Improving Extreme Wind Prediction with Frequency-Informed Learning
Authors: Chenrui Xu
Xi Huang
Ying-Jun Angela Zhang
Jianwei Huang
Venue: ICLR 2026 Poster (2026)
Keywords: Extreme Weather Forecasting
Meteorological Analysis
AI for Science
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)

Abstract: Accurate prediction of extreme wind velocities has substantial significance in industry, particularly for the operation management of wind power plants. Although the state-of-the-art data-driven models perform well for general meteorological forecasting, they may exhibit large errors for extreme weather—for example, systematically underestimating the magnitudes and short-term variation of extreme winds. To address this issue, we conduct a theoretical analysis of how the data frequency spectrum influences errors in extreme wind prediction. Based on these insights, we propose a novel loss function that incorporates a gradient penalty to mitigate the magnitude shrinkage of extreme weather, and we theoretically justify its effectiveness via a PDE-based energy–enstrophy analysis. To capture more precise short-term wind velocity variations, we design a novel structure of physics-embedded machine learning models with frequency reweighting. Experiments demonstrate that, compared to the baseline models, our approach achieves significant improvements in predicting extreme wind velocities while maintaining robust overall performance.

Published as a conference paper at ICLR 2026
IMPROVING EXTREME WIND PREDICTION WITH BREQUENCY-INFORMED LEARNING
Chenrui Xu', Xi Huang', Ying-Jun Angela Zhang', Jianwei Huang
The Chinese University of Hong Kong 2 AAround Tech (Shenzhen), Ltd. School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen 4 Shenzhen Institute of Artificial Intelligence and Robotics for Society (AIRS)
j i anwe aaweiihuange ihuange cubk. edu. en
ABSTRACT
Accurate prediction of extreme wind velocities has substantial significance in in- dustry, particularly for the operation management wind power plants. Although the state-of-the-ae data-driven models perform well for general metcorological forecasting, they may exhibit large errors for extreme weather- for example, sys- tematically underestimating the magnitudes and short-term variation of extreme winds. To address this issue, we conduct a theoretical analysis of how the data frequency spectrum influences errors in extreme wind prediction. Based these insights, we propose a novel loss function that incorporates gradient penalty mitigate the magnitude shrinkage of extreme weather, and we theoretically justify its effectiveness via PDE-based energy-enstrng more precise short-term wind velocity variations, we design a novel structure of physics-emeddded machine learning models with frequency reweighting. Experi- ments demonstrate that, compared to the baseline models, our approach achieves significant improvements in predicting extreme wind velocities while maintaining robust overall performance.
analysis. To capture
INTRODUCTION
Wind velocity field prediction is crucial to both academic research and industrial practice (Tas- cikaraoglu Uzunoglu 2014, Hanifi et al., 2020). For instance, the wind power plants require accurate predictions of the wind speed magnitude to support accurate real-time production estima- tion and safe operational control, since the output power approximately proportional to the wind magnitude cubed (e²), and the wind turbines will cease to work for extremely large wind (Sabzehgar ct al. 2020 Wan et al., 2010). Traditionally, solving dynamic systems by mathematical methods, Numerical Weather Prediction (NWP) has been the workhorse of wind velocity prediction (Coiffier) 2011]. However, recent advances in deep learning have revolutionized weather prediction, with models like FourCastNet (Pathak et al. (022) and Pangu-Weattie (Bi al. (022) significantly out- performing traditional NWP methods (Coiffier) 2011). Based on an extensive amount of data, these models are specialized in producing accurate overall predictions of the wind velocity field.
One of the key challenges in wind velocity prediction is to accurately predict the amplitude changes of the extreme wind. General data-driven models may be struggling with this challenge if they are trained by common loss functions (like MSE) based on regular wind speed datasets (Olivetti) & Messori 2024). For example, many data-driven forecasters systematically underestimate the amplitudes of extreme winds (Rychlik & Mao] 2019). This bias persists even when overall (non- extreme) skill is strong (Shi et al. 2025), leading to underestimated risks and missed rapid ramps in practical operations. Therefore, addressing this challenge in extreme wind velocity prediction is the main focus of this paper.
There exist several data-driven models specifically designed for extreme weather predictions. Some models employ classical deep learning models such as RNN (Prasetya & Djamal 2019), CNN
"Corresponding author. Jianwei Huang is also affiliated with the Shenzhen Key Laboratory of Crowd In- telligence Empowered Low-Carbon Energy Network, CSURI Joint Research Centre on Smart Energy Storage, The Chinese University of Hong Kong, Shenzhen, and Shenzhen Loop Area Institute (SLAI), Shenzhen, China.

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Zhang al. 2 and LSTM (Gao al. 2018 capture spatiotemporal dependencies in weather data. Many other models utilize generative data augmentation methods, including variational au- toencoder (VAE) (Vega-Bay
al. 2024) and diffusion models (Zhong et al.) 2024), to address data scarcity through weather pattern simulation. Despite these advances, critical gaps remain for extreme wind predictions: (i) most approaches offer little theoretical (or even intuitive) explanation of why errors arise significantly for extreme winds; (ii) many methods implicitly rely on abundant training data that include extreme cases, whereas such cases are intrinsically scarce in real datasets; (iii) to better capture the dynamics of sharply-changing pattern, the data-driven models may re- quire much more complicated model structure than overall prediction; and (iv) depending mainly on large data, some models may be insufficient for capture the intrinsic dynamics and still suffer from uncontrollable regional errors for extreme weather prediction (Zhou et al. 2024).
To resolve the challenge, we conduct a theoretical analysis of the error behavior in the frequency domain. Based on proper simplification, we separate the traditional mean-squared error (MSE) into three terms: amplituee shrinkage error, pattern translation enror, and noise. We show that while train- ing with standard MSE as a loss function, small pattern deviations will lead to significant amplitude shrinkage for the high-frequency wind field components, causing the underestimation of extreme amplitude and blurred short-term variability in prediction. Inspired by the analysis, we propose a gradient-penalized loss function upweighting the amplitude shrinkage error. We further provide PDE-based energy-enstrophy interpretation showing that the new loss function enforces enstrophy matching and controls small-scale vorticity, thereby theoretically explaining its effectiveness in mit- igating amplitude shrinkage for extreme winds. To more effectively reduce the pattern translation error, well as improve parameter efficiency. we design a physics-embeddee structure for the neu- ral network, with a backbone of the Navier-Stoker (NS) equation. The equations reveal how the motion of fluid, such as the atmosphere, is affected by a combination of external forces, pressures within the fluid, and viscous effects (Marion & Temam 1998). Moreover, we utilize a frequency separation and reweighting mechanism to coordinate the impact of high- and low-frequency compo- nents to the loss function. Based on the above frequency inffoomeed modification of the loss function and neural network structure, our model overcomes the amplitude shrinkage challenge in wind pre- diction and achieves significant improvement in extreme wind velocity prediction accuracy. Our research makes the following significant contributions to the field of extreme wind prediction:
trequency-theoretic explanation of underestimation. We provide Fourier-domain anal- yses showing how small spatial shifts and scaling yield frequency-depender MSE, theo- retically accounting for underestimation of extreme amplitudes of short-term variability.
Erradient-penaiizn objective for extremes. We propose a simple, implementation loss that augments MSE with gradient matching, equivalently reweighting nighfrreuency errors to mitigate spectral shrinkage and recover sharp ramps.
Frequency separation & reweighting with physics-embedded backbone. We design a spectral pipeline (Fourier masking, Frequency separation) atop simplified NS block, targeting precise short-horizon dynamics while preserving stability and data efficiency.
Empirical validation on regional extremes. Across diverse regions and baselines (in- cluding CNN, ConvLSTM, and PINN), our method substantially improves extreme-wind prediction while maintaining robust overall performance under normal conditions.
The remainder of this paper is structured as follows: Section 2 iicludes problem formulation, phys- ical backgrounds and theoretical insights. 33 our methodology, including the novel loss function formulation and network design. Section 4 describes our experiments and correspond- ing results. Section 5 contains the conclusion with limitations and future directions. Related works to this paper can be found in Appendix Other details are provided in the appendices.
2 PRELIMINARIES AND INSIGHTS
2.1 PROBLEM FORMULATION
In this paper, we mainly consider the wind velocity field prediction within certain rectangular gions, which can be discretized into N x M points. Let u(x, t) = [o(2, y, t), w(x, y, t)] de- note the wind velocity field in this region, where x (x, is the two-dimensional spatial do-
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main and is the temporal domain;u and represent the velocity components in the longitude and latitude directions, respectively. We denote historical wind data sequences by U[ = (u(2, ys (1), (2x, y, 12)....... u(x; y, Lx)].
Let ú(x,t) t) denote the prediction of u given by a certain model at time t. Then our objective to predict the wind velocity field at the next time can be expressed as follows:
1 ū(a, y, (++1) Lo (M[c::ty]: other data),
where fo is the model we intend to train, and "other data" contains other data sequences that might also contribute to wind velocity prediction (like surface pressure, which will explained later). The error between u and evaluates the performance of the predictive model.
Temporal and Spatial Scale for Extreme Prediction. In atmospheric forecasting, temporal and spatial scales are tightly coupled: short-term predictions are typically associated with short-range dynamics +Jug & Broadwater] (2014) (Zhu et al.) 2019). In this paper, we adopt the convention of extreme wind velocity prediction that focuses on short-period and regional prediction, while the temporal and spatial resolutions are also higher compared to global weather forecasting to resolve rapidly evolving, small-scale features.
2.2 PHYSICAL BACKGROUNDS
As fundamental assumption in meteorology, atmospheric systems' dynamics generally satisfy the Navier-Stokes (NS) equations together with the continuity equation (mass-conservation) con- straint(Holton & Hakim 2013). The NS equations are of nonlinear partial differential equations (PDEs) that describe the relationship between the motion of a fluid and the forces acting upon it. For vo-ddmenniooall domain with wind velocity field [u(z, y. t), w(x, y, t)], the NS equation and continuity equation are shown as follows:
du 28
Vu- - - V FE 1>
(2)
O,
x 2, >> 0,
where Vu is the advective acceleration; V refers the pressure gradient force; 2 denotes the viscous friction (v: kinematic viscosity); and F is the external body forces. The body force term may vary in different scenarios, with typical examples including gravity and the Coriolis force (Holton & Hakim 2013).
2.3 INSIGHTTS FROM FREQUENCY DOMAIN ANALYSIS
When solving and analyzing PDEs, standard method is to apply the Fourier transform (often with respect to spatial domains) to convert the PDEs into ODEs (Evans 2022). Let : denote the Fourier operator, and k = (kz, ky) denote the frequency domain coordinates. For example, applying the Fourier transform to simplified version of the NS equation equation 2 with advection and diffusion:
0,u(x,t) U- UVVx(X t) u-V2u(x t) + f(x,t),
we will get
0,ū(k,) f) + (k. U) ū(k,t) - ~ t) f(k,t),
(3)
3 shows that the advection of air may appear as a phase shift e U corresponding to a spatial translation physical space. Moreover, diffusion may induce amplitude damping a rate proportional to ||k| (stronger for high frequency).
which is ODE with respect to ú. Equation equation
Motivated by this idea, we apply a 2-dimensional Fourier transform to the wind velocity fields and analyze how the frequency spectrum affects the prediction error. By equation equation 3 and statistical convention, we assume that the prediction error is mainly caused by three factors: scaling, translation, and noise. Therefore, the relationship between û and can be illustrated follows:
û(x) = au(x + A) + =xx),
(4)
where a is the scaling magnitude wind speed amplitude, and A corresponds to the deviation amount of the data pattern. We may also assume ~N )²) be Gaussian noise.
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Now, let's consider the mean squared error (MSE) of the prediction and the ground-truth. By Rayleigh's energy theorem (Temple et al. 2004), we can show that the MSE the original data in the spatial domain is equivalent to the MSE of the fourier-transformer data in the (double) fre- quency domain:
MSE(U, û)
NM Σ u
(NM)² Σ i i
We denote 0x=
2
|.
M and we assume 0g is sufficiently small. Then the Fourier
transform of the prediction is û(k) th ú(k) =kk), and the expectation of the MSE will be:
E|MSE u, ú)] C1 Σ (1 (1- (1-ae" "h [|ū(k)]1 + +0²;
=1 Σ (a² -aaa-aB/cos ||ū(k)]| + +0²;
(S)
k
C1 Σ {a Eloos | + +(1 EE [cos
ccos 0k) (()))
scaling
translation
where C1 is constant depend on N and M.
As the last line of equation shows, the MSE has been separated into three components: The scaling error reflects the magnitude difference between the prediction and the ground-truth; the translation error is caused by the pattern deviation the noise assumed be independent of both a and A.
The Cause of Amplitude Shrinkage. The scaling error term is highly related to the pattern de- viation factor A, and the theoretically optimal amplitude scaling will be a = E(cos 0k]- Given the existence of the derivation A, we will have Elcos 0k] < 1, causing the shrinkage in the amplitude of the predicted wind speed. Therefore, we also name the scaling error as shrinkage error.
Moreover, if we assume that A is small enough, then the optimal will be:
Aopt = Elcos 0x] = C2(k A)2
+ o(llk(1))
(6)
2
which is decreasing as k becomes higher and C2 is scalar. Therefore, the amplitude shrinkage phenomenon will theoretically tend to be more severe for high-frequency spectrum data.
When trained with MSF, the estimator reduces the squared discrepancy between the prediction and the target, effectively acting on a decomposition of error into translation, scaling, and stochastic noise. If model capacity optimization is insufficient avoid the translation component (A 0), gradients can still decrease the objective by attenuating the field's amplitude (i.e., driving a < 1). This mechanism explains why general MSE-trained models may underestimate wind-speed ampli- tudes and dampen short-term variability, thereby degrading performance on extreme-wind predic- tion. These results yield three practical insights for improving extreme-wind prediction:
Upweight scaling error. Increase the relative weight of the amplitude (scaling) component in the loss to counteract shrinkage.
Reduce translation error. Incorporate mechanisms that explicitly address misalignment
so the optimizer need not compensate by damping amplitudes.
Frequency-aware weighting. Reweight residuals by frequency spectrums mitigate high-frequency attenuation and preserve short-term variability.
3 METHODOLOGY
Guided by the insights from Section on frequency-domain error behavior, we propose a new gradient-penalized loss function that mitigates MSE-induced amplitude shrinkage under pattern deviation, and we design neural framework that combines a quency separation & reweighting. The model architecture is shown in Figure] 1
structure and fre-
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3.1 GRADIENT- PENALIIED LOSS FUNCTION
Building on the previous analysis in Section 2.3 the amplitude shrinkage phenomena under MSE mainly arise from pattern deviation between the predicted and true wind fields. One idea to solve the problem is to modify MSE by a correction term, which should be insensitive to such deviation, but capture the field's general spatial change. One of the intuitive approaches is encouraging the norm of the prediction gradient | to match that of the ground-trutt |/uul Therefore, we propose our novel Gradient- Sradient-Penalized Loss Function follows:
Cúp(ú. = MSE(U, u) + X |ū| |ul]
The coefficient À balances pointwise fit against global variation matching: larger À more strongly discourages amplitude shrinkage and preserves high-frequency variability; a smaller ap- proaches plain MSE.
Connection with error decomposition. Due to the amplitude shrinkage phenomena studied in [ulla is likely less than Vull2 in practice. When this happens, minimizing equation equation is equivalent to minimizing the following simplified version:
Section 2.3
Egp (, u) MSE(ū, u) A||Vū||
Applying the Fourier transform to Vū and using the Rayleigh's 's energy theorem in Section 2.3
know V| is proportional to Σx |k| [kk 2 On the other hand, suppose that A is sufficiently
small, then the decomposition equation equation 5 will also yield
[[SSE] 2J C1 D (a- - [kk]] + C2 cos (k, A)|||| | [[(k)]] + o² k
|k [k) and thus proportional to || Therefore, the essential effect of the madient--ennlizatiin can be explained as follows: By tuning X, the loss Lgp increases the effective weight on shrinkage error relative to pattern translation error. Consequently, when optimization hits bottleneck in reducing the translation mismatch, the model will attempt to optimize on the shrinkage error and thus improve the extreme prediction.
where the second term is proportional to En cos (k, A) All
3.2 ENERGY-ENSTROPE INTERPRETATION OF THE GRADIENT- PENALIZED Loss
The motivation of our pradient-penalizee loss function is strongly connected to the energy structure of the incompressible NS equations (Leslie & Shvydkoy 2016). Conceptually, the loss function can be viewed trading off energy matching (via the MSE term) and enstrophy matching (via the gradi- ent term) (Dascaliue et al. 2005 Palha & Gerritsmal (017). This perspective explains why gradient penalization is theoretically effective in mitigating amplitude shrinkage of high-gradient, extreme wind events. More detailed explanations and proofs to all theorems can be found in Appendix C.2
In fluid dynamics, vorticity and enstrophy are often used characterize the rotation and turbulent behavior of fluid. The formal definition of them is shown as follows (Foias et al.) 2001).
Definition (Vorticity). For an incompressible velocity field u(u1, u2) Ω R²,, the vorticity is defined to be the curl of the fluid velocity, mathematically.
w(x, t) == (V x u)(x, t) = O., u2 t) Org u (x,
Definition 2 (Enstropohy).. The enstrophy & of the flow is defined the I n?-moomm of the vorticity,
E (u(()) = / [w t)1 dæ
v u(x, dx.
According to Foias et al. (2001), for incompressible NS equations, there exist an important connec- tion between enstrophy and the norm of the velocity field, illustrated by the following theorem. Theorem (Enstrophy of 2D incompressible flow). Let be sufficiently regular, divergence-free 2D velocity field on Ω with periodic or no-slip boundary conditions. Then its enstrophy is equivalent the L norm of the velocity gradient:
E(u) / [Vuu())) = ||Vu|
(10)
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Thus, |/V can be interpreted the total strength of rotation and shear in the flow: it is domi- nated by fronts, shear layers, and small-scale vortices where Vull is large.
Energy balance for 2D incompressible NS equations. Denote the kinetic energy of velocity field by E(t :- 2 ( i Taking the L inner product of NS equation 2 with u(t) and integrating by parts (see Appendix C.2p. we obtain the following basic energy balance theorem.
Theorem 2 (Energy balance for incompressible NS). Let be sufficiently regular solution of equation 2 with periodic or no-slip boundary conditions. Then, for almost every 2 0, the kinetic energy satisfies
1 d 2 dt
[u)II2 + VIV (F(),
(11)
This theorem shows that the enstrophy Vu(tt controls the rate at which kinetic energy dissi- pated by viscosity, and therefore plays central role in the evolution of kinetic energy.
Spectral representation of energy and enstrophy. k. By Rayleigh's theorem (see appendix B). we have
Let E(k, t) be the spectral energy at frequency
/.20
1
E(t) = 2 |u((i 2
E(k,t) dk,
(12)
E(I) = / Vu(x, t)ll dx dx=
1 0o
k.2 El: d)
Thus, the same spectrum E(K, 1) generates both the kinetic energy (an unweighted integral) and the enstrophy (a weightede integral) (Fischer et al. 2007). In particular, enstrophy is heavily biased toward high frequencies (small scales), while energy treats all scales equally.
caudient-penaliize loss as an energy-enstrophy trade-off. Based on the above analysis, the continuous form of the gradient-penalized loss function can be interpreted as follows:
Cgp ūu u) =
[[(r)) u2 dx + ||ūū|2 Vulli
(13)
energy matching (MSE)
enstrophy matching
where the "energy matching" here means reducing the energy of the error field.
Here, the first term minimizes the kinetic energy of the error field, while the second term matches the enstrophy, which controls the temporal decay rate of kinetic energy. Consequently, the gradient- penalized loss does not only suppress the error energy, but also constrains the predicted flow to exhibit a physically consistent rate energy change. practice, the learned model thus maintains an overall accurate prediction of the velocity field, while restoring sufficient gradient and vortic- ity strength in high-impact regions so that the total enstrophy remains comparable to the ground truth. Therefore, the mudiient-penalzeer loss makes uniform amplitude shrinkage inefficient way to reduce the objective, encouraging the network to preserve the magnitude of physically relevant small-scale structures.
3.3 PHYYIIC- EMBEDDED STRUCTURE
To more effectively reduce the translation error highlighted at the end of Section 2.3 we introduce a chysice-embedded structure that leverages the NS equations inductive bias. The translation er- for predominantly stems from uncertainty in the direction and magnitude of the wind-fied field shift at the next time step. Traditional neural networks do not impose explicit constraints on such pattern transport: they attempt to learn it implicitly via the loss. In contrast, using an physics embedded backbone provides a first-principles estimate of the dominant transport and deformation of the field (e.g., advection and diffusion), yielding rough but informative pattern forecast. This explicit phys- ical guidance both constrains translation error more directly and reduces the burden on the learnable components, thereby lowering parameter and training costs.
Inspired by equation equation [ we embed the Navier-Stokes equation into our neural network and name it as NS Operator. The operator is decomposed into four components:

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HS Operttors
Sody
Kike
Openator
Prestre
Loca Detambherr
Faqquncy
Adaid on
Adrectvve
Figure 1: The full architecture of our model. The imput data are the wind velocity field u and the pressure field P. The wind velocity field will be successively processed by the Frequency filter, temporal attention module, and the NS operator to obtain the prediction of high- and low-frequency data. Then the model will combine the two predictions to produce the final prediction results.
1. Advective Operator: implements the nonlinear transport term u Vu.
2. Viscous Operator: implements viscous diffusion 2 u arising from internal friction.
3. Pressure Operator: accounts for the pressure-gradien force / V.P. Here, we will utilize the pressure data P(t1 tv.. However, if the pressure data not obtainable, may con- sider the pressure force as implicit and merge this operator into the Body-Force operator.
4. Body-Force Operator(conventionn neural networks): because explicit short-term formu- lations of external forces are often imprecise or unavailable, we model the body force with learnable neural networks that capture dynamics not explained by the above three operators.
Remark that the body force operator is equivalent to conventional neural networks and can adopt various structures as other pure data-driven models (but likely contains fewer layers and parameters). During training, the first three operators will learn to generate rough pre-prediction based on dynamic properties and to ensure the pattern translation magnitude and direction lie a reasonable range, while the Body-Force operator will learn how to capture the exact wind field dynamics based on the pre-prediction given by the other three operators.
3.4 FREQUENCY DOMAIN SEPARATION AND REWEIGHTING
Guided by the insights from Section 2.3 we adopt a frequency-aware weighting strategy to counter high-frequency attenuation and preserve short-term variability. Concretely, we design frequency filtering & reweighting scheme that (i) splits the wind field into low- and high-frequency compo- nents and (ii) processes and reweights these components respectively the model can retain rapid, localized dynamics without sacrificing large-scale coherence.
Fourier Filter. We employ Fourier filter (Alleyne & Cawley 1991) Münch et al. 2009) to de- compose wind velocity data into low-frequency (u²) and high-frequency (u"² components. The filter consists of three main steps: 1) Fourier Transform: Converts wind velocity data from the po- sitional domain to frequency domain. 2) Frequency Masking: Separates high- and owwffrquency components using appropriate masks: ú = u(k) M(K), where M(K) denotes the frequency mask (high or low). 3) Inverse Fourier Transform: Transforms the filtered components back to the positional domain. This decomposition enables the model to focus on distinct frequency compo- nents, enhancing its ability to capture both large-scale trends and rapid, localized variations.
Frequency- -Based Temporal Attention To refine the dynamic modeling, we design temporal tention mechanisms for both high- and low-frequency data sequences. Inspired by SENet (Cheng ctt al. 2016), temporal attention contains two operations: Squeeze, which compresses the data of each time slot into a value; and Excitation, which produces weight sequences that reflect the relative importance of each time slot for future predictions. The temporal attention is applied at different res- olutions for high- and low-freque fre-uuuye components, respectively. Since high-frequenge data are more critical for short-term dynamics, they are processed with higher temporal resolution (shorter time intervals). Conversely, low-frequency sequences, which correspond to long-term trends, han-

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dled at lower temporal resolution. This differentiation ensures that the model effectively captures the unique characteristics of both short-term and long-tew Longtterm dynamics.
4 EXPERIMENTAL RESULTS
We evaluated approaches through three key experiments: 1. Efffect of Gradient Penalized Loss Function: Our novel loss function effectively resolves the amplitude shrinkage problem in extreme wind prediction. 2. Main Prediction Results: Our model outperformed baselines in both overall accuracy and predictions in extreme wind regions. 3. Different Frequency Masking Levels: The results showed that intermediate masking thresholds achieved the best balance between high- and low-frequency information, leading to more accurate predictions.
Data. We evaluate our approaches on meteorological data sampled from the 5th generation of the ECMWF reanalysis (ERA5) database (Hersbach et al. 2020p. The dataset includes three key meteorological variables related to wind prediction: the eastward and northward components of 10- meter wind and surface pressure. Guided by the prediction scales stated in Section 2.1 the data of each variable is represented as a time series of two-dimensional latitude-longiuude fields over the study region, temporally ordered and co-registered on a common grid. The temporal resolution of the data is I hour, while the spatial resolution is 0.25%. For convenience, define each 24-hour period as a prediction unit, where the first hours are used as inputs to predict the 24th hour.
Baseline Models. To study the effect of the gradient penalized loss function, we utilize the struc- ture of multivariate meteorological data fusion wind prediction network called MFWPN (Zhang jtt al. 2025). We further compare our full model with several state of the-art regional weather pre- diction approaches, including CNN, Convolutional LSTM (Tan et al.) 2023), and Physics-Inie InformesInformed Neural Network (PINN) (Eivazi et al.] (0022). We remark that the PINN model is designed with a revised form of the Navier-Stokes (NS) equations (the Reynolds-averaged Navier-Stokes (RANS)
equations) (Ling & Templeton 2015 Cai & Wang 2024
Evaluation Metrics. We assess the performance of models using Root Mean Squared Error (RMSE), one of the most commonly used metrics for overall predictions. We also evaluate the Ex- treme Attentive RMSE (Ex-RMSE), which is a modified version of RMSE focusing on regions with extreme wind velocities. The detailed explanations of the two metrics are shown in Appendix E.2
4.1 EFFECT OF GRADIENT PENALIZED Loss FUNCTION
To quantify the impact of the proposed wradient-penalizer ob- jective function, we first compare the performance of mod- els trained by equation 7 and MSE over the same base- line structure, and study the impact of different hyperpa- rameter on the model performance. The baseline model here adopts the same structure MFWPN (Zhang al. 2025), which is a machine learning model for short-term wind speed prediction using spatial-temporal fusion and CNN units. We use second-orde central differences along both axes to represent spatial gradients. All other settings (op- timizer, learning rate, augmentations, and early stopping criteria) are kept identical to the baseline for fair com- parison. The choices of hyperparameters include X {0, 0.01, 0.02, 0.03, 0.05, 0.07, 0.10, 0.15, 0.20, 0.253.
6.6
6.4
0.0
031
022
032
040
690
Figure 2: Effect of the gradient- penalized loss across À values.
Results. Figure demonstrates the effectiveness of the gra- dient penalized loss function and the trade-off between amplitude error and translation error. With a proper À value, the models trained by gradient- penalized loss outperform the baseline trained by MSE in general. We also observe consistent U-shaped curve w.r.t. A: small positive values markedly reduce extreme attentive error while preserving overall accuracy: too-large values over- weight high-frequency residuals and harm stability. In particular, the best performance is achieved
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Table Comparative error results cross models. The best values are shown in bold.
Lead time: 1h
Lead time: 3h
Lead time: 5h
Model
RMSE
EX-RMSE
RMSE
Ex-RMSE
RMSE
Ex-RMSE
CNN
0,4639
0.3183
1.0442
0.7355
2.0757
1.0693
ConvLSTM
0,3471
0.2294
0.7834
0.5357
1.0644
0.8097
PINN
0,3946
0.2541
0.8283
0,5646
1.1434
0.7347
Ours
0.3287
0.1868
0.6622
0.4329
0.9076
0.6158
Ground Thuth
Prect Beokine
Fredi Modill
Exteemc
Baudinc
Ext Our Modld
The first line: the ground truth and prediction results of the baseline (PINN) and our models. The first sub-figure in the second line: regions where wind velocity exceeds specified threshold, highlighted as "Extreme Regions" (in white). The last two sub-figures: comparative prediction errors between our model and the baseline, where bluer indicate lower prediction errors.
Figure 3:
at A X", which is 0.02 in this experiment. Moreover, when À 2 0.15, optimization becomes unstable and the models fail converge within the prescribed training budget.
The empirical trend aligns with our frequency-domain analysis in Section 2.3 and Section 3.1 The gradient term in equation 7 effectively penalizes the amplitude shrinkage trend and there- fore improves the accuracy for extreme wind velocity prediction. However, the penalized term ||Vūβ|= Vull?] itself does not contain any information regarding positional alignment. There- fore, beyond a threshold of X, the learned model may intend to generate predictions with large spatial fluctuations, regardless of the positional pattern mismatch. Moreover, we also provide a convergence objective in Appendix D which theoretically explains why the
analysis for the model performance will degrade for large X.
4.2 EXTREME WIND VELOCITY PREDICTION
In this section, we evaluate the performance of our final model, which integrates all components in the methodology section with architecture shown in Figure [ We compare against several representative baselines on the same regional wind velocity prediction task. The baseline models include: CNN, ConvLSTM, and PINN. We train all models with initial learning rate of 1 10-5 and the SGD optimizer. All approaches consume the same wind-velocity inputs, except that our model additionally utilizes surface pressure P auxiliary field.
Results. Table I reports both overall prediction errors and extreme wind attentive errors. Com- pared to the CNN, LSMM and PINN baselines, next-frame prediction, the overall RMSEs of our model decrease by 29.19 19.1%, %, 5.3%. and 16.7%, respectively; while the extreme attentive RMSEs decrease by 41.3%, 18.6%, and 26.5%. Moreover, we evaluate the performances for predictions. Our method consistently attains the lowest RMSE and Ex Ex-RMSE across all longer forecasting lead times (3h, and 5h). In weather forecasting, lead time is the period between issu- ing forecast and the occurrence of the predicted weather (Fasterling & Mjelde 1987). Overall,

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these results demonstrate that the proposed model delivers stable substantially enhances extreme wind prediction without sacrificing overall accuracy.
improvements and
Figure provides visual comparison of regional wind velocity amplitudes. Compared with the PINN bascline, our predictions exhibit larger and more realistic amplitudes that are closer to the ground truth, particularly in the most extreme zones (highlighted by red boxes). This aligns with our frequency-domain analysis in Section 2.3 and the first-stage results in Section [4.] The above results show our model's better performance for both extreme and overall wind velocity predictions, mitigating the critical amplitude shrinkage problem in extreme weather prediction.
4.3 ABLATION STUDY
To provide insights into the effectiveness of the main modules in our model, we conduct an ab-
Table Ablation study of the proposed model.
lation study quantify their contributions. We compare the performance our final model with
Model
RMSE
EX-RMSE
the following cases: only NS operator (NS op); without gradient-penalized loss (W/O grad-loss); without NS structure, or in other words, only body force operator (W/O NS); and without fre-
NS op
0.7061
0.4577
W/O grad-loss
0.3351
0.2632
quency separation (W/O freq-seq). The results of
W/O NS
0.3754
0.2363
the ablation study is shown in Table 2
W/O freq-sep
0.4199
0.2703
When we retain only the NS operator, the errors become substantially larger, indicating that sim- ply learning a neural NS operator alone is insuf- ficient for the predictions. Removing the gradient-penalized term leads to only a negligible change in the global RMSE, but causes a pronounced degradation EX-RMSE, confirming that the gra- dient penalty has targeted effect on reconstructing sharp gradients and vorticity in high-impact regions and thus sharpening extremes predictions. When the NS- WS-based structure is removed, both RMSE and EX-RMSE deteriorate, suggesting that encoding the NS prior into the model architecture provides a beneficial inductive hias that improves the background velocity field. Moreover, the frequency separation mechanism is discarded, the model yields significant increase in both RMSE and EX-RMSE, highlighting that explicitly disentangling low- and high-frequency components is crucial for capturing both large- and small-scale structures of the velocity fields.
Ours
0.3287
0.1868
Overall, these results demonstrate that all the above components (the NS-informed architecture, the gradient-penalized loss, and the frequency separation module) make complementary and nontrivial contributions for our full model.
CONCLUSIONS
We conducted comprehensive frequency-informeed learning to address a key obstacle in wind ve- locity prediction: the amplitude misalignment (particularly underestimation) of extreme wind ve- locity prediction. From a frequency-dome perspective, we showed that small spatial pattern de- viations combined with standard MSE training induce frequency-depende disproportionately suppressing high-frequency components. Guided by this insight, we separated the error into different components and proposed gradient-penalized loss function that encour- ages models to emphasize amplitude misalignment. Furthermore, we provided a PDE-based en- ergy-enstrophy analysis explain the proposed loss function's 's effectiveness in restoring mmall-scale gradients and vorticity. We proposed a frequency separation and reweighting framework with physics-embedded backbone further enhance the capture of extreme wind dynamics. Empirically, the proposed methods outperform baselines on regional datasets, significantly improving extreme- wind prediction accuracy while keeping robustness of overall wind prediction.
amplitude shrinkage,
Limitations Our analysis relies on a simplified assumption on the factors (scaling, shifting, and noise) that cause prediction errors, and comprehensive study more complex error-causing fac- tors might be a promising direction. Moreover, generalizations to longer lead times, 3-dimensional scenarios, and other weather variables may also be interesting.
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ACKNOWLEDGMENTS
This paper is supported by Longgang District Shenzhen's "Ten Action Plan" for Supporting Inno- vation Projects under Grant LGKCSDPT 2024004.
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A RELATED WORKS
A.1
NUMERICAL AND DATA-DRIVEN WEATHER PREDICTION
Numerical Weather Prediction (NWP) (Bauer et al. 2013) represents traditional physical weather forecasting methods, which rely computational techniques to solve the physical equations gov- eriig atmospheric dynamics. For wind velocity prediction, the Navier-Stokes equations and the continuity equation are pivotal in describing the wind velocity field dynamics within region. NWP models discretize these equations over a computational grid and solve them numerically using meth- ods such as finite difference, finite volume, or spectral techniques. Despite their widespread use, NWP models face significant limitations, including a reliance on precise initialization data and high computational costs. These challenges make real-time predictions and extreme weather scenario forecasting particularly difficult.
In contrast, purely data-driven models leverage machine learning algorithms predict wind speed by identifying patterns in historical data. Examples include CNNs (Liu et al. 2016). LSTMs YYu al. 2019). ConvLSTMs (Kim et al., 2017), GANs (Li et al., 2021), and transformers (Bi et al.) 2022). These models excel at capturing complex wind patterns and local variations, demonstrat- ing flcxibility and adaptability in learning from data. However, they lack the physical constraints required to ensure realistic predictions, which can sometimes result in unreasonable outputs.
A.2 PHYSICS-INFORMET DATA-DRIVEN WEATHER PREDICTION
Recent advancements in weather prediction have introduced hybrid approaches that integrate phys- ical laws with machine learning. For example, Physics-Informed Neural Networks (PINNs) (Cai al. 2021 Cai & Wang 2024) incorporate differential equations into the training process to en- force physical realism. These methods reduce dependency on large datasets and computational resources, ensure predictions adhere known physical laws, and enhance robustness in complex environments.
notable example of this approach is DeepPhysiNet (Li et al. 2024), developed by W. Li et al. This model combines physics-guides guided machine learning with weather prediction by constructing physics networks based on multilayer perceptrons for meteorological variables. Partial Differential Equations are incorporated as part of the loss function, while hyper-network based on deep learning directly learns weather patterns, contributing to the weights of the physics networks. This hybrid design ensures both physical consistency and the ability capture intricate weather patterns.
A.3 EXTREME WEATHER PREDICTION
In this paper, extreme weather refers to the outlying values of specific weather properties. For exam- plc, wind power plants require precise predictions of wind speeds at turbine locations. particularly the extreme values of wind speeds (Sabzehgar al. 2020p. According Waan al. 2010). when wind speeds exceed the cut-out speed, wind turbines cease operation to prevent damage. As such, accurate regional wind speed forecasting, especially for extreme wind speeds, critical for optimiz- ing wind power generation.
Recent advancements in machine learning have significantly improved predictive capabilities for extreme weather conditions. For instance, Fuxi Extreme, developed by X. Zhong et al. (Zhong) al. 2024), leverages Denoising Diffusion Probabilistic Model (DDPM) to enhance accuracy and detail in extreme weather predictions. This model combines base weather prediction framework with DDPM, capturing fine-scale features through a two-step process: adding noise in a forward step and refining details in a reverse denoising step. This innovative approach has demonstrated ex- ceptional accuracy and detail restoration, making it highly effective for forecasting extreme weather conditions.
B ANALYSIS ON THE IMPACT OF FREQUENCY ON LOSS FUNCTION
In this section, we study how the frequency of the transforedd data affects the MSE loss function. Suppose discretize the 2-dimensional region by N x M points (corresponding to
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longitudinal and latitudinal directions, respectively). Then the discrete Fourier transform of the wind velocity field u on the region is given by:
N- M-1
- Agp + *x²)
ú(kes kyy) = F(u) = Σ
Σ
()
==
where is the Fourier operator, h = (kg. kyy Kul simplicity, we let (x, y) = e 2m [x-= 37 can be written by:
are the mode index in the frequency domain. For denote the Fourier basis. Then the inverse transform
1 NM Σ [((K)ox
u(x, v)
Suppose we have prediction of denoted by ú. The mean square (MSE) of the prediction is 1
given by MSE(U, u) NM Eay Iu ú/² By the discrete orthogonality of the Fourier basis Φx s, we have Ez.y OR N N. Therefore, we can derive the equivalent form of the MSE in the frequency domain:
MSE(U, ū)
Σ 2,9
lu
úi2
NM
(((k)) - i(x:)) on (2r, y)
NM
NM
3.9
1
((()) - ū(()) (ū((')) û(()) (x, y) ow (0,z)
NM
(
2,9
((k)) - ū(k)) (a(k') -(0)))(())))
Σ
or (z,, v) ox (x,
(N
k'
N.
M 62.
(2 Σ
Now, let's analyze the potential impact of frequency on MSE. Statistically, the prediction of the wind velocity field can be modeled as translation of the ground truth plus white noise:
ū(x) = u(x + A) + e(x).
where v): A (A.. (g); ~ N/.00. oo) is Gaussian white noise that may may not be invariant with respect to x. should be clarified that the real prediction scenario can be much more complicated than this formula. However, it will be extremely difficult even impossible to study the real cases detail. Besides, this simplification is enough to show some insights about the
impact of frequency on the loss function. denote 0z = 2n transform of the prediction is:
K.
then the Fourier 17
ū(k) e0 ū(k) + =(k).
Therefore, the mean square error of the prediction is:
MSE(U, ú) (NM)² Σ (1 thh E(K)
Since e is Gaussian white noise, we have [[E] = 0 and B(E(k)E(K)) = 2M Therefore, the expectation of the MSE will be:
1 111- -(0))))))) 21ū(A))² +
E[MSB(u, ū)]
(NM)²
Σ 4sin (c) -1((k))2 +
(NM)2
2
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Suppose the magnitude of the translation |A] is sufficiently small such that 0g. < 1. Then we approximate the ( term by 8/2:
EIMEE] R
(NM)² Σ 6 [(())) +
|k| +o²
where constant.
According to the above formula, we can separate the MSE into two parts: one is attributed to the translation of the overall wind velocity field, and the other is attributed to the essential error from the white noise. The error from translation is approximately proportional to the square of the magnitude of the frequency.
The above analysis provides us with insights that the high-frequency data is more likely to affect the MSE.
C PHYSICAL INTERPRETATION OF THE GRADIENT- PENALIZED Loss: ENERGY-ENSTROPHY TRADE-OFF
C.I ENSTROPHY AND ITS RELATION TO THE SQUARED GRADIENT NORM
In fluid dynamics, vorticity and enstrophy are often used to characterize and quantify the rotation and turbulent behavior of fluid. The former definition of them is shown follows.
Definition 3 (Vorticity and Enstrophy). For an incompressible velocity field u(u1, u2) : - IR3, the vorticity is defined to be the curl of the fluid velocity, mathematically,
w(I,t)= = (V x (I 1) Ox, u2(2,, L) Ong ui (zr, D).
The enstrophy & of the flow defined as the 1.2-norm of the vorticity,
E(u(t)) = [wo 6)1 dx
V u(x, dr.
(14)
According to Foias et al. (2001), for the incompressible Navier-Stoker equations, there exist an im- portant connection between enstrophy and the norm of the velocity field, illustrated by the following theorem.
Theorem (Enstrophy of 2D incompressible flow), ueru= (u1, u2) sufficiently smooth veloc- ity field solving the 21) incompressible Navier-Stors Stokes equations domain Ω. Then the enstrophy of the velocity field is equal to its I?2 gradient energy, or mathematically
E(u(()) / Vu((,, 1)12
(15)
Since the book (Foias et al. 2001) didn't provide the full justification of the theorem, here we prove it in details.
Proof. We first express | terms of the components of Vu. By definition,
w Ōz, Or,21, u1,
so that
| = (0z, U2 - 0z, 21) (0z, 2z) + u1) 2 pz, un Oz
On the other hand, the squared Frobenius norm the gradient
Vulle Σ uz)= (0r, (0r, w1)² 2 (O,, u2)² + (02, 22)
111=1
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A short algebraic rearrangement shows that,
w Σ (2z, ui)
Σ dz. uy Oz u4
(16)
(Indeed, expanding the right-hand side and cancelling terms yields exactly the expression for [w| above.) Integrating equation 16 over & gives
J W/l
|u|2==== dz
/
Σ i,f=1
(0x, u:) dx /
Σ
Oz ug On us dr.
(17)
i,j=1
We show that the second integral on the right-hand side vanishes by integration by parts and incompressibility. For clear notation we denote
= Σ J L.j=1
Oz, u O2y uz dx.
Using integration by parts in the varvaiieble and the assumption that boundary terms vanish, obtain
2
I=
Σ
Ug Uy Oz u: dr
ij==1
Σ
Uj Oz Oz, us dr.
1,j=1
Since partial derivatives commute, 0z 0z = Oz, Sz. we may rewrite this as
1 Σ 1 u Oz, Σ
0z, Ui dr.
By incompressibility, Σ 0z,u, V u 0, and hence 0.y (V u) = 0 for j = 1,2. It follows
that the integrand in the last expression vanishes identically, and therefore
= O.
Returning to equation
17 we conclude that
J ||| dx = /
Σ b===1
2
(Dx, u.)² dx = / || /u
This shows that, for 2D incompressible flows under the stated boundary conditions, the enstrophy defined via the vorticity coincides with the gradient energy of the velocity field.
which is precisely equation 15
C.2 ENERGY BALANCING FOR THE 2D INCOMPRESSIBLE NAVIER-STOKES EQUATIONS
We consider the incompressible Navier-Stokes equations on a bounded domain C R2, with viscosity 0 and density p > 0. As comvention in PDE analysis, combine the NS equation with the continuity equation, which stands for the incompressibility of the fluid system. We have
dn Br
u
Vu Vu- -V Ω 1>
(18)
Q
>
where u(x, t) E ? the velocity field, P(x,t) t) E R the pressure, and F(x, t) E R an external forcing (in context, represented by the neural network). We may also assume either periodic
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boundary conditions no slip boundary conditions ulan 0, so that all boundary terms arising from integration by parts vanish.
We denote the L inner product and norm follows:
(a. b) = / a(x) b(xr) da,
|| J [u(r){ dr.
Following the classical energy method dealing with PDEs, we take the inner product of equation 18 with u over 92 which yields
(u u) -(u Vju. uj - (VP. +v + (F,
(19)
We now treat each term in equation 19 separately.
Time derivative term. tion, we have
Assuming sufficient regularity to interchange differentiation and integra-
[u())]1 / |u(r, t)l² dx,
and thus
|lu(t) = / (,((((²) dx
2u u.u = 2(0,u, u).
dt
S
Hence
1
(0.u, u) = d [tt]
(20)
Advection term.
Write u (u1, U2) and note that (u V)u (ur un u1, u1 u2 + uzd u2).
Then
((u V)u, u3 / (u- udæ
2
2
uu v V)u u
ur us u,)u; dr.
1=1
Notice that uzO2e u4 10, (u, )
1
1
u,O., U,
02g
ui
0ry |u
2
2
i=1
Therefore,
2
((u V)u,u)
/
Σ
Uj 2
0z |u] dr
32 j=1
2 2 u v([u| 2 dz.
./
Using the identity
u v((u() = vu(ul|(3 - |u|2(v u),
we obtain
2 / v (ulul?) de / | (
(u V)u, u)
u) da.
Since V u = 0, the second term vanishes. Applying the divergence theorem to the first term gives
[ v (ulul?) dx f [u|2(u - dS,
where n is the outward unit normal on 0Q. Under periodic boundary conditions, this boundary integral cancels out; under no-slip conditions, case,
on 0n so the integrand vanishes. In either
(uu Vuu, u) = 0.
(21)
Thus the convection term does not contribute the evolution of the energy.
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Viscous term. We now consider the viscous term
VV u) / V udæ
V u; Ui dr.
Since Viu V (Vu.), integration by parts yields
1 2 us dx = / V (Vus) us dx = Jaa
/
(Ō, (8nui)us dS
[uuilaa dx.
12
12
Therefore,
//
VVV u = v >)
(dnu;)ud
Vuil dr
i=1
/ 52
Vu,
2r
dr +
Σ
/
(Onus)us dS.
==1 85
As before, under periodic boundary conditions the boundary integral vanishes by periodicity; under no-slip conditions u, = 0 on 02, so the boundary term again zero. Thus
u. u) -- Σ / Vul - / Vu dr - -ul|V
(22)
This term represents viscous dissipation of kinetic energy.
Pressure term. For the pressure term we use integration by parts:
/ VP udr = _ / P(V u) dz + / P(u n) ds.
(VP.u) p n
an
Since 0, the volume integral vanishes. Under periodic boundary conditions, the boundary integral cancels; under no-slip conditions = 0 on M,, so n 0 and the boundary integral is zero. Consequently,
VVP.u u) = =0.
(23)
Substituting equation following theorem.
20 equation
21
equation
22 and equation
23 into equation
19
we obtain the
Theorem 4 (Energy balance for NS equation). Ler Ω C R2 be a bounded domain with smooth boundary, u be velocity field defined on Ω satisfying the imcompressible Navier-Stokes equation 78 with corresponding boundary conditions. Then the velocity field u satisfies the energy balance equality
1 [1(0))))))) + Vu] = uF(),
(24)
2 dt
for all t in the interval of existence.
Physical interpretation. Equation equation 24 can be interpreted as an exact balance of kinetic energy (per unit density) for the incompressible flow:
1 d
(
(F(), u())
dt.
viscous dissipation
power input by forcing
change of energy
The first term measures how fast the total kinetic energy E({) |lu(()) changes in time. The second term is nonnegative and represents viscous dissipation: it is proportional to ||Vul which in two dimensions is equivalent (up to constants) to the enstrophy |/ Xull 1 and therefore quantifies the strength of small-scale shear and vortical structures. The right-hand side (F, u) is the rate at which the external forcing F does work on the flow, injecting kinetic energy into the system. In our setting, F is represented by the neural network and thus encodes the part of the dynamics that is not explicitly captured by the analytic Navier-Stoker operator.
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C.3 PREQUENCY-DOMAR PERSPECTIVES OF ENERGY AND ENSTROPHY
We now consider the energy balance in frequency domain. This makes explicit how kinetic energy and enstrophy are distributed across spatial scales, and will later be used interpret the purposed gradient-regularized loss.
Let Et) and Et) denote the total kinetic energy and enstrophym, respectively.
( = 1 [l(())
E(t) = [V
According to Rayleigh's 's theorem (Temple et al. 2004 illustrated in appendix B these quantities can be written in terms of the Fourier coefficients of u:
(tt)
ū(kk t)1²,
E(t)
Σ keZ
11 ū(k, t)1²
kEZ
For isotropic flows, it is convenient to introduce a one one mic-dimesiionl energy spectrum El,, such that
E(t) = 2 2 t(k, dk,
E(t) =(t)= f. k2 E²E(,, dk.
(25)
Given this representation, yielded by the same spectra {E(k, h) the energy is an unweighted integral, while the enstrophy includes k.2 weight that emphasizes high frequency.
may consider the energy and enstrophy follows: they both
Analogizing the analysis of theorem to spectral energy, we obtain the following theorem in the frequency domain.
Theorem 5 (Frequency-domall energy balance). Let C R² be a bounded domain with smooth boundary, u be a velocity field defined on satisfying the imcompressible Navier-Stokes equa- tion 18 with corresponding boundary conditions. Let E(k, t) be the isotropic energy spectrum de- fined above, and let G(k, 1) denote rute of energy injection by at frequency k. Then the total kinetic energy E(L) sutisfies the frequency- energy balance equation
2²2 E2 1) dk dk Jo 00 E(t) = -2v 1 ह.
d
G(k, f) dk.
(26)
Proof. Taking the spatial Fourier transform of the Navicr- equations yields
0,ū(k, -v|k/2((kk t) (lk, F(k,t),
(27)
where (k: t) collects the contributions of the advection and pressure terms. Because the flow is incompressible, the pressure term acts as an orthogonal projection onto divergence-free modes and does not contribute directly the energy balance.
Taking the inner product in C² of equation 27 with ū(k, t) and then keeping the real part (denoted as R())), we obtain spectrum energy balance:
1 d ū(k, 1 R(a, ((,,ū(k))) ū(k, d
(kk ((V( t(k,t) + ((( i(k,)))
(28)
The nonlinear term [kk t) redistributes energy among spectra. However, when summed over all k its contribution cancels due the skew-symmetry of the advection operator, which is the spectral counterpart of ((u V)u, = 0 in physical space. Therefore,
Σ E ((((k) t(k,t)) 0.
Summing equation 28 "x ūkk, 1 and Ett
over lo and using Rayleigh's energy theorem to identify E(t) = Σx [ ? we obtain
d
Jl E(t = -
Σ R(F(()t)--((K kEZ
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Passing from the discrete sum over l G(k,t) t) yields equation 26 with
Z to the isotropic E(k,t) and power injection density
{()) !. he2 E(k, dk,
[ G(k,l) Σ (F((,))) ()):
kEZ
This completes the proof.
Special case without external force. The simple unforced case F already offers an intuitive viewpoint on why. data-driven predictors are particularly prone to amplitude shrinkage when a flow 's energy is heavily concentrated in tigh-freeuency spectra, as implied by equation the frequency-domain balance equation 26) reduces to
such cases,
d
F(() 22 dt
k² E(KK dk -1)
(29)
Using equation 25 we now introduce an mnergy-weighted average frequency
k2 E(K 1) dk
{())
(1)
(30)
Et)
E(k, t) dk
Then equation 29 may be written as the scalar ODE
E(1) 2 E(I).
(31)
This linear ODE has the exact solution
E(t) E()) exp
2y (s)(s) ds
(32)
(() exp(
2 t),
the average frequency (t) can be regarded as approximately constant (for instance, in a statis- tically steady regime).
Equation equation 32 highlights the joint role of energy and enstrophy in controlling the evolution of the flow: the decay rate of Ft(t) is governed by the cumulative integral of k2 (f), i.e. by how strongly the energy is biased toward small or large scales. When the spectrum is concentrated at large frequency (large (t)))), kinetic energy decays rapidly; when most energy sits large scales (small (t)), the decay is much slower. These results consist with the implication of equation 6 in the main context.
C.4 GRADIENT-PENALIIE LOSS AN ENERGY-ENSTROPHY TRADE-OFF
We finally interpret our consider the continuous version of loss function help mitigate the amplitude shrinkage. We
loss function 7 on the domain Q:
Cgp (ū, u)) J [(x)) u(x)]l2 dx + X ||Vūli: - ||
(33)
where is the prediction of the ground-truth u, and > is a tunable hyperparameter. For the vector fields Vū and Vu, have
/ [Vū(z)] dr,
Vu i / ||Vu())]] dr.
Vullz=
Enstrophy matching. velocity field v on Ω
By Theorem
we know for a sufficiently regular and divergence-free 2D
[ [IV(()) de =/ (w)
E(v)
where wiy V denotes the vorticity and E(-) denotes the enstrophy. Consequently, the second
term in equation 33 can be interpreted as global enstrophy matching constraint:
| Vull f.2
[[ 8()),,
The loss Egp therefore penalizes predictions whose overall vorticity and shear intensity is signifi- cantly weaker stronger than that of the reference flow.
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MSE average energy of the error field. eet ū(x)) u(x) the MSE term in equation 33 can be written as
denote the error field. Then
MSE(ū, u) [|(())]] dr - ||
This quantity is naturally interpreted the average kinetic energy of the enror field. Therefore, the original gradient-penaliin loss function can be interpreted from energy perspective as follows:
/ |Í(()) u()) de
Eg Ū, u) =
(34)
energy matching (MSE)
castrophy matching
where the "energy matching" means reducing the energy of the enror field.
The MSE measures how large the velocity error in an L sense, without distinguishing across spatial scales. In contrast, the enstrophy weights regions with strong gradients and vorticity much more heavily, and is therefore especially sensitive shear layers, fronts, and small-scale vortices.
Why enstrophy matching mitigates amplitude shrinkage. We first conclude the two perspective of the energy-enstrophy connection analyzed before.
1. By energy balance of the 2D incompressible NS equation
dl [u()]]? + w/l vu((il) = (F((), u(t))
we know that the enstrophy governs the viscous dissipation kinetic energy.
2. By the spectral interpretation of energy and enstrophy 25
1 EEt) = f E(k, dk,
E()) - |0 k2 E2 dk,
we know that the enstrophy is biased average of spectral energies leaning towards high- frequency spectra.
The above two perspectives jointly provide the properties of enstrophy. A high enstrophy corre- sponds to strong small-scale activity and rapid energy decay, while low enstrophy corresponds to smoother flows and weaker dissipation. Moreover, enstrophy can be regarded as a measure of "how strongly the flow is twisted and sheared". Since enstrophy weights regions with strong gradients and vorticity much more heavily, it especially sensitive to shear layers, fronts, and small-scale vortices- the common representation of extreme winds.
In our extreme wind prediction setting, amplitude shrinkage manifests precisely in these high- gradient regions: the model tends to underestimate peak wind speeds and over-smooth sharp struc- tures. If training uses only the MSE term, such behavior can be relatively cheap in terms of the objective: the extreme regions typically occupy a small fraction of the domain Q2 so the error intro- duced by flattening sharp peaks may not dominate the global 2 norm |/e|| As result, model that systematically reduces gradients and peak amplitudes can still obtain a low MSE, leading to pronounced amplitude shrinkage in practice.
The enstrophy matching term equation 13 directly counteracts this tendency. Since the global enstrophy Vull2= 2 is concentrated in regions where || [Vu(z)] is large, any systematic smoothing of extreme events and fronts immediately drives [ significantly below / This leads to large penalty in the regularization term.
Therefore, by using the gradient penalized loss, the model is encouraged both to:
maintain accurate average energy of the velocity field via the MSE term, and
restore sufficient gradient and vorticity strength high-impact regions so that the total enstrophy [ remains comparable //Vul
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In other words, the loss Cep enforces a trade-off between matching the energy of the flow and matching its enstrophy. This trade-off explicitly discourages solutions that achieve small 1,2 error by globally smoothing sharp structures and suppressing extremes. Instead, biases training to- ward predictions that preserve the correct overall intensity of shear and vorticity, thereby mitigating amplitude shrinkage and yielding more faithful reconstructions of extreme wind events.
D CONVERGENCE ANALYSIS OF THE GRADIENT-PENALIZE Loss MODEL
In this section, we provide convergence analysis sketch for the simplified model with the gradient- penalized loss function shown in equation 7 The main goal is to explain why excessively large values of over-amplify high-reeuency modes and can destabilize discrete gradient-basse badeent-based opti- mization, as shown in figure 2
Recall the gradient-penalinnn loss
(ú, ( u) = MSE(Ú, u) + X 1VV|| |||l|
(6 revisited)
where ū is the predicted wind field at given lead time and is the ground truth. As discussed
in Sec 3.1 in the typical amplitude-shrinkage regime we have | < [/Vul? in this case, minimizing the original loss function is approximately equivalent to minimizing the following:
Egp (ū, u) MSE(ū, u) A Vūl
(8 revisited)
which means that, up to an additive constant independent of ü, the gradient term contributes negative Sobolev-type penalty on Vūl We also recall that
Vull 2
Σ |A: [()))
where ū(k) denotes the discrete Fourier transform of ū and (key ky) is the (double) frequency index.
According to Appendix B the MSE can be equivalently written (up positive constant factor) in the frequency domain:
MSF(Ú, u)
-(((())
(35)
Therefore, by absorbing positive constants into À for notational simplicity, we obtain the following normalized local surrogate for Lgpi
uū
- a(+) - |||| [
(36)
Thus the loss is decomposed into sum of independent vector-valued quadratics over frequency spectra.
For the clarity of notations, we now denote Ps := .(kk) and qı := ú(k) E Rd for every frequency spectrum k. Then the contribution each spectrum & to Egp will be
E (px) | - 101l A || |
(37)
D.1
VECTOR HESSIAN FIGENVALUES ANALYSIS
We now compute the gradient and Hessian of & with respect to P Expanding the squared norms in equation 37 gives
(px)) = P Pa - 29 D + qr q - A||A1/ P Pa.
Differentiating with respect to Pı. yields
Pp l 2p: 2q - 2X| 2[(1 >1]b))))p qk)
Differentiate once again, we get the Hessian of Ex:
2
la 2(1 TT
(38)
Da
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where La is the identity matrix. Thus, all d eigenvalues of the Hessian block for mode & are equal to
0, if A||+ 1,
(x) = 2(1 2(1-~|
= O, A| = 1,
(39)
< O, if X| > 1.
The above results imply the theoretical convexity of the simplified gradient-penalized loss function. For low frequencies (small |[ and moderate X, we have hy (A) > and the loss remains locally convex in those directions. However, for sufficiently large A, modes with \|| > 1 exhibit nega- tive curvature and the local quadratic approximation becomes a saddle (or even local maximum) along those fighffreguency directions. Practically, the extremely high-frequ nigh-frequency components of the data are negligible. Therefore, we may assume that the frequency support is bounded by some con- stant maxx i.e. |||||| s K Kmxx. In these cases, sufficient condition for (block-diagonal) positive definiteness of the Hessian of LEp is
1 - -K- max 0
(40)
K2 max
D.2 GRADIENT DESCENT DYNAMICS AND INSTABILITY FOR LARGE À
We now analyze vanilla gradient descent dynamics on the per-spectrum objective function (<(:)) with step size 1) > 0. The update rule is
P& P (p
(++1) (c)
(41)
(t) (8)
Suppose there exists stationary point P for &x. Defining the error eg Pk following lemma:
P we have the
Lemma (Spectrum stability condition). Under the update rule 41 the error only if
e converges to zero
< n11 - 21116113) <1
(42)
or equivalently,
21 >0. 0,
(43)
0 <<ク< 1- \||1112
Proof. Plugging the gradient of the spectral loss function Vp Ls into 41
we have
(++1) Pk P V la (pk))
(2) -2n((1 A qk
(44)
Therefore, stationary point P should satisfy
(1 >||B|)))p - q =
P = 1 - A|| 9k-
provided 2|| + 0. Therefore, the recurrence formula of cf is
((+1) Ta 222(11 X1 e pk pk (2, e
(45)
with the linear recurrence scaling factor
Pa (,(2,)) n) = 1 (A) = - 2n - -
Since the update matrix is a scalar multiple of L, each component of e follows the same one- dimensional recurrence. The recurrence formula 45 converges only if Ipl 1, which is equivalent
to
-1 < 1 2n(1 \|||1))
- 2n -1 2,
which finishes the proof.
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Interpretation. Lemma reveals two key effects of À in the vector-valued Fourier setting:
(i) Theoretical upper-bound X. For any spectrum & satisfying \|kk|| 2 2 1, we have 1 1- >||a|| I 0 and no choice of 1 > 0 can satisfy 0 ny(1 case the per-mode quadratic is flat or concave and gradient descent cannot converge to a local minimum along that high-frequency direction. The above analysis suggests that there exist a theoretical upper bound Amax Σ / K2 max: where K Kmax denotes a high-frequency cutoff chosen so that the majority of the spectral energy is concentrated in spectra with ||:]| < maxx- If À is chosen larger than the upper-bound, there is no guarantee that the training will converge, As necessary condition for À and the learning rate should be
X | < 1. In this
A A Amax
and < < 1 AK?
(46)
K2
max
max
As a result, the performance as a function of À naturally exhibits a U-shaped curve, as illus- trated by figure 2 small to moderate À improves the fit by correcting amplitude shrinkage, while overly large À will degrade or even unstabilize the optimization and affect final ac- curacy.
(ii) Hrigufrryeenccy spectra suffers more. Given that À is fixed, the effective spectral- curvature (A) = 2(1 - A {||
in more likely to become negative for high-frree geh-frequency spectra. Therefore, is À is not chosen properly, it's possible to witness the instability of training in regions with large energy of higher-frequene spectra, such as turbulent regions.
Remark. This convergence sketch is based on simplified model of the gradient-penalinee loss rather than the full nonlinear network and training pipeline. Although real world optimization dy- namics more complex, the above analysis reveals the key structural effect of our method, and provide the intuitive insights of the U-shape performance curve with respect to X, in line with the empirical experiments.
E DATASETS AND EVALUATION METRICS
E. ERA5 DATASET
We mainly used the ERA5 datasets for our model training and testing processes. The ERAS datasets(Hersbach et al., 2020), developed by the European Centre for Medium-Range Weather Forecasts (ECMWF), is a fifth fitth-generation reanalysis of the climate and weather covering data from 1940 to the present. Although the datasets contain detailed reanalysis data globally, it provides flex- ibility to select and obtain data in rectangular spatial region in different scales and locations. There- fore, the datasets is suitable for studying is created through data assimilation, which combines model data with observations from various sources worldwide, resulting in a globally consistent and comprehensive datasets. ERA5 provides hourly estimates for wide range of atmospheric, ocean-wave, and land-surface variables, including uncertainty estimates using a 10-member ensemble three-hour intervals. The data is available on regular latitude-longitude grid, with horizontal resolution of 0.25 x 0.25 for atmospheric reanalysis. The temporal resolution of ERA5 is hourly, and the data is accessible in GRIB format, providing high-resolution information for many climate and weather applications.
work on regional weather prediction. This datasets
In this study, we focus on specific variables from the ERA5 dataset relevant to wind speed prediction, namely the 10-meter wind components and surface pressure. The 10-meter u-component of wind represents the eastward component of horizontal wind speed at 10 meters above ground level, while the v-component represents the northward component at the same height. These components are measured in meters per second (m/s) and can be combined to calculate the speed and direction of the horizontal wind. Surface pressure, given in Pascals (Pa), is the atmospheric pressure at the Earth's 's surface, which reflects the weight of the air column above a specific point. These parameters together provide essential information for modeling and predicting wind dynamics in the atmosphere.
E.2 DETAILS OF EVALUATION METRICS
Root Mean Squared Error (RMSE) The RMSE quantifies the overall accuracy of the predicted wind velocity field by measuring the difference between the predicted and ground truth values. It is
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defined as:
1
RMSE
i. u, 11
where:
ii and u, are the predicted and ground truth wind velocity vectors the i-th grid point,
N is the total number of grid points.
Extreme Region Error (EX-RMSE) The Extreme Region Error (Ex-RMSE) focuses on the model's accuracy in predicting extreme weather regions, characterized by high wind velocities. assigns larger weights to regions with extreme wind values to emphasize their importance. Mathe- matically, defined as:
E wi
û; u: Ei=1
ExtremeErr
N
Wi
where:
w; is the weight assigned to the i-th grid point, with higher values for extreme wind velocity regions,
ú; and u, are the predicted and ground truth wind velocity vectors at the i-th grid point.
These metries collectively assess the model's accuracy, adherence to physical principles, and capa- bility to predict extreme weather conditions effectively.
F ADDITIONAL EXPERIMENTS
F.1 IMPACT OF FREQUENCY ON TEMPORAL DATA
Short-term
Long-term
High- feah-ffrquencc Component Impact 1% of total dynamics)
40
30
20
10
Prediction Horizon (hours)
Figure 4: Impact of high-frequency components wind field dynamics decreases prediction horizon extends from to 21 hours.
We conducted experiments to investigate how high- and low- wwwffreuency components of wind data contribute to future dynamics across different temporal scales. Using Fourier filtering techniques
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(detailed in Section 4), we decomposed the two-dimensional wind velocity field time series into their respective frequency components.
Our correlation analysis examined the relationship between these decomposed components and tual future wind patterns across prediction horizons ranging from 1 to 24 hours. The results, il- lustrated in Figure I, reveal clear temporal dependency pattern: For longer prediction horizons (approaching 24 hours), low-frequency components demonstrate dominant predictive power in wind speed pattern evolution. Conversely, at shorter intervals (approaching hour), high-frequency com- ponents become increasingly significant in determining wind pattern changes.
This observation be theoretically explained as follows. Suppose u(a,, y) is the wind velocity field. Then the Fourier transform of the spatial gradient of wind velocity Vu is:
Vū(k) = ik /f u(z, y)e +2m(kzz+ Rgg) dr dy,
where & = (ky, represents frequency domain coordinates and denotes the Fourier transform. This relationship demonstrates that higher frequencies (larger (k:) correspond larger spatial gradi- ents Vul).. Consequently, high-frequener components capture small-scale features characterized by sharp gradients and abrupt changes in the wind velocity cold-characteristiis typically associated with turbulence and extreme weather events.
F.2 DIFFERENT FREQUENCY MASKING LEVEL
In this subsection, we investigate how different frequency masking levels affect the model's wind velocity prediction performance using the Fourier Frequency Filter. The results show that exces- sively high or low masking thresholds degrade accuracy, while optimal performance is achieved at intermediate levels, where balance between high- and low-frequency information is maintained.
To explore this, we varied the threshold for dividing high- and low-frequency components and an- alyzed its effect wind speed prediction accuracy. Experiments masking levels of 0.1, 0.3, 0.5, 0.7, and 0.9, which represent the proportion of the highest frequen- cies ineluded in the high-frequency data. These experiments were performed wind velocity field data from four distinct regions, with the results summarized in Figure
conducted using frequency
Regions
EA
EU
RMSE
61
6.1
0.
29
Frequency Masking Level
Figure 5: Impact of high-frequency components on wind field dynamics decreases as prediction horizon extends from 21 hours.
The results demonstrate that both excessively high and excessively low-frequency masking thresh- olds negatively impact the model's prediction accuracy. When the masking level is too high, critical low-frequency information excluded, leading to incomplete data representation. Conversely, when the masking level is too low, significant high-frequency details are overlooked, impairing the model's ability to capture rapid variations in wind speed. Optimal prediction performance is achieved when the frequency masking level lies between 0.3 and 0.7, this range effectively balances the inclusion of high- and low-frequent frequency information, enabling the model to better capture both large-scale and small-scale dynamics.
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Ground Truth
Pred: Baseline
Pred: Our Model
Prediction Scale
Region: East Asia
Extreme Region
Err: Baseline
Our
2.0
15
Scale
Ground Truth
Pred: Baseline
Pred: Our Model
Prediction Stale
Region: North America
Extreme Region
Err: Baseline
Er: Our Model
13
-1a
Error
Ground Truth
Pred: Baseline
Pred: Our Model
Prediction Scale
Region: Austria
Extreme Region
Err: Baseline
Our Model
-15
Errrr Scale
Figure 6: Results of cross-regional experiments.
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Table 3: Comparative error results across regions and models.
Lead time: 1h
Lead time: 3h
Lead time: 5h
Region
Model
RMSE
Ex-RMSE
RMSE
Ex-RMSE
RMSE
Ex-RMSE
CNN
0,9559
0.6011
3.3106
2.2872
5.1008
3,7931
ConvLSTM
0,8356
0,4217
2,4391
1.5702
3,0405
2.5460
EA
PINN
0.8156
0.3983
2.0649
1.4828
2.8327
2.2321
Ours
0.6947
0.3236
1.8657
1.3334
2.5840
1.9037
CNN
0.8309
0,5691
1.9977
1,5617
3.6052
2,4897
ConYLSTM
0.6569
0,4237
1,2097
1.0331
1.7852
1,5028
NA
PINN
0.6630
0,4863
1.3394
1.1201
1.8592
1.6391
Ours
0.6065
0.3872
1.1935
0.9305
1.6862
1.2477
CNN
1.1043
0,6830
2.3372
1,4968
3.0311
2.1449
ConYLSTM
0.8584
0.3292
2.0674
1.0791
2.4728
1,7153
AU
PINN
0.9342
0.4409
2.2171
1.0831
2.3327
1,5570
Ours
0.7411
0.3244
1.8853
0.8901
2.0521
1.1371
F.3
CROSS REGIONAL EXPERIMENTS
To further evaluate the robustness and generalizability of our model, we conducted additional ex- periments using datasets from different geographical regions. The experimental setup. including the fundamental parameters and evaluation metrics, remained consistent with those described in the main text to ensure comparability.
Among the various results obtained, we present the most representative findings in Table 3 and Fig- ure 6 these experiments, EA, NA, and AU correspond to datasets from East Asia, North Amer- ica, and Australia, respectively. The results consistently demonstrate that our model outperforms the baseline and comparative methods in predicting extreme wind speeds across all tested regions. Specifically, our model achieves the lowest RMSE while maintaining high accuracy in capturing the most significant variations in wind speed. Moreover, the improvements are particularly evident in regions with frequent extreme weather events, further validating the effectiveness of approach in handling complex wind dynamics.
29


Reviews:
[{"review_id": "jto8Ooai6m", "reviewer": "Reviewer_SvAa", "summary": "This paper investigates the underestimation of extreme wind speeds in data-driven weather forecasting models, a persistent issue in both academic research and industrial applications such as wind power management. The authors propose a frequency-aware learning framework that integrates theoretical analysis in the Fourier domain with innovative model and loss function design.\n\nThe study first demonstrates through analytical proof that traditional mean squared error training induces frequency-dependent amplitude decay, where high-frequency components are systematically weakened due to spatial translation errors. To address this, the authors propose:\n(1) A gradient penalty loss function to enhance sensitivity to amplitude errors and mitigate high-frequency signal attenuation.\n(2) A physically embedded neural network architecture: Employing a simplified Navier-Stokes core, it integrates physically grounded convection, diffusion, and pressure modules with a learnable “volume force” network.\n(3) Band separation and weight rebalancing mechanism: Input signals are decomposed into high- and low-frequency components, processed separately via Fourier filters, and equipped with time-domain attention mechanisms for each band.\n\nExperiments on the ERA5 meteorological reanalysis dataset demonstrate that this model consistently outperforms CNN, ConvLSTM, and PINN benchmark models in RMSE metrics, achieving significantly improved accuracy in extreme wind regions. Analysis further confirms the stabilizing effect of gradient penalties and the optimal trade-off relationship regulated by their coefficient λ.", "questions": "（1）How does the proposed framework perform under longer-term forecasts or coarse-resolution settings where statistical noise dominates over high-frequency content?\n（2）Can the gradient-penalized loss lead to overfitting sharp gradients or instability in turbulent regions, and how is λ chosen or adapted dynamically during training?\n（3）How generalizable is the physics-embedded structure? Could it extend effectively to 3D atmospheric models or other variables (temperature, humidity) without major redesigns?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "(1) Strong theoretical insight and motivation: The paper provides a clear Fourier-domain error decomposition explaining why high-frequency components are underestimated by standard MSE. The analytical backing is mathematically sound and bridges physical intuition and machine-learning loss design, a commendable improvement over purely empirical approaches in weather forecasting.\n(2) Innovative frequency-informed hybrid framework: The integration of gradient-penalized loss, Navier–Stokes-based backbone, and frequency-domain reweighting demonstrates an elegant blend of physics knowledge and deep learning. The model architecture and spectral treatment are well-motivated, providing an interpretable mechanism for addressing extreme-event underestimation.\n(3) Comprehensive experimental design and convincing results: Clear comparisons with strong baselines such as CNN, ConvLSTM and PINN. Analysis of the λ hyperparameter and frequency masking ablations demonstrates methodological robustness. Results on both overall and extreme-attentive errors substantiate the theoretical claims.", "weaknesses": "(1) Limited experimental diversity and scale: The study relies on ERA5 data from selected regions, with no cross-region or cross-time validations to assess generalization ability. Suggestion: Extend evaluations to multiple climate zones and temporal ranges (e.g., 6h, 48h forecasts) to assess robustness and transferability of the frequency-informed model.\n(2) Computational overhead and implementation detail gaps: The physics-embedded backbone and frequency separation introduce heavy computation during both training and inference. Quantitative data on training time, convergence speed, or complexity trade-offs are missing. Suggestion: Provide complexity analysis versus baseline models, and discuss deployment feasibility for operational forecasting.\n(3) Limited connection between frequency-domain theory and physical interpretability: The gradient-penalized term and frequency reweighting are theoretically justified, but it remains unclear how these modifications alter learned spectra or physical consistency over time.\nSuggestion: Include frequency-spectrum visualizations pre- and post-training, or energy distribution comparisons to ground truth, to confirm the mitigation of amplitude shrinkage empirically.", "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": "mZitu7efHB", "reviewer": "Reviewer_Koi8", "summary": "This paper introduces an improved physics-informed approach for extreme wind prediction.", "questions": "- Please give a comprehensive introduction of your sampled data, such as how many hours in total, across which regions, etc.\n- The frequency masking level seems to be a critical hyperparameter. Did you find an optimal masking threshold to be consistent across different geographic regions and weather regimes, or does it require case-specific tuning? Is there a potential to make this threshold learnable?\n- Besides, the forecasting horizon and lookback length could be importance factors to see the robustness of the proposed approach.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- An interesting theoretical view of frequency-domain error behaviors to enhance the model design for extreme wind prediction\n- Significantly improved performance demonstrated in experiments", "weaknesses": "- Limited domain: the proposed method is only applicable to extreme wind prediction\n- Limited evaluation: the experiments are based on sampled data, and the forecasting horizon is fixed as one-hour and the lookback includes 23 hours.", "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": "z0mlFham8Y", "reviewer": "Reviewer_AeXV", "summary": "This paper tackles the persistent \"amplitude underestimation\" problem in extreme wind speed prediction. Through a detailed frequency-domain theoretical analysis, it demonstrates that standard data-driven models (trained with MSE loss) systematically underestimate the magnitude and short-term variability of high-frequency wind components. To address this, the authors propose a novel loss function with gradient penalization to mitigate amplitude shrinkage, and design a physics-embedded architecture (leveraging the Navier-Stokes equation) along with frequency separation and reweighting modules. Experiments on ERA5 meteorological datasets show that the proposed method significantly outperforms classic models (such as CNN, ConvLSTM, PINN), especially for extreme wind scenarios, and captures both overall and extreme case prediction more accurately.", "questions": "For gradient penalization's optimization instability at large λ, can the authors provide a detailed theoretical convergence analysis? Is there a risk of uncontrolled high-frequency oscillations? Are there any additional regularization measures?\n\nCan the authors supply full ablation experiments for the Navier-Stokes physics module to quantitatively show its standalone benefit compared to standard neural network baselines?\n\nHow robust is the model when extreme wind samples are extremely scarce or when transferring to new regions? Can the authors add few-shot or cross-region generalization experiments?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The work skillfully combines rigorous frequency-domain analysis (revealing frequency-dependent amplitude shrinkage) with tailored engineering solutions in both loss function and model architecture—providing an effective, innovation-driven answer to a real-world pain point in extreme wind forecasting.", "weaknesses": "The core innovation (gradient-penalized loss) lacks thorough theoretical exploration regarding its effect during optimization. The observed instability with large λ is only empirically described (U-shaped curve), with no detailed theoretical analysis as to why or how excessive λ causes non-convergence, possible oscillation, or even gradient explosion, nor are additional regularization strategies discussed.\n\nAlthough the model's backbone is described as a combination of physics-embedding and neural networks, experiments do not present clear ablation studies to isolate the contribution of the Navier-Stokes module. Only coarse comparisons (\"Ours\" vs \"NS-Op\") are given, so the true benefit of embedding physics remains ambiguous.\n\nExtreme wind samples are inherently rare, but the paper does not clarify how many such cases exist in the data, nor does it analyze the trade-off between extreme case data volume and model robustness. The generalization capability (e.g., performance when transferring to new regions or under few-shot settings) is not systematically evaluated.", "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)