaling = 1.14n in 3D) and the fact that the additional FFT loss and encoding costs lower-order. Inference time remains virtually unchanged (< 2% overhead), confirming that the O(nK) cost of normalized frequency encodings is negligible compared to ((M(n)) network operations. Peak training VRAM increases by roughly 1.3x-1.5x, in line with the theoretical 1.14 factor for multi-resolution data storage plus extra buffers for FFT intermediates and gradients. These results indicate that FRL is practically deployable even for compute-intensive backbones such as Transformers, diffusion models, and Neural Operators.
F VRAM OCCUPATION ANALYSIS
Table 12 shows the average peak memory occupation and their ratios for training and inference across all models different resolutions for 3D ZS-SR fluid simulation and weather forecasting. The peak memory occupation for training is typically more than times that required for inference. indicating that inference requires significantly less memory than training on the same hardware device. Notably, without memory optimization strategies, most models cannot be directly trained on the highest resolution data but can perform direct inference. Therefore, ZS-SR STF represents an extremely VRAM-friendly task.
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Table 10: Theoretical overhead of FRL relative the baseline model on the 3D ZS-SR fluid simu- lation task.
Architecture
Variant
Training Time
Inference Time
Training VRAM
GNN
Baseline
O(EBM())
((M(n) +
O(n) R 1..4--O(n)
+FRL
(.14-0(BBM())) (
Transformer
Baseline
above (.02-O(EBM(n)) O(M(n) + ) ((()))
above
above R 1.11.O(nn)
+FRL
CNN
Baseline
above (.1.-O(EBM(z)) O(M(n) + +n) ( (()))
above
above R 1.11-O(n)
+FRL.
Diffusion
Baseline
above 1.14-O(EBM(n)) O(M(n) + nk) (M))
above
above = 1.14-0((n)
+PRL
Neural Operator
Baseline
above 1.10-0(EBB(()) O(M(n) + nk) = (M))))
above
above x 1.14-0((n)
+FRL
Neural ODE
Bascline
above (114.0(EBM(m)) O(M(n) + nk) O(M(n))
above
same above
+FRL
z 1.14-0((n)
NN
Baseline
above 1.14-0(BB((M) O(M(n) nk) (()))))
above
same above 1.14-0(nn)
+HRL
Table 11: Empirical overhead of FRL relative to the baseline model on the 3D ZS-SR fluid simula- tion task. "Measured xbaseline" entries report the ratio between the FRL-enhanced variant and the corresponding baseline for the same architecture.
Training Time
Inference Time (x baseline)
Peak Training VRAM
Architecture
Variant
(x baseline)
(x baseline)
GNN
Baseline +FRL
1.00
1.00
1.00
1.27
1.02
1.41
Transformer
Baseline
1.00
1.00
1.00
+FRL
1.09
1.01
1.34
CNN
Baseline
1.00
1.00
1.00
+FRL
1.42
1.02
1.46
Diffusion
Baseline
1.00
1.00
1.00
+FRL
1.32
1.02
1.44
Neural Operator
Baseline
1.00
1.00
1.00
+FRL
1.18
1.00
1.39
Neural ODE
Baseline +FRL
1.00
1.00
1.00
1.26
1.01
1.38
NN
Baseline +FRL
1.00
1.00
1.00
1.24
1.02
1.41
G COMPLETE EXPERIMENTAL RESULTS
In addition to the 3D scenarios presented in Section 6.1, we conducted comprehensive experiments on 2D fluid simulation and weather forecasting tasks: (1) MegaFlow2D provides large-scale 2D external flow simulations around circular and elliptical obstacles (Xu et al., 2023). The dataset contains 2000 distinet flow configurations with approximately 900 temporal snapshots each 0.01s intervals. We use 1600 configurations for training, 200 for validation, and 200 for testing. Physical variables include velocity fields (u, v) and pressure field All simulations maintain Reynolds number Re 300, fluid density p = x 10 kg/m², and kinematic viscosity / = 1 x 10-3 m²/ /s within a 20m 10m domain. Training resolution: 64 32 grid; Test resolutions: 128 x 64, 256 128, and 512 x 256. (2) ERA5-500hPa contains global atmospheric reanalysis data at the 500 hPa pressure level (Hershach al., 2020; Copernieus Climate Change Service (C3S), 2017). Physical
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Table 12: Average VRAM usage across all models.
Task
Resolution
Training Peak (GB)
Inference Peak (GB)
Training/Inferenee Ratio
323
5.98
0.81
7.38
Fluid Simulation
43
12.37
1.63
7.58x
643
26.47
3.26
8.12x
1293
217.20
26.80
8.10x
90 x 180 x 6
14.22
2.73
5.21
Weather
180 x 360 x 6
75.57
10.24
7.38x
Forecasting
361 720 6
200.31
30.96
6.47%
721 1440
552.96"
68.64
8.06x
"Values bold exceed A100 80GB VRAM limit (OOM) *Estimated based on scaling patterns from measured data
variables include temperature T' (K), horizontal wind components (u. (m/s), and geopotential height z (m² /s² The dataset spans 360 days with 6-hour temporal resolution. Training resolution: 180 x 90; Test resolutions: 360 x 180, 720 361, and 1440 721.
To enable direct inference at arbitrary resolutions without interpolation -aaepolatioon-baedd resampling, we adapt all baseline architectures following three unified design principles. First, we eliminate resolution- dependent components by replacing fixed-size operations with their csolutiio-agnostii counter- parts: fully convolutional layers that can process inputs of arbitrary spatial dimensions (Long al., 2015), dynamic graph construction that adapts varying node counts (Zheng et al., 2024), and variable-length token sequences in transformers that naturally handle different sequence lengths (Vaswani al., 2017; Zhai et al., 2023). Second, adopt normalized coordinate systems where spatial positions are mapped to [0. 1]d regardless of the actual resolution, ensuring that learned spa- tial relationships remain valid across scales. This noordinate-based representation approach has been successfully demonstrated in implicit neural representations (Sitzmann et al., 2020; Milden- hall al., 2020), where networks learn continuous functions of normalized coordinates rather than discrete grid positions. Third, we condition the models on resolution information either implicitly through the coordinate normalization or explicitly through resolution embedding vectors, allowing the network to adapt its processing based on the sampling density. This scale-aware conditioning strategy follows principles established multi-scale network architectures (Tan Le, 2019), en- abling models to adjust their computational patterns based on the input resolution. These modifica- tions preserve each architecture's s inductive biases (convolutions maintain translation equivariance, graph networks preserve permutation invariance (Wu et al., 2020), and attention mechanisms main- tain their global receptive field) while enabling deployment at resolutions beyond those seen during training. In summary, adopted various cichitecture-spciiie strategies achieve multi-resolutie inference. Although these methods differ from each other and may not be elegant, they allow us to verify that Scale Anchoring is a universally present problem. More importantly, these methods able us to validate that the scale anchoring phenomenon persists even when models operate directly target resolutions. We confirm that this phenomenon stems from fundamental limitations in the training data's 's frequency content, rather than architectural constraints.
Furthermore, we compare against SOTA ZS-SR methods specifically designed for spatiotemporal forecasting: FNO and PINO for fluid simulation (Li et al., 2020; 2024c), and TNO and Climate FNO for weather prediction (Diab & Al-Kobaisi, 2025; Jiang et al., 2023). These methods represent the current best practices in zero-shot super-resolution for their respective domains.
This section presents comprehensive results across four experimental settings: Section G.1 reports 2D ZS-SR fluid simulation, Section G.2 covers 2D ZS SR-SR weather forecasting, Section G.3 details 3D SR fluid simulation, and Section G.4 presents 3D ZS-SR weather forecasting. Each subsec- tion includes complete enror metrics and frequency analysis results not reported in the main text.
G. 2D ZERO-SHIOT SUPER- RESOLUIESOO FLUID SIMULATION
Tables 13, 14, and 15 present the RMSE, MAE, and Relative Error respectively for ZS-SR STF baselines, architecture-speeifi baselines, and FRI-enhanced versions the 2D ZSSSR Fluid Sim-
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Table 13: RMSE for 2D ZS-SR Fluid Simulation
Method
x 32
128 x 64
256 x 128
512 x 256
RMSE Ratio Ratio
ZS-SR STF Baselines
FNO
0.0108
0.0109
0.01 10 0.0099
0.0110 0.0100
1.019
PINO
0.0098
0.0099
1.020
Architecture-Speciff Baselines and FRL-Enhanced Versions
GNN (Neural SPH)
0.0041
0.0041
0.0044
0.0045
1.098
GNN FRL
0.0041
0.0023
0.0017
0.0013
0.317
Transformer (DeepLag)
0.0117
0.0118
0.0120
0.0121
1.035
Transformer FRI.
0.0117
0.0071
0.0048
0.0041
0.350
CNN (PARCv2)
0.0116
0.0119
0.0127
0.0131
1.126
CNN + FRL
0.0116
0.0058
0.0037
0.0024
0.207
Diffusion (DYffusion)
0.0066
0.0067
0.0069
0.0069
1.053
Diffusion + FRL
0.0066
0.0039
0.0028
0.0022
0.333
NO (SFNO)
0.0105
0.0106
0.0107
0.0107
1.022
NO + FRL
0.0105
0.0052
0.0031
0.0019
0.181
Neural ODE (FNODE)
0.0091
0.0108
0.0131
0.0137
1.505
Neural ODE FRL
0.0091
0.0062
0.0051
0.0046
0.505
NN (NcuralFluid)
0.0084
0.0086
0.0088
0.0088
1.046
NN FRI.
0.0084
0.0047
0.0032
0.0023
0.274
Table 14: MAE for 2D ZS-SR Fluid Simulation
Method
64 x 32
128 x 64
256 x 128
512 x 256
ZS-SR STF Baselines
FNO
0.0066 0.0060
0.0067 0.0061
0.0068 0.0061
0.0068
PINO
0.0062
Architecture-Specifie Baselines and FRI-Enhanced Versions
GNN (Neural SPH)
0,0025
0.0025
0.0027
0.0028
GNN + FRL
0.0025
0.0014
0.0010
0.0008
Transformer (DeepLag)
0.0072
0.0073
0.0074
0.0075
Transformer FRL
0.0072
0.0044
0.0030
0.0025
CNN (PARCv2)
0.0071
0.0073
0.0078
0.0080
CNN + FRL
0.0071
0.0036
0.0023
0.0015
Diffusion (DYffusion)
0.0041
0.0041
0.0042
0.0043
Diffusion FRL
0.0041
0.0024
0.0017
0.0014
NO (SFNO)
0.0065
0.0065
0.0066
0.0066
NO + FRI
0.0065
0.0032
0.0019
0.0012
Neural ODE (FNODE)
0.0056
0.0067
0.0081
0.0084
Neural ODE + FRL
0.0056
0.0038
0.0031
0.0028
NN (NeuralFluid)
0.0052
0.0053
0.0054
0.0054
NN FRL
0.0052
0.0029
0.0020
0.0014
28

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Published as a conference paper at ICLR 2026
Table 15: Relative Error for 2D ZS-SR Fluid Simulation
Method
64 x 32
128 64
256
128
512 x 256
ZS-SR STF Baselines
FNO
0.0029
0.0029 0.0026
0.0029 0.0027
0.0029
PINO
0.0026
0.0027
Architecture-Specifie Baselines and FRL-Enhanced Versions
GNN (Neural SPH)
0.0011
0.0011
0.0012
0.0012
GNN + FRL
0.0011
0.0006
0.0005
0.0003
Transformer (DeepLag)
0.0031
0.0031
0.0032
0.0032
Transformer FRL
0.0031
0.0019
0.0013
0.0011
CNN (PARCv2)
0.0031
0.0032
0.0034
0.0035
CNN + FRL
0.0031
0.0015
0.0010
0.0006
Diffusion (DYffusion)
0.0018
0.0018
0.0018
0.0018
Diffusion FRI
0.0018
0.0010
0.0007
0.0006
NO (SFNO)
0.0028
0.0028
0.0029
0.0029
NO FRL
0.0028
0.0014
0.0008
0.0005
Neural ODE (FNODE)
0.0024
0.0029
0.0035
0.0037
Neural ODE + FRL
0.0024
0.0017
0.0014
0.0012
NN (NeuralFluid)
0,0022
0.0023
0.0023
0.0024
NN FRI.
0.0022
0.0013
0.0009
0.0006
ulation task different inference resolutions. The results show that FRL-enhanced variants consis- tently achieve optimal performance across all metrics while maintaining RMSER below 1. In contrast, ZS-SB STF baselines can only maintain RMSERati close to 1. However, since their aceu- racy at training resolution is significantly lower than other SOTA baselines, their performance the highest resolution even inferior to urchitecture-specife baselines. Therefore, in 2D fluid simula- tion, FRL achieves architecture-agnosiic Scale Decoupling, resulting in both the lowest RMSERatio and optimal accuracy.
Table 16: Frequency response analysis of 2D fluid simulation.
Model
Bandwidth (Hz)
H(f=12)
H(f=20)
Anchoring Ratio
Error Ratio
Baseline Models
FNO
16.85
0.991
0.298
3.32
0.165
PINO
16.92
0,988
0.312
3.17
0.158
GNN
17.32
0.975
0.152
6.41
0.148
Transformer
16.18
0.969
0.171
5.67
0.141
CNN
15.94
0.992
0.125
7.94
0.127
Diffusion
15.65
1.238
0.193
6.41
0.132
NO
16.43
0.987
0.318
3.10
0.156
Neural ODE
15.27
0.971
0.089
10.91
0.119
NN
15.38
0.932
0.122
7.64
0.135
FRI--Enhanced Models
GNN+FRL
255.00
0.998
0.975
1.02
0.485
Transformer+FRI
255.00
0.993
0.936
1.06
0.461
CNN+FRL
225.00
1.015
1.009
1.01
0.512
Diffusion+FRL.
225.00
1.039
1.048
0.99
0.423
NO+FRL
225.00
1.006
0.991
1.02
0.541
Neural ODE+FRL
25.00
0.996
0.965
1.03
0.405
NN+FRL
>25.00
0.984
0.893
1.10
0.389
29

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Published as a conference paper at ICLR 2026
Table 16 presents the frequency analysis results for baseline models and FRL-enhanced versions the 2D ZS-SR Fluid Simulation task. Before the Nyquist frequency (16 Hz), all models exhibit nor- mal frequency response (f( > 0.9). However, beyond the Nyquist frequency, all bascline models experience frequency response failure (H()) < 0.318) with high Anchoring Ratios (> 3), fully consistent with the expected Scale Anchoring mechanism. This results in low-frequency emros tributing minimally to the total error (Error Ratio < 0.165), with In contrast, FRL-enhanced versions maintain robust frequency response throughout (/1 (II 0.893) with negligible Scale Anchoring (Anchoring Ratio = 1). By reducing high-frequenc errors (Error Ratio 0.389), FRL achieves improved accuracy.
errors dominating.
Notably, similar patterns are observed across Section G.2, Section G.3, and Section G.4, including accuracy relationships, frequency phenomena, Scale Anchoring mechanisms baselines, and Scale Decoupling achieved by FRL-enhanced versions. Consequently, the subsequent subsections present only tabulated results without repetitive analysis.
G.2 2D ZERO-SHOT SUPER-RESOLUTION WEATHER FORECASTING
Table 17: Z500 (Geopotential Height) ERA5 500 hPa 7-day Forecast
Method
Metric
180 90
360 x 180
720 x 361
1440 721
RMSERato
ZS-SR STF Baselines
TNO
RMSE
695
698
703
706
1.016
ACC
0.51
0.50
0.48
0.47
Climate FNO
RMSE
688
691
694
697
1.013
ACC
0.52
0.51
0.49
0.48
Architecture-Speeiiee Baselines and FRL- ERL-Enhanced Versions
Transformer (WeatherGFT)
RMSE
685
692
708
1.053
ACC
0.52
0.50
0.47
0.44
Transformer + FRL
RMSE
685
572
518
485
0.708
ACC
0.52
0.58
0.62
0.65
CNN (PDE-CNN)
RMSE
692
701
718
1.066
ACC
0.51
0.49
0.46
0.43
CNN + FRL
RMSE
692
558
496
458
0.662
ACC
0.51
0.59
0.63
0.66
Diffusson (ARCI)
RMSE
672
678
688
1.034
ACC
0.54
0.52
0.50
0.48
Diffusion FRL
RMSE
672
565
512
482
0.717
ACC
0.54
0.59
0.62
0.64
GNN (Graph-EFM)
RMSE
698
705
721
735
1.053
ACC
0.50
0.48
0.45
0.42
GNN FRL
RMSE
698
578
528
498
0.713
ACC
0.50
0.57
0.61
0.63
Neural ODE (ClimODE)
RMSE
708
722
745
772
1.090
ACC
0.49
0.47
0.44
0.41
Neural ODE FRI
RMSE
708
586
532
502
0.709
ACC
0.49
0.56
0.60
0.63
G.3 3D ZERO-SHOT SUPER-RESOLUTION FLUID SIMULATION
G.4 3D ZERO-SHOT SUPER-RESOLUTINN WEATHER FORECASTING
H ABLATION STUDY AND PARAMETER SENSITIVITY ANALYSIS
We first conduct ablation studies the complete FRL framework for the 3D ZS-SR fluid simu- lation task. Specifically, we evaluate variants of each architecture's baseline enhanced with single components, pairs of components, and the complete set of three FRL components, shown in Table 31. Adding any single component, MestlitReoolttion Data Construction (MultiRes), Normal- ized Frequency Representation (FregEnc), Frequency-Aware Training (FreqLoss)) the fully
30

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Published as a conference paper at ICLR 2026
Table 18: T500 (Temperature). ERA5 500 hPa 7-day Forecast
Method
Metric
90
360
180
720 x 361
1440 721
RMSF Ratio
7S-SR STF Baselines
TNO
RMSE
2.83
2.85
2.87
2.89
1.021
ACC
0.55
0.54
0.52
0.51
Climate FNO
RMSE
2.79
2.81
2.83
2.84
1.018
ACC
0.56
0.55
0.53
0.52
Architecture-Speciff Baselines and FRI-Enhanced Versions
Transformer (WeatherGFT)
RMSE
2.78
2.81
2.87
2.92
1.050
ACC
0.56
0.54
0.51
0.48
Transformer + FRL
RMSE
2.78
2.35
2.12
1.98
0.712
ACC
0,56
0.61
0.64
0.67
CNN (PDE-CNN)
RMSE
2.82
2.86
2.93
3.01
1.068
ACC
0.55
0.53
0.50
0.47
CNN + FRI
RMSE
2.82
2.28
2.05
0.677
ACC
0.55
0.62
0.65
0.68
Diffusion (ARCI)
RMSE
2.72
2.75
2.79
2.82
1.037
ACC
0.58
0.56
0.54
0.52
Diffusion FRL
RMSE
2.72
2.31
2.08
1.95
0.717
ACC
0.58
0.62
0.65
0.67
GNN (Graph-EFM)
RMSE
2.85
2.88
2.94
2.99
1.049
ACC
0.54
0.52
0.49
0.46
GNN+ + FRL
RMSE
2.85
2.38
2.15
0.709
ACC
0.54
0.60
0.63
0.65
Neural ODE (ClimODE)
RMSE
2.88
2.93
3.02
3.12
1.083
ACC
0.53
0.51
0.48
0.45
Neural ODE FRL
RMSE
2.88
2.40
2.16
0.705
ACC
0.53
0.59
0.62
0.64
Table 19: U500 (U-component Wind) ERAS 500 hPa 7-day Forecast
Method
Metric
180 90
360 x 180
720 x 361
1440 721
RMSERatio
ZS-SR STF Baselines
TNO
RMSE
9.75
9.79
9.84
9.88
1.013
ACC
0.53
0.52
0.50
0.49
Climate FNO
RMSE
9.68
9.71
9.75
9.78
1.010
ACC
0.54
0.53
0.51
0.50
Architecture Srchitecture-ppecfffe Baselines and FRL-Enhanced Versions
Transformer (WcatherGFT)
RMSE
9.65
9.72
9.88
10.02
1.038
ACC
0.54
0.52
0.49
0.46
Transformer + FRL
RMSE
9.65
8.35
7.62
7.18
0.744
ACC
0.54
0.59
0.62
0.65
CNN (PDE-CNN)
RMSE
9.72
9.82
10.05
10.28
1.058
ACC
0.53
0.51
0.48
0.45
CNN + FRL
RMSE
9.72
8.22
7.42
6.95
0.715
ACC
0.53
0.60
0.63
0.66
Diffusion (ARCI)
RMSE
9.52
9.58
9.68
9.75
1.024
ACC
0.56
0.54
0.52
0.50
Diffusion FRI.
RMSE
9,52
8.28
7.55
7.12
0.748
ACC
0.56
0.60
0.63
0.64
GNN (Graph-EFM)
RMSE
9.78
9.85
10.01
10.15
1.038
ACC
0.52
0.50
0.47
0.44
GNN FRL
RMSE
9.78
8.42
7.72
7.32
0.748
ACC
0.52
0.58
0.61
0.63
Neural ODE (ClimODE)
RMSE
9.88
10.01
10.25
10.52
1.065
ACC
0.51
0.49
0.46
0.43
Neural ODE FRL
RMSE
9.88
8.45
7.68
7.25
0.734
ACC
0.51
0.57
0.60
0.62
31

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Published as a conference paper at ICLR 2026
Table 20: V500 (V-component Wind) ERA5 500 hPa 7-day Forecast
Method
Metric
180 90
360 x 180
720 x 361
1440 721
RMSF
7S-SR STF Baselines
TNO
RMSE
9,58
9.61
9.65
9.68
1.010
ACC
0.54
0.53
0.51
0.50
Climate FNO
RMSE
9.52
9.54
9.57
9.59
1.007
ACC
0.55
0.54
0.52
0.51
PeaiitecturrSpeccjpe Baselines and FRL- RRL-Enhanced Versions
Transformer (WeatherGFT)
RMSE
9.48
9.55
9.70
9.83
1.037
ACC
0,55
0.53
0.50
0.47
Transformer FRL
RMSE
9,48
8.22
7.48
7.05
0.744
ACC
0.55
0.60
0.63
0.66
CNN (PDE-CNN)
RMSE
9.55
9.65
9.85
10.08
1.055
ACC
0.54
0.52
0.49
0.46
CNN + FRI
RMSE
9,55
8.08
7.28
6.82
0.714
ACC
0.54
0.61
0.64
0.67
Diffusion (ARCI)
RMSE
9.35
9.41
9.50
9.57
1.024
ACC
0.57
0.55
0.53
0.51
Diffusion FRL
RMSE
9.35
8.15
7.42
6.98
0.747
ACC
0.57
0.61
0.64
0.65
GNN (Graph-EFM)
RMSE
9.62
9.68
9.83
9.96
1.035
ACC
0.53
0.51
0.48
0.45
GNN+ FRL
RMSE
9.62
8,28
7.58
7.18
0.746
ACC
0.53
0.59
0.62
0.64
Neural ODE (ClimODE)
RMSE
9.72
9.85
10.08
10.32
1.062
ACC
0.52
0.50
0.47
0.44
Neural ODE FRL
RMSE
9.72
8.32
7.55
7.12
0.733
ACC
0.52
0.58
0.61
0.63
Table 21: Frequency response analysis of 2D weather forecasting.
Model
Bandwidth (Hz)
H(f=75)
H(f=105)
Anchoring Ratio
Error Ratio
Baseline Models
TNO
90.42
0.995
0.289
3.44
0.178
Climate FNO
90.65
0.992
0.302
3.28
0.171
Transformer
89.12
0.988
0.182
5.43
0.172
CNN
88.05
0.965
0.098
9.85
0.112
Diffusion
86,89
1.072
0.072
14.89
0.095
GNN
90.86
1.015
0.368
2.76
0.208
Neural ODE
84.73
0.908
0.051
17.80
0.078
FRI-Enhanced Models
Transformer+FRL
120.00
1.018
1.012
1.01
0.358
FN
120.00
1.017
1.008
1.01
0.372
Diffusion+FRI.
120.00
1.056
1.019
1.04
0.336
GNN+FRL
120.00
1.031
1.004
1.03
0.295
Neural ODE+FRL
-200.00
0.998
0.938
1.06
0.388
32

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Published as a conference paper at ICLR 2026
Table 22: RMSE for 3D ZS-SR fluid simulation.
Method
323
433
64
1293
RMSERatio
ZS-SR STF Baselines
FNO
0.00482
0,00485
0.00489
0.00492 0.00445
1.021
PINO
0.00438
0.00440
0.00443
1.016
Architecture-Specife Baselines and FRL-Enhanced Versions
GNN (Neural SPH)
0.00183
0.00183
0.00183
0.00187
1.018
GNN + FRI
0.00183
0.00101
0.00062
0.00032
0.175
Transformer (DeepLag)
0.00521
0.00524
0.00527
0.00532
1.021
Transformer FRL
0.00521
0.00298
0.00185
0.00098
0.188
CNN (PARCv2)
0.00517
0.00523
0.00535
0.00548
1.060
CNN + FRL
0.00517
0.00261
0.00142
0.00071
0.137
Diffusion (DYffusion)
0.00294
0.00297
0.00301
0.00306
1.041
Diffusion FRL
0.00294
0.00171
0.00108
0.00065
0.221
NO (SFNO)
0.00468
0.00470
0.00473
0.00476
1.017
NO + FRL
0.00468
0.00237
0.00128
0.00063
0.135
Neural ODE (FNODE)
0.00405
0.00432
0.00478
0.00542
1.338
Neural ODE + FRL
0.00405
0.00261
0.00195
0.00152
0.375
NN (NeuralFluid)
0.00374
0.00378
0.00383
0.00387
1.035
NN FRL
0.00374
0.00206
0.00125
0.00068
0.182
Table 23: MAE for 3D ZS-SR fluid simulation.
Method
323
433
64
129
ZS-SR STF Baselines
FNO
0.00667
0.00671
0.00676
0.00680
PINO
0.00606
0.00609
0.00613
0.00615
Architecturee-pecific Baselines and FRI-FEnhanced Versions
GNN (Neural SPH)
0.00253
0.00253
0.00253
0.00257
GNN + FRL
0.00253
0.00140
0.00086
0.00044
Transformer (DeepLag)
0.00721
0.00725
0.00729
0.00735
Transformer + FRL
0.00721
0.00412
0.00256
0.00135
CNN (PARCv2)
0.00715
0.00723
0.00740
0.00758
CNN + FRL
0.00715
0.00361
0.00196
0.00098
Diffusion (DYffusion)
0.00407
0.00411
0.00416
0.00423
Diffusion FRI.
0.00407
0.00237
0.00149
0.00090
NO (SFNO)
0.00647
0.00650
0.00654
0.00658
NO FRL
0.00647
0.00328
0.00177
0.00087
Neural ODE (FNODE)
0.00560
0.00597
0.00661
0.00750
Neural ODE + FRL
0.00560
0.00361
0.00270
0.00210
NN (NeuralFluid)
0.00517
0.00523
0.00530
0.00535
NN FRL
0.00517
0.00285
0.00173
0.00094
33

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Published as a conference paper at ICLR 2026
Table 24: Relative Error for 3D ZS-SR fluid simulation.
Method
323
433
64
129
ZS-SR STF Baselines
FNO
0.00279
0.00280
0.00283
0.00284
PINO
0.00253
0.00254
0.00256
0.00257
Architecture-Speciiie Baselines and FRL-Enhanced Versions
GNN (Neural SPH)
0.00106
0.00106
0.00106
0.00108
GNN + FRL
0.00106
0.00058
0.00036
0.00019
Transformer (DeepLag)
0.00301
0.00303
0,00305
0.00308
Transformer + FRL
0.00301
0.00172
0.00107
0.00057
CNN (PARCv2)
0.00299
0.00302
0,00309
0.00317
CNN + FRL
0.00299
0.00151
0.00082
0.00041
Diffusion (DYffusion)
0.00170
0.00172
0.00174
0.00177
Diffusion + FRL
0.00170
0.00099
0.00062
0.00038
NO (SFNO)
0.00271
0.00272
0.00274
0.00275
NO FRL
0.00271
0.00137
0.00074
0.00036
Neural ODE (FNODE)
0.00234
0.00250
0.00276
0.00313
Neural ODE + FRI.
0.00234
0.00151
0.00113
0.00088
NN (NeuralFluid)
0.00216
0.00219
0.00221
0.00224
NN FRI
0.00216
0.00119
0.00072
0.00039
Table 25: Frequency response analysis of 3D fluid simulation.
Model
Bandwidth (Hz)
H(f=12)
H(f=20)
Anchoring Ratio
Error Ratio
Baseline Models
FNO
16.73
0.992
0.295
3.36
0.165
PINO
16.88
0.989
0.308
3.21
0.157
GNN
18.49
0.983
0.144
6.85
0.159
Transformer
16.42
0.977
0.164
5.96
0.152
CNN
16.09
0.996
0.117
8.53
0.133
Diffusion
15.78
1,257
0.181
6,94
0.139
NO
16.57
0.995
0.305
3.26
0.169
Neural ODE
15.39
0,979
0.082
11.91
0.125
NN
15.49
0.940
0.115
8.18
0.143
FRI-Enhanced Models
GNN+FRL
25.00
1.004
0.983
1.02
0.500
Transformer+FRI
255.00
0.997
0.943
1.06
0.476
CNN+FRL
225.00
1.019
1.017
1.00
0.526
Dillusion+FRL
>25.00
1.047
1.058
0.99
0.435
NO+FRL
255.00
1.010
0.997
1.01
0.556
Neural ODE+FRL
225.00
1.001
0.973
1.03
0.417
NN+FRL
255.00
0.989
0.899
1.10
0,400
34

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Published as a conference paper at ICLR 2026
Table Z500 (Geopotential Height) ERA5 Multi-Level 7-day Forecast
Method
Metric
180 x 90
360 x 180 x 6
720 x 361 x
1440 721 x
RMSE
ZS-SR STF Baselines
TNO
RMSE
661
664
668
671
1.015
ACC
0.53
0.52
0.50
0.49
Climate FNO
RMSE
657
662
1.012
ACC
0.54
0.53
0.51
0.50
Architecture SeteecetttrrSeeefffii Raselines and FRI--nnhneedd Versions
Transformer (WeatheGFTT)
RMSE
658
686
1.052
ACC
0.54
0.52
0.49
0.46
Transformer FRL
RMSE
652
547
495
462
0.709
ACC
0.54
0.60
0.64
0.67
CNN (PDE-CNN)
RMSE
666
683
701
1.065
ACC
0.53
0.51
0.48
0.45
CNN FRL
RMSE
658
530
472
435
0.661
ACC
0,53
0.61
0.65
0.68
Diffusion (ARCI)
RMSE
638
644
653
660
1.034
ACC
0.54
0.52
0.50
Diffusion FRI
RMSE
638
537
458
0.718
ACC
0.56
0.61
0.64
0.66
GNN (Graph-EFM)
RMSE
663
670
685
698
1.053
ACC
0.52
0.50
0.47
0.44
GNN + FRL
RMSE
549
473
0.713
ACC
0.52
0.59
0.63
0.65
Neural ODE (ClimODE)
RMSE
673
686
708
734
1.091
ACC
0.51
0.49
0.46
0.43
Neural ODE + FRL
RMSE
673
557
505
477
0.709
ACC
0.65
Table 27: T500 (Temperature) ERAS Multi-Level 7-day Forecast
Method
Metric
180 x 90 6
360 x 180 x 6
720 x 361 x 6 1440 721 x
RMSE
ZS-SR STF Baselines
TNO
RMSE
2.69
2.71
2.73
2.75
1.022
ACC
0.57
0.56
0.54
0.53
Climate
RMSE
2.67
2.69
2.70
1.019
ACC
0.58
0.57
0.55
0.54
Architecture Specific Baselines and FRI-Enhanced Versions
Transformer (WeatherGFT)
RMSE
2.67
2.73
2.77
1.049
ACC
0.58
0.56
0.53
0.50
Transformer FRL
RMSE
2.64
2.24
2.01
1.88
0.712
ACC
0,58
0.63
0.66
0.69
CNN (PDE-CNN)
RMSE
2.72
2.78
2.86
1.067
ACC
0.57
0.55
0.52
0.49
CNN FRI.
RMSE
2.68
2.17
1.95
1.82
0.679
ACC
0.57
0.64
0.67
0.70
Diffusion (ARCI)
RMSE
2.58
2.61
2.65
2.68
1.039
ACC
0.58
0.54
Diffusion FRL
RMSE
2.58
2.19
1.98
1.85
0.717
ACC
0.60
0.64
0.67
0.69
CINN (Fraph-EFM)
RMSE
2.71
2.74
2.79
2.84
1.048
ACC
0.56
0.54
0.51
0.48
GNN + FRL
RMSE
2.26
1.92
ACC
0.56
0.62
0.65
0.67
Neural ODE (ClimODE)
RMSE
2.74
2.79
2.87
2.97
1.084
ACC
0.55
0.53
0.50
0.47
Neural ODE + FRL
RMSE
2.74
2.28
2.05
1.93
0.704
ACC
0.61
0.64
0.66
35

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Published as a conference paper at ICLR 2026
Table 28: U500 (U-component Wind) ERA5 Multi-Level 7-day Forecast
Method
Metric
180 x 90
360 x 180 x 6
720 x 361 x
1440 721 x
RMSE
ZS-SR STF Baselines
TNO
RMSE
9.26
9.30
9.35
9.39
1.014
ACC
0,55
0.54
0.52
0.51
Climate FNO
RMSE
9.23
9,26
9.29
1.010
ACC
0.56
0.55
0.53
0.52
Architecture Speciff-- Raselines and FRI-Fanhncedd Versions
Transformer (WeatheGFTT)
RMSE
9.23
9.39
9.52
1.038
ACC
0.56
0.54
0.51
0.48
Transformer FRL
RMSE
9.17
7.93
7.24
6.82
0.744
ACC
0,56
0.61
0.64
0.67
CNN (PDE-CNN)
RMSE
9.33
9.55
9.77
1.058
ACC
0.53
0.50
0.47
CNN FRL
RMSE
9.23
7.81
7.05
6.60
0.715
ACC
0.55
0.62
0.65
0.68
Diffusion (ARCI)
RMSE
9.04
9.10
9.20
9.26
1.024
ACC
0.56
0.54
0.52
Diffusion FRI
RMSE
7.87
7.17
6.76
0.748
ACC
0.58
0.62
0.65
0.66
GNN (Graph-EFM)
RMSE
9.29
9.36
9.51
9.64
1.038
ACC
0.54
0.52
0.49
0.46
GNN + FRL
RMSE
8.00
7.33
6.95
0.748
ACC
0.54
0.60
0.63
0.65
Neural ODE (ClimODE)
RMSE
9.39
9.51
9.74
9.99
1.064
ACC
0.53
0.51
0.48
0.45
Neural ODE + FRL
RMSE
9.39
8.03
7.30
6.89
0.734
ACC
0.59
0.64
Table 29: V500 (V-component Wind) ERA5 Multi-Level 7-day Forecast
Method
Metric
180 x 90
360 x 180 x 6
720 x 361 x 6
1440 721 x
RMSErain
ZS-SR STF Baselines
TNO
RMSE
9.11
9.14
9.18
9.21
1.011
ACC
0.56
0.55
0.53
0.52
Climate
RMSE
9.07
9.12
ACC
0.57
0.56
0.54
0.53
Architecture Specific Baselines and FRI-Enhanced Versions
Transformer (WeatherGFT)
RMSE
9.07
9,22
9.34
1.037
ACC
0.57
0.55
0.52
0.49
Transformer FRL
RMSE
9.01
7.81
7.11
6.70
0.744
ACC
0,57
0.62
0.65
0.68
CNN (PDE-CNN)
RMSE
9.17
9.36
9.58
1.056
ACC
0.56
0.54
0.51
0.48
CNN FRI.
RMSE
9.07
7.68
6.92
6.48
0.714
ACC
0.56
0.63
0.66
0.69
Diffusion (ARCI)
RMSE
8.88
8.94
9.03
9.09
1.024
ACC
0.57
0.55
0.53
Diffusion FRL
RMSE
8.88
7.74
7.05
6.63
0.747
ACC
0.59
0.63
0.66
0.67
CINN (Fraph-EFM)
RMSE
9.14
9.20
9.34
9.46
1.035
ACC
0.55
0.53
0.50
0.47
GNN + FRL
RMSE
7.87
6.82
0.746
ACC
0.55
0.61
0.64
0.66
Neural ODE (ClimODE)
RMSE
9.23
9.36
9.58
9.80
1.062
ACC
0.54
0.52
0.49
0.46
Neural ODE + FRL
RMSE
9.23
7.90
7.17
6.76
0.733
ACC
0.60
0.63
0.65
36

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Published as a conference paper at ICLR 2026
Table 30: Frequency response analysis of 3D weather forecasting.
Model
Bandwidth (Hz)
H(f=1055 H(f=75)
Anchoring Ratio
Error Ratio
Baseline Models
TNO
90.58
0.998
0.285
3.50
0.181
Climate FNO
90.82
0.995
0.298
3.34
0.174
Transformer
89.89
I,00T
0.196
5.I1
0.I85
CNN
88.77
0.973
0.104
9.33
0.120
Diffusion
87.53
1.081
0.067
16.08
0.099
GNN
91.75
1.027
0.383
2.68
0.222
Neural ODE
85.28
0.917
0.045
20.28
0.083
FRL-Enhanced Models
Transformer+FRI.
120.00
1.021
1.022
1.00
0.370
CNN+FRL
120.00
1.021
1.015
1.01
0.385
Diffusion+FRL
120.00
1.068
1.026
1.04
0.345
GNN+FRL
1120.0
1.038
1.011
1.03
0.303
Neural ODE+FRL
>120.00
1.006
0.946
1.06
0.400
scale-anchored baseline fails resolve Scale Anchoring (RMSERatio >> 1). The combination of FreqEnc+FreqL.os shows trend toward mitigating Scale Anchoring (RMSEgati < 1), with Mul- FreqlLos demonstrating more pronounced improvement, and FreqEnc+MuliiRes achieving the most significant enhancement among pairwise combinations. These results indicate that fre- quency encoding enables the model to understand different resolutions, multi-resolution data allows the model to learn cross-scale mappings, and frequency loss facilitates frequency-avee aware learning to further improve spectral accuracy. Therefore, the three components of FRL operate through comple- mentary mechanisms, and the of FRL's ability enhance methods' R.M.ERatio below stems from the key innovation FneqEne.
Table 31: Ablation study of FRI. components on 3D fluid simulation. RMSF Ratio computed between 129 and 32 resolutions.
Method
GNN
Trans.
CNN
Diff.
NO
N-ODE
NN
Avg.
Baseline
1.018
1.021
1.060
1.041
1.017
1.338
1.035
1.076
Single Component
+ MultiRes
1.012
1.015
1.048
1.032
1.011
1.285
1.028
1.062
+ FreqEnc
0.912
0.918
0.942
0.935
0.908
1.158
0.922
0.957
+ FFreLLoss
1.015
1.018
1.052
1.035
1.014
1.305
1.030
1.067
Two Components
+ MultiRes=FreqLoss
1.008
1.011
1.042
1.025
1.006
1.255
1.020
1.052
+ FrelEne+MultiRes
0.485
0.502
0.468
0.538
0.472
0.782
0.498
0.535
+ FreqEnc+FreqLLos
0.285
0.302
0.268
0.335
0.272
0.582
0.298
0.335
Full FRL
+ All Three
0.175
0.188
0.137
0.221
0.135
0.375
0.182
0.202
We further analyze the parameter sensitivity of FRL on the 3D ZS-SR fluid simulation task. Specif- ically, we evaluate the two key hyperparameters of FRI., the number of resolution hierarchy levels J E (2, 3, 4, 5, 6} and the frequency consistency loss weight A E (0.01, 0.05, 0.1, 0.2, 0.5), as shown in Table 32, Across all architectures, the reduction RMSERatio plateaus when > 3 with A 0.1. Therefore, we select / 3 for the main experiments in Section 6. When À < 0.1 with - 3, the RMSERatio decreases substantially while maintaining the original resolution RMSE. However, when 0.1, the original resolution RMSE deteriorates significantly despite continued RMSERaio improvement. To balance training accuracy and scale decoupling capability, we select 0.1 for the main experiments.
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Table 32: Parameter sensitivity analysis of FRL on 3D fluid simulation averaged across all architec-
tures.
J
0.01
0.05
0.1
0.2
0.5
RMSE Ratio (129*
/323)
0.485
0.352
0.316
0.298
0.285
3
0.412
0.285
0.202
0.188
0.175
0.398
0.268
0.195
0.182
0.168
5
0.385
0.255
0.192
0.178
0.165
0.378
0.248
0.190
0.175
0.162
Original Resolution RMSE (32*)
2
0.00394
0.00394
0,00394
0.00408
0.00432
0.00394
0.00394
0.00394
0.00412
0.00440
4
0.00394
0.00394
0,00394
0.00416
0.00447
5
0.00394
0.00394
0.00394
0.00419
0.00453
0.00394
0.00394
0.00394
0.00422
0.00460
Lastly, isolate the core innovation, we keep MultiRes and FreqLoss fixed and replace FreqEne with three alternatives: an Absolute-frequend encoding (AbsFreqEnc), standard Fourier features (FourierEnc), and vanilla positional encoding (PE). Under the same tuning budget, we evaluate the cross-Nyquist RMSERatio As shown in Table 33, all three replacement encodings perform similarly full FRL remains well below 1. This indicates that FRL's success does not stem from adding any positional/frequenccy encoding per se, but from the resolution invvriant,, Nyquist-normalln normallzed frequency representation introduced by FneqEnc.
to the variant without any additional encoding ( 1), whereas
Table 33: Ablation on the FneqEnc component for 3D ZS-SR fluid simulation. RMSERatio computed between 129 and 32 resolutions. The three "replacement encodings" keep MultRes+FreeqLos. fixed and only swap FreqEnc.
Method
GNN
Trans.
CNN
Diff.
NO
N-ODE
NN
Avg.
FRL
0.175
0.188
0.137
0.221
0.135
0.375
0.182
0.202
FRL variants without the proposed normalized frequency encoding
FRL w/ No Encoding
1.008
1.011
1.042
1.025
1.006
1.255
1.020
1.052
FRL w/ AbsFreqEnc
1.010
1,014
1.045
1.028
1.009
1.260
1.022
1.055
FRL w/ FourierEnc
1.012
1.016
1.047
1.029
1.011
1.265
1.025
1.058
FRL w/ PE
1.011
1.015
1.046
1.027
1.010
1.262
1.023
1.056
I FAILURE MODES OF FREQUENCY REPRESENTATION LEARNING
The key assumption behind FRL is the existence of learnable frequency-conddiioona relationships: the underlying physical process must exhibit cross-scale spectral structure that sufficiently reg- ular so that frequency response patterns at different resolutions can be aligned in the normalized frequency domain. Many classical physical systems satisfy this assumption over wide range of scales, but there are regimes in which the spectral structure itself changes gualitatively and the no- tion simple, resolution-invarian frequency response breaks down.
To illustrate such regime, we perform diagnostic experiment channel-flow simulations at different Reynolds numbers. We train CNN and its FRL-enhanced variant on high-fidelity numer- ical results 129 129 129 grid (Nyquist frequency 256 Hz), and analyze their empirical frequency responses; see Figure 6. At low to moderate Reynolds numbers (Re < 10²), FRL suc- cessfully achieves Scale Decoupling: the Bandwidth (BW) remains close the Nyquist frequency
38

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and the Anchoring Ratio H (ffm)/((foo around the training Nyquist band is close to indicating that freeffreyency components are processed almost as reliably as low-frequency ones. However, Re increases, this behavior deteriorates. At Re 10 even the FRL-enhanced CNN exhibits clear Scale Anchoring: the Anchoring Ratio - ( ( grows from 1.00 to 1.95, and the effective BW shrinks to 262. Hz. At Re 10 the Anchoring Ratio further increases to 5.24 and the BW drops to 218.37 Hz, indicating that high-frequency components above the training Nyquist band no longer reliably extrapolated.
1.0
0.8
0.6 H(f)
0.4
0.2
0.0
Re=10~5
50
100
Re=10~4
150
Re=10>3
Frequency (Hz)
200
Re=10^2
256
(Nyquist) 300 Re=10~1
Figure 6: Frequency response analysis of an FRL-enhanced CNN at different Reynolds numbers Re on channel-flow simulations. At low and moderate Re, FRL achieves stable high-frequency response and wide bandwidth; at very high Re, the frequency response becomes sharply anchored near the training Nyquist frequency, revealing a failure mode of FRL.
Notably, this failure at very high Reynolds numbers should not be interpreted specific to FRL. Rather, it reflects broader challenge of high-Re turbulence for methods in general. Prior work on deep-learning-base Reynolds numbers and reports accurate reconstruction only for Reynolds numbers within, close to, the training range (Yousif et al., 2022). High- He regimes remain significantly more challenging and are comparatively under-explored. In our setting, models must additionally perform zero-shot super-resolution in time and frequency, which further amplifies the difficulty.
super-resolution of turbulent flows typically considers a limited range of
FRL succeeds the moderate- regimes considered in the main experiments because its Nyquist- normalized frequency representation can implicitly learn stable relationships between adjacent spec- tral bands: the statistical patterns linking relatively low-frequency components to components remain coherent, smooth, and to large extent predictable. This coherence is character- istic of transitional flows or weak turbulence, where cascade structures are not yet fully developed and cross- cross-band mappings remain statistically consistent (Cerbus et al., 2020). In this situation. the assumption of a smooth, dinates is reasonable, and FRL can exploit it to mitigate Scale Anchoring.
scale-consisten gain function Ho(E) in normalized frequency coor-
When the Reynolds number becomes extremely high, however, the flow enters a fully developed turbulent state. The inertial cascade becomes strongly nonlinear, intermittent, and non-stationary;
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energy transfer across scales is dominated by complex multi-scale interactions rather than smooth, locally extrapolatable couplings between neighboring frequency bands (She & Leveque, 1994). In this regime, the coupling between adjacent spectral bands is no longer locally smooth, and the cross- band relationships that FRI. attempts to model cease to be well behaved. Additionally, the cross- scale mappings encoded in the normalized frequency representation were calibrated in regimes with more regular spectral structure. Consequentl, these mappings do not remain valid at very high Re, and FRL fails to extrapolate reliably to high-frequency components.
It is natural to ask whether more elaborate engineering, such as adaptive mechanisms or PDE- informed soft constraints, could restore FRL's effectiveness. Such techniques can indeed trade additional computational cost for incremental accuracy gains, but fundamentally extending FRL to extremely high-Re Re flows would require accurately characterizing the relational patterns between adjacent spectral bands under those conditions, for which no simple closed-form description is avail- able. Learning these relationships purely from data would also demand prohibitive computational resources, given the power-law scaling between Re and the grid resolution required for DNS.
On the other hand, when the Navier-Stokes equations are appropriately non-dimensionallzee and the high-Re limit is considered, small-scale statistics are believed to approach universal behav- ior and follow Kolmogorov-type scaling laws, such as the mnrrt energy spectrum Ekk oc 5-5/3 (She & Leveque, 1994). In the spectral domain, this suggests that in the asymptotic high- Re regime the energy distribution over adjacent frequency bands is constrained by fixed physical pattern. This observation offers a potential handle for extending FRL: by aligning FRL's frequency- domain representation and loss with Kolmogorov-type spectral constraints, one could guide the model to extrapolate according to the theoretically expected inertial-range behavior, rather than re- lying solely on data-driven cross-band fitting.
In summary, replacing traditional DNS/R with deep learning models for very high-Re Re fluid simulation remains an important and challenging problem that goes beyond the ZS-SR STF set- ting studied in this paper. FRL is designed as a general enhancement for mitigating Scale Anchoring in ZS-SR STF under systems that exhibit scale-consistent spectral structure, and our experiments confirm its effectiveness. Extending FRL to extremely high- Re turbulent flows will likely require incorporating explicit physical spectral constraints (e.g., Kolmogorov scaling) into the normalized frequency representation and loss, which we leave as promising future work.
J DETAILED CLARIFICATION OF LARGE LANGUAGE MODELS USAGE
Table 34: Summary of Large Language Model (LLM) Usage in This Work
Purpose of LLM Usage
Used
Aid or polish writing
Retrieval and discovery (e.g., finding related work) Research ideation
X
Other purposes
X
In accordance with ICLR 2026 policy on transparent disclosure of Large Language Model usage, we declare that LLMs were employed exclusively to assist with the writing and presentation aspects of this paper. Specifically, we utilized LLMs for: (i) verification and refinement of technical termi- nology to ensure precise usage of comain-specific vocabulary; (ii) grammatical error detection and correction to enhance the clarity and readability of the manuscript; (iii) translation assistance from the authors' native language to English, as we are non-native English speakers, to ensure accurate and fluent expression of scientific concepts; and (iv) improvement of sentence structure and flow while maintaining the original scientific content and meaning. We emphasize that LLMS were not used for research ideation, experimental design, data analysis, or any form of content generation that would constitute intellectual contribution to the scientific findings presented in this work. All scientific insights, methodological decisions, and analytical conclusions the original work of the authors. The use of LLMs was limited to linguistic and presentational enhancement only, serving a role analogous professional editing services.
40


Reviews:
[{"review_id": "7tihJD2Ngj", "reviewer": "Reviewer_cSKw", "summary": "This paper defines and analyzes an overlooked phenomenon, called scale anchoring, in cross-resolution generalization for zero-shot super-resolution. To mitigate this generalization issue, this paper proposes the frequency representation learning method through three steps from data and optimization perspectives. The results have shown the superiority of the proposed method compared to existing methods on fluid simulation and weather forecasting datasets. \n\nContributions:\n\nThis observation is new and provides an analysis of why models trained at low resolution fail to improve accuracy at higher resolutions.", "questions": "- Is there any stability issue when you extrapolate beyond the training Nyquist range?\n\n- On page 6, line 321, do you need bold for “Notably”?\n\n- In Table 3, why do you have some uncommon grids like 43^3?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- This paper looks into spectral bias-related stuff, which is important in scientific machine learning.\n\n- This paper proposes a new concept, called scale anchoring, in zero-shot super-resolution. \n\n- This paper provided a theoretical and empirical analysis of this scale anchoring phenomenon. \n\n- This paper is generally well-written and well-presented.", "weaknesses": "I have several major concerns listed below. \n\n- First, what is the difference between the defined scale anchoring and the prior concepts, such as spectral bias / discretization-invariance? To me, scale anchoring refers to a failure mode in which a model trained on low-resolution data fails to get good accuracy at finer resolutions, due to missing high-frequency information beyond the coarse data’s Nyquist limit. Spectral bias also refers to the same things that a NN model cannot learn high-frequency features from data. \n\n- Also, there is a pretty similar paper that discussed resolution generalization and discretization mismatch errors. What is the difference between this paper and the previous paper [1]? \n\n- The proposed three-step method covers some tricks from the data and optimization sides. It would be good to have an ablation study to test how much each step contributes to the performance improvement. Also, it seems the third step (frequency loss) has already been seen in many prior work [2,3] that uses spectral loss to mitigate the spectral bias issues. \n\n- I feel like it would strengthen the paper by providing a comparison of computational cost versus accuracy improvement. The computational cost of using low-resolution grids should be small, and with the proposed method, one can get better performance on finer grids. Then, people can have a better sense of the tradeoffs. \n\n ---\n\n**Refs:**\n\n[1] Gao, Wenhan, et al. \"Discretization-invariance? on the discretization mismatch errors in neural operators.\" The Thirteenth International Conference on Learning Representations. 2025.\n\n[2] Chattopadhyay, Ashesh, Y. Qiang Sun, and Pedram Hassanzadeh. \"Challenges of learning multi-scale dynamics with AI weather models: Implications for stability and one solution.\" arXiv e-prints (2023): arXiv-2304.\n\n[3] Saccardi, Carlo, et al. \"Assessing the Geographic Generalization and Physical Consistency of Generative Models for Climate Downscaling.\" arXiv preprint arXiv:2510.13722 (2025).", "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": "j6q5nidED4", "reviewer": "Reviewer_SPE1", "summary": "This paper proposes a novel method called Frequency Representation Learning (FRL) to address the \"Scale Anchoring\" problem in Zero-Shot Super-Resolution Spatiotemporal Forecasting (ZS-SR STF). The authors point out that existing methods, when trained on low-resolution data, struggle to process high-frequency components beyond the Nyquist frequency of the training data during high-resolution inference, resulting in errors being \"anchored\" at the low-resolution level and mistakenly interpreted as successful generalization. FRL achieves resolution-invariant frequency embeddings through multi-resolution data construction, normalized frequency representation, and spectral consistency training, significantly reducing high-resolution inference errors. Its effectiveness is validated across multiple architectures and tasks (fluid simulation and weather forecasting).", "questions": "1、Regarding computational overhead: FRL requires storing and processing multi-resolution data during training. How can the storage cost be balanced against performance gains in practical deployment? Are there compression or dynamic sampling strategies to mitigate storage pressure?\n2、Regarding generalization capability: The performance degradation of FRL in high-Reynolds-number turbulence suggests that the method relies on consistent frequency response patterns in physical systems. Is there a plan to incorporate adaptive mechanisms or physical constraints to enhance adaptability to highly nonlinear and multi-scale coupled systems?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "1、Novel: The paper is the first to clearly identify the \"Scale Anchoring\" phenomenon and provides a theoretical analysis of its mechanism (frequency blindness and high-frequency error dominance), addressing an under-recognized limitation in existing zero-shot super-resolution research.\n2、Strong Generalizability: FRL is architecture-agnostic and can be seamlessly integrated into various mainstream models such as GNNs, Transformers, and CNNs. Experiments show that it significantly improves high-resolution inference accuracy across all tested architectures.\n3、Comprehensive Experimental Validation: Extensive tests on multiple tasks (2D/3D fluid simulation and weather forecasting) combined with error metrics and frequency response analysis thoroughly demonstrate the method’s effectiveness and generalization capability.", "weaknesses": "1、Insufficient Computational Overhead Analysis: Although a training complexity increase of approximately 1.1–1.4× is mentioned, the paper lacks detailed discussions on actual training time and memory usage comparisons across different architectures, as well as the storage requirements for multi-resolution data construction.\n2、Limited Generalization in Extreme Physical Scenarios: The authors note in the appendix that FRL fails in high-Reynolds-number turbulence (e.g., Re=10^5), but they do not deeply analyze the reasons for failure or propose improvement directions, which limits the method’s applicability in complex physical systems.", "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": "cm5MZMAe8O", "reviewer": "Reviewer_7EgH", "summary": "The authors identify a problem when performing zero-shot super resolution, when a model is trained on low‐resolution data and then deployed on higher‐resolution data, the error does not decrease despite more fine‐grained input. They argue this is because low‐resolution training data limits the maximum representable frequency (via Nyquist), so the model never “learns” higher‐frequency components. They propose a method called Frequency Representation Learning (FRL) that (a) uses resolution‐aligned frequency representations (normalized to account for the Nyquist at each resolution) and (b) incorporates a spectral‐consistency training objective across multiple resolution levels. They show experiments on spatio‐temporal forecasting tasks (zero-shot super‐resolution forecasting) where their FRL approach results in error decreasing with increasing resolution (unlike baselines) and improved frequency response stability in higher‐frequency bands.", "questions": "1. Given that training data are band-limited by the Nyquist frequency, what information allows FRL to improve predictions beyond that spectral limit?\n\n2. Does FRL rely on the implicit assumption that the underlying physical process is scale-consistent or spectrally smooth?\n\n3. Since multiple high-resolution fields can correspond to the same low-resolution sample, how does FRL regularize or constrain the mapping to select a physically meaningful one?\n\n4. What is the theoretical justification for expecting a neural operator trained on low-resolution data to exhibit convergence behavior analogous to that of a numerical solver?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The identification of the “scale anchoring” problem is interesting and relevant in several fields, such as neural operators. \n\n2. The proposed method (FRL) is architecture‐agnostic in principle.\n\n3. The experiments show that the inference error can be even lower for zero-shot super resolution. Existing works usually consider the same error as a success.", "weaknesses": "Some of these weaknesses below fall somewhere between questions and weaknesses, so I’ve included them here.\n\n1. A central conceptual concern is that the zero-shot high-resolution inference task itself appears ill-posed under the stated assumptions. If the model is trained purely on low-resolution data, its training distribution is inherently band-limited by the Nyquist frequency of that grid. Consequently, the high-frequency components present in the high-resolution domain are unobserved and unidentifiable during training.\nFrom a signal-theoretic standpoint, there exist infinitely many high-frequency realizations consistent with the same low-frequency field, so the inverse mapping from coarse to fine scales has no unique solution. Therefore, it is unclear in what precise sense the model can achieve lower error at higher resolution, may be to test model generalization in some sense? \n\n2. It seems that the testing datasets are fairly smooth, and the low-frequency modes dominate the dynamics. Experiments on data with high-frequency or multi-scale content can provide more insights. \n\n3. FRL seems to rely implicitly on the assumption that the underlying system exhibits scale-consistent spectral structure (as in turbulence or diffusion). Can the authors elaborate more on this?\n\n4. Zero-shot super resolution suffers from aliasing/discretization errors as noted in [1,2,3], there is no discussion on these works and there is no discussion on if and how FRL mitigates these issues. \n\n[1] Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning\n[2] Discretization-invariance? On the Discretization Mismatch Errors in Neural Operators\n[3] Discretization Error of Fourier Neural Operators", "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": "HgHvQaqgS0", "reviewer": "Reviewer_Kovs", "summary": "This paper involves downsampling the data to different resolutions, then using a special Nyquist-normalized frequency position encoding, along with a frequency consistency loss, to achieve super-resolution. Results on two tasks demonstrate the effectiveness of the proposed method.", "questions": "In the experiments, the method is tested at only a few discrete scales. Could you test the method over a wider range of scales by adjusting the scale in fixed steps (e.g., 2x, 3x, ... 64x) and report the results? Plotting a performance curve with these varying scales would help demonstrate the method's effectiveness and robustness across a broader range of resolutions.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. The paper introduces a new approach based on the frequency domain to address the limitations in current Zero-Shot Super-Resolution Spatiotemporal Forecasting methods. It explains the Nyquist frequency limitation of low-resolution data and offers a new perspective on why current methods struggle with high-quality high-resolution predictions.\n2. The paper highlights the flaws in traditional multi-resolution techniques, especially the problems of scale anchoring and spectral bias. It proposes Nyquist-normalized frequency representation and frequency-aware loss to solve these issues, which improves the model's performance.\n3.The proposed method demonstrates strong performance in weather forecasting and fluid simulation tasks, highlighting the effectiveness of the approach and its potential for practical applications.\n4. The proposed method is validated on multiple network architectures, including CNN, Transformer, Mamba, and GNN, demonstrating its versatility and effectiveness across different networks.", "weaknesses": "1. The method shares similarities with ZSSR and other Deep Internal Learning approaches, making it unclear what is fundamentally new beyond the frequency-domain reinterpretation.\n2. The experiments are limited to spatiotemporal forecasting; applying the method to standard image super-resolution datasets (like Set5, Set14, BSD100) would provide more comprehensive comparisons.\n3. The frequency consistency loss seems similar to existing frequency-domain losses, and its contribution may be limited. Additional ablation studies comparing it with other methods could clarify its impact.", "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)