Generalizing Weather Forecast to Fine-grained Temporal Scales via Physics-AI Hybrid Modeling
Authors: Wanghan Xu
Fenghua Ling
Wenlong Zhang
Tao Han
Hao Chen
Wanli Ouyang
LEI BAI
Venue: NeurIPS 2024 poster (2024)
Keywords: weather forecast
physics-AI hybrid model
partial differential equation
machine learning
Primary area: machine_learning_for_physical_sciences

Abstract: Data-driven artificial intelligence (AI) models have made significant advancements in weather forecasting, particularly in medium-range and nowcasting. However, most data-driven weather forecasting models are black-box systems that focus on learning data mapping rather than fine-grained physical evolution in the time dimension. Consequently, the limitations in the temporal scale of datasets prevent these models from forecasting at finer time scales. This paper proposes a physics-AI hybrid model (i.e., WeatherGFT) which generalizes weather forecasts to finer-grained temporal scales beyond training dataset. Specifically, we employ a carefully designed PDE kernel to simulate physical evolution on a small time scale (e.g., 300 seconds) and use a parallel neural networks with a learnable router for bias correction. Furthermore, we introduce a lead time-aware training framework to promote the generalization of the model at different lead times. The weight analysis of physics-AI modules indicates that physics conducts major evolution while AI performs corrections adaptively. Extensive experiments show that WeatherGFT trained on an hourly dataset, effectively generalizes forecasts across multiple time scales, including 30-minute, which is even smaller than the dataset's temporal resolution.

Generalizing Weather Forecast to Fine- Fine-grained Temporal Scales via Physics-AI Hybrid Modeling
Wanghan Xu"
Fenghua Ling
Shanghai Jiao Tong University Shanghai Al Laboratory su_nanghan ju, edu.
Shanghai AI Laboratory lingf wng.gggggga@pjaabb.omm org. cn
Wenlong Zhang
Tao Han
Shanghai AI Laboratory
Shanghai Al Laboratory
changwenlongg@pjlabb. org.
hantao dispatchhhjjab org.
Hao Chen
Wanli Ouyang
Lei Bait
Shanghai AT Laboratory @pjhaao@@pjabb org.cn
Shanghai AT Laboratory ouyangvanli@p) jppjla org.
Shanghai Al Laboratory bailei@pjlab. org.
Abstract
Data-driven artificial intelligence (AI) models have made significant advancements in weather forecasting, particularly in medium-range -range and nowcasting. However, most data-driven weather forecasting models are black-box systems that focus learning data mapping rather than fine-grained physical evolution in the time dimension. Consequently, the limitations in the temporal scale of datasets pre- vent these models from forecasting at finer time scales. This paper proposes physics-AI hybrid model (i.e., WeatherGFT) which Generalizes weather forecasts Finer-grained Temporal scales beyond training dataset. Specifically, we employ a carefully designed PDE kernel to simulate physical evolution on small time scale (e.g., 300 seconds) and use a parallel neural networks with a learnable router for bias correction. Furthermore, we introduce a lead time-aware training framework to promote the generalization of the model at different lead times. The weight analysis of physics-Al AI modules indicates that physics conducts major evolution while Al performs corrections adaptively. Extensive experiments show that WeatherGFT trained on an hourly dataset, effectively generalizes forecasts across multiple time scales, including 30-minute, which even smaller than the dataset's temporal resolution. Code is available at http:/ggihhub.coo/baakk/////////
Introduction
Weather forecasting plays a vital role in modern society, impacting a wide range human activities. For example, minute-level precipitation nowcasting is particularly valuable for short-term planning, such as outdoor activities, while medium-range forecasts that offer daily predictions play a crucial role in long-ten term strategic decisions like maritime trade. This field has witnessed remarkable advancements in recent years, largely attributed to the rapid progress of machine learning-basd based (ML) weather forecasting models (38] spanning from nowcasting medium-range forecasts.
"This work done during his internship Shanghai Artificial Intelligence Laboratory. Conresponding author.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Prior studies tackle the problem of weather forecasting by leveraging data-driven models trained on benchmark weather forecasting datasets, such as WeatherBench [4] and ERA5 [22]] Prevalent medium-range forecasting models (e.g., FourCastNet [2]. GraphCast 33 and FengWu (7) are
commonly trained on the aforementioned hourly datasets to generate global forecasts with a time
interval of 6-hour, can not offer finer predictions like 30-minute forecasts for nowcasting.
significant limitation of current ML-based weather forecasting models 32
3 33
[8] lies in
their black-box training paradigm [53 171, that is, primarily focusing on learning the mapping of data pairs with a fixed lead time (e.g.. hours), without explicitly incorporating the laws of atmospheric dynamics which govern finer-grained grained physical evolution processes. Consequently, this training paradigm brings a significant challenge for weather forecasting: existing black-box Al models are unable to generalize at finer temporal scales beyond the inherent time resolution of the training datasets due to the absence of fine-grained physics modeling.
To address this challenge, we propose WeatherGFT, a physics-Al Al hybrid model capable of simulating weather changes on fine-grained time scales through set of partial differential equations (PDEs) [50]. WeatherGFT consists of an encoder, multiple stacked Hy HybridBlocks and a decoder. As the core of our model, HybridBlock contains two branches: One utilizes PDE kernels to conduct physical evolution over small time scales, while the other employs neural networks to learn unresolved atmospheric processes and perform bias correction on the physical evolution. These two branches are adaptively fused through a learnable router initialized as 0.5:0.5. Unlike existing models [32] (7] trained with fixed lead time, we introduce lead time-aware framework through multi-lead time training strategy and a lead time conditional decoder + I. enabling the model to generalize to finer-grained temporal scales. Experiments demonstrate that our method capable of forecasting at different lead times within one single model and one unified framework, overcoming the limitations of the dataset's s temporal resolution and enabling 30-minute forecasts with an hourly dataset.
0.6
Additionally, we find two interesting insights by examining the learnable route weight of the hybrid physical-A. Al modules at different lead
Physics
AI
Learnalle router weight
Physical weight reduction due to PDE error accumulation
times, as depicted in Figure [ a) The physi- cal weight is consistently higher than the AI,
indicating the significant role played by the
0.5
PDE kernel. b) As the lead time increases, the weight of Al gradually increases. We attribute this increment to the errors accumulation of PDE kernel during the evolution process, ne- cessitating more Al corrections. In summary, when there is training data available at the lead time, such at 1:00:00, the fitting abil- ity of Al is enhanced. Conversely, at the lead time without training data, such as at 0:30:00, the importance of physical evolution becomes more pronounced, which confirms our moti- vation: WeatherGFT can benefit from both physies and AI adaptively.
Initialization 0.5:0.5
Al correction increment
0.4
0:15:00
0:45:00 forecast lead time
0:30:00
1:00:00
Physical evolution time < time resolution Figure 1: Learnable router weight. The role of physics and AI at different lead times: major evolu- tion and adaptive correction (details in Sec.
We summarize the contributions of this paper as follows:
We propose a physics- AI hybrid model that incorporates physical PDEs into the networks, enabling the simulation of fine-grained physical evolution through its forward process.
With the flexible PDE kernel and new lead time-aware training framework, our model per- forms multiple lead time forecasts, which bridges the nowcast and medium-range forecast. For the first time, our model extends the forecasting ability learned from an hourly dataset to make accurate predictions at finer time scale, i.e., 30 minutes.
Our model exhibits strong generalization ability while maintaining prediction errors compa- rable to those of pure AI and physical models.
2

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2 Related Work
Data-driven Weather Forecasting. In recent years, data-driven weather forecasting models based on machine learning have developed rapidly [2], especially for medium-range weather forecast [54], which provides weather variables for the next few days. Clare et al. propose weather forecasting approach using stacked ResNets [2T. but their model only considers geopotential and tempera- ture, which is limited for real-world forecasting applications. FourCastNet [32 expands the model to include additional variables such as wind at different heights, and employs Adaptive Fourier Neural Operator (AFNO) [16] networks for prediction. Pangu-Weathsr Weather (3) utilizes the 3D Swin Transformer [6]] and introduces hierarchical temporal aggregation to minimize iterations in the autoregressive forecasting, followed by Feng Wu 7 [77. FuXi [9] and other Transformer hased [52] prediction models. Apart from Transformers, GraphCast [3] and Keisler [29] adopt graph repre-
sentation of the Earth and employ Graph Neural Network (GNN) [62 for weather prediction.
In addition to medium-range weather forecast, nowcast [4 55 is another important field in weather forecast, which usually provides 30-minute forecasting of severe convective weather like thunder- storms. OFAF [4], Preciplstm [1]] Sim [2] use convolutions to capture spatial information and model temporal information through networks such as Long Short-Term Memory [60] Recurrent Neural Network [59]. Earthformer 133 and CasCast [14] use Transformer-base
models for nowcast- ing. The former proposes cuboid attention to efficiently model space-time information, and the latter uses the diffusion model [49] to address the problem of blur output. These nowcast models focus on minute-level forecasts for specific regions, and is difficult forecast for long-term such 5-day.
Consequently, there exists a significant gap (global vs. regional, day-level vs. minute-level, long-term vs. shot-term) between medium-range forecasts and nowcasts. Integrating AI models with physical guidance to make finer-grained predictions can bridge this gap.
Physical Neural Networks. Most data-driven models commonly neglect the incorporation of physics and treat networks as black-boxes. In order enhance the consistency of predictions with respect to physical laws, PINNs (.) PINO (37). and DeepPhysiNet (5 add PDE loss to overall training loss. Nevertheless, these methods of changing loss functions often require balancing the weights between different PDEs, and the training results are heavily affected by hyperparameters. PI- HC-MoE [6), ClimODE (53] integrate physical processes into the networks, but they do not explicitly simulate the physical evolution of distinct variables based on PDEs. Instead, they implement the evolution using general kernels, such as Euler kernels [1]] NeuralGCM 331 employs neural networks to parameterize dynamic core. However, it is primarily designed for medium medium-range range forecasting. These works typically focus on forecasting fixed lead times, rather than leveraging physical laws to generalize to finer-grained time scales beyond the training datasets.
3 Method
3.1 Problem Formulation
Weather forecasting aims to predict future weather states X, given current weather states Xg:
(X6) = P(X((A))
(1)
where 0 represents the parameters of the model and t denotes the lead time. The weather state X' E RC*HXW consists of C' atmospheric variables across different pressure levels. Each variable is characterized by an H x W matrix that corresponds to the projection of the Earth's 's plane.
Assuming that the time resolution of the dataset is lataa the lead time for data-driven models can only be equal to or greater than ldata, because these models are trained using data pairs (Aa, X,) sampled from the dataset. Consequently, black-box Al models 32 33
20 19 [5] are unable to
forecast at finer lead times such as Edate, indicating a lack of temporal generalization ability.
3

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Attention Block
Router Init 0.5:0.5
Cony
duu
dr
du
Conv
Attention Block
SpdE
PDE Kernel x
PDD Kernel
AP
Upsampling
Altention Block
PPN
5q
dq
Aq
Upaampling
0t
300s
Eg
tu
fooock 3
Block
37'
15min
PDE Kernel X
1h
Network features
Physical variables
Lead time embedding
+ Add
Concatenate
HybridBlock
Lead Time Conditional Decoder
Stacked HybridBlocks
Sjuck
1.5min
JUNICS
Dataset time resolution
Panchify
Training Inference
Fanta
time
WeatherGFT with Multiple Lead Time Training
Figure 2: Overview of WeatherGFT. HybridBlock serves the fundamental unit of the model, consisting three PDE kernels, parallel Attention Block, and subsequent learnable router. A lead time conditional decoder is employed to generate forecasts for different lead times.
3.2 WeatherGFT Overview
As shown in Figure our model consists of encoder to patchify the weather states into tokens (52). multiple (specifically, 24) stacked HybridBlocks to preform weather evolution via PDE modeling, and decoder generate predictions under specific lead-time conditions.
Specifically, to enable our model to generalize at a finererained temporal resolution, we employ PDEs to model the evolution finer time scale:
X, KC(X)), where t, -ddat m +
m
We simulate the physical evolution from Xg to X,, through uniquely designed PDE kernel (details in Section where t, is much smaller than the time resolution tatta of the dataset, allowing model to capture fine-grained weather changes. By stacking PDE kernels K. the longer evolution can be achieved like X Km K2 Ki Xg. In this paper, set to 12, that is, t. f
To mitigate the issue of error accumulation as the number of evolutionary steps increases, we introduce a parallel Attention Block [52) that performs bias correction for every iterations of K. Additionally, learnable router initialized as 0.5 0.5, is employed adaptively fuse features from PDE kernels and the Attention Block. We encapsulate three PDE kernels K. and one parallel Attention Block = = Ldata-
within a HybridBlock, whose evolution time btack
Our model can not only forecast at lead times equal to or greater than fdota. but also generalize finer-grained time scale such ddtaa even in the absence of corresponding training data pairs.
This is achieved by modeling the physical evolution of latock = Ldlatta rather than simply learning from data pairs (2o, Xr sampled from the dataset. Notably, these generalized finer-grained predictions of our model outperform temporal interpolation on multiple metrics, as shown Table 3 emphasizing the advantages of fine- fine-grained physical evolution over black-bor box models.

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3.3
PDE Kernel
We employ a set five PDEs including the motion equation, the continuous equation and others to establish a closed system, which simulate the physical evolution of 5 essential atmospheric variables: u Katitude-direction wind), v (longiude-direction wind), 2 (geopotential), q (humidity). T (temperature). The partial derivative of each atmospheric variable with respect to time can be separated mathematically (details Appendix A), denotes as SpDE, which takes current weather state as input and produces the derivative of each variable with respect to time. We define PDE kernel K. as the evolution of the variables over short period of time a. as demonstrated Equation] 3
du
Su(u, v, u.
16
at
0:
S,
16
PDE Kernel (X) Spog(X),,
SPDE (X)
2: U,,,,
24 1,,,,,,,,,,77722
where t. 12 Edata
DE
ST
ST,,,
Calculating SpDE requires the use of differential and integral operations. For example, for temper- ature T, its derivative with respect time is shown in Equation 4 In order efficiently calculate SPDE and enable loss backward [25], we designed a fast implementation of differentiation and integration through convolution and matrix multiplication respectively. Equation presents the
implementation of the differential and integral of X' in the x direction (latitude direction).
0T 0t
8p
0p
W
OT
0T
OT
where = =-
Ou
Ov Oy
dp
(4)
Cp
Oz
0p"
0x
0
0
0
I
4x de 1 Cono (2, Kz)
0
0
0
K2,
1
8
-1
M₂
=
E RRW
(5)
J X'd = XMM
0
0
0
0
0
0
Similarly, K2 and My can be constructed to perform differential and integral operations in the direction (Iongitude direction). For differential and integral operations in the p direction (pressure level direction), we first reshape X E xCxH*W to 3D space X3D IR fg x based on the variables' pressure layers, and then implement corresponding operations through Kp and Mp.
3.4 HybridBlock with Adaptive Router
HybridBlock is module that combines physics and Al. Firstly, employs neural networks to address the issue of eттоr accumulation resulting from the stacking of PDE kernel K. Secondly, it utilizes the PDE kernel to guide the neural networks to learn the physical evolution of specific time step. The structure of HybridBlock consists of three PDE kernels K. and one parallel Attention Block. Consequently, the time step corresponding to Hybrid- Block is bMock 3 x L, 1 Idata-
X,
BXLXD
XN R SXLXD
Learnable 7 E RD
0.8
0.1 0.1 0.9 0.2 0.8
r rx Xp + (1 - r) x XN
HybridBlock has two branches, as depicted in Figure one is physics and the other is AI. The neural networks features X' are aligned with physical features Xp through convolutional layer, followed by three PDE kernels. Sub- sequently, the PDE kernel output is projected back to the latent space of X' through another convolutional layer. Finally, features fusion is performed through the learnable router shown in Figure
Fusion MLP
Router
Figure 3: Router in HybridBlock. B represents batch size, is the number of tokens with D dimension.
In the router, the features XN obtained from the neural networks and the features Xp derived from the PDE kernels are initially linearly fused along the
5

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feature dimension D, with the learnable factor r initialized as 0.5 0.5. Subsequently, the preliminary fused features will go through an Multilayer Perceptron 40 layer containing a ReLU 36 activation function to accomplish nonlinear feature fusion.
3.5 Lead Time Conditional Decoder
HybridBlock provides the smallest time scale of model evolution, which is bblock ddtaa. Through
Ix HybridBlocks, we can predict the weather lead time of 4 lataa- To enable the model to generalize its prediction capabilities to finer-grained time scales, we design lead time conditional decoder generate forecasts varying lead times from the output of the corresponding HybridBlock.
In order to promote the expression of the condition, we embed the lead time i into a high-dimensional vector Lemb through learnable Fourier embedding 48 as shown Equation 6
temb snn (T -t W) @ coS(IT t W) @ t, where is lead time
(6)
where W learnable vector of size 16, and denotes concatenation. Furthermore, temb will be concatenated with the output of HybridBlock and input to the decoder together. The decoder structure utilizes Swin Transformer
39
with hierarchical upsampling, as illustrated in Figure 2
3.6 Multiple Lead Time Training
For dataset like ERA5 22 or WeatherBench 47). their time resolution dataa = 1h. We set the time step of the PDE kernel f,- ddtta 300s. Consequently, the time step of each HybridBlock is bMock = 3 x 1,4 900s, equivalent to 15 minutes, By cascading 24 HybridBlocks, model can generate lead time of 24 x 15min 6h. To encourage the model to learn evolution for different lead times and generalize forecasting to finer-grained time scales, during training, we not only use the output of the last HybridBlock but also include the outputs of the 4th and 12th HybridBlocks. These outputs are passed through the lead time conditional decoder with corresponding temb to predict the weather states x 15min 1h and 12 15min- 3h.
forecasts
During inference, we can take the output of the second HybridBlock and pass it through the decoder with corresponding temb get 2 x 15min 30min forecasts, which are not present in the dataset. In the Section 4.3 we provide a comprehensive demonstration showcasing the accuracy of these generalized prediction results for time scales smaller than the dataset's time resolution.
4 Experiment
Through the design of HybridBlock mixed with physics & Al and the multi-lead time training method, model is capable of simultaneously conducting short-term forecasting and long-term forecasting
without additional finetuning [42] on different forecasting tasks. In the experiments, we will showcase the superior performance of our model and try to answer the following questions:
(1) How does the model perform on the medium-range forecasting task?
(2) How does the model perform on the generalized 30-minute nowcasting task?
(3) As hybrid expert model of AI and physics, what roles do they each play?
(4) How do PDE kernel and multi-lead time training contribute the overall performance?
4.1 Experimental Setup
Dataset. We use WeatherBench 47 as our training dataset, whose time resolution is
Dataset
Train
Test
Time resolution
WeatherBench NASA
1-hour
Edata = 1h and spatial resolution is 128 x 256.
30-minute
The dataset spanning from 1980 to 2015 serves
as training set, while the data of 2017 is the val- idation and test sets. Our model processes 4 sur- face variables and 5 upper-air variables across 13 pressure levels, as shown Table 2
Table 1: Datasets. NASA dataset only contains pre- cipitation, which will he used as the ground truth for precipitation noweast.

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Given that WeatherBench lacks data at finer tem- poral resolutions, we use the 30-minute satellite observations downloaded from NASA as ground truth to quantitatively assess the model's s gen- eralizability. NOTE: Data from NASA is only used for testing and not for model training.
Name Description
Levels
u10
x-direction wind 10m height
Single
v10 y-direction wind 10m height
Single
12m
Temperature 2m height
Single
tp
Hourly precipitation
Single
Geopotential
13
q
Specific humidity
13
Tasks. We conducted experiments on two typ- ical weather forecasting tasks: medium-range forecasting and precipitation nowcasting. The forecast range for medium-range forecasting spans from 6 hours 5 days, while the nowcast- ing is set to range of 30 minutes to 2 hours.
u
x-direction wind
13
V
y-direction wind Temperature
13
T
13
Table 2: Atmospheric Variables Considered. The 13 levels are 50, 100, 150, 200, 250, 300, 400, 500, 600, 700, 850, 925, 1000 hPa.
Baseline Methods. We compare WeatherGFT with four forecast approaches: FourCastNet 32 uses
AFNO [6] networks to simulate the nonlinear relationship between weather variables, Keisler [29 models global atmospheric data through GNN, ClimODE 53] adds ordinary differential equations (ODE) 26 the neural networks, and ECMWF-IFS 46 a physical dynamic model.
The above three data-driven models cannot generalize forecasting to finer-grained time scales due the absence of 30-minute labels. Therefore, in nowcasting tasks, interpolate the 30-minute forecast results through SOTA frame interpolation models Flavr [28) and UPR [27]. In contrast, our model can conduct 30-minute predictions inherently without interpolating.
Implementation Details. We implemented the model with PyTorch [25] and trained 50 epochs 8 NVIDIA A100 GPUs (T0) for 3 days, with learning rate of cosine schedule starting from 5e-4.
4.2 Skillful Medium-Range Forecasts by WeatherGFT
Autoregression is commonly employed in medium term forecasting, where the model output serves the input for the subsequent forecast step, allowing for longer lead time predictions. However, prediction errors tend to accumulate during the autoregression, leading to an increase in the root mean square error (RMSH). As result, smaller RMSE indicates
accurate prediction.
Figure 4 illustrates the changes in prediction RMSE of different weather variables lead time increases. Our model demonstrates competitive performance across various lead times with Al or physical dynamics models, especially the prediction of surface temperature (12m) and surface wind speed (u10) is significantly better than other models. The geopotential of the 500hpa pressure layer (z500) a crucial weather variable in weather forecasting, as reflects atmospheric circulation [5), subtropical high-pressure systems [34] and other significant phenomena. Due to the modeling of geopotential the PDE z500 prediction of our model outperforms the physical dynamic model ECMWF-IPP as visualized Figure
500
25
2.5
RMSE
15
0.8
0.5
50
75
100
125
25
50
100
125
25
50
75
100
125
25
50
75
100
125
12m
u10
2500
18.50
FourCastNet
ClimODE
Keisler
ECMWE-IPS
WeatherGFT
Figure 4: Medium-Range Forecast. The x-axis represents the lead time in hours, while the y-axis represents the RMSE for different variables. The smaller RMSE the better.
From the visualization in Figure our model is more accurate in predicting the subtropical high, as indicated by the highlighted box. In addition, the prediction error of our model the lead time of 6-hour is significantly smaller than that of the physical dynamic model ECMWF-IFS.
7

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#50002017 01 01106: 00:
Prediction etror
2500 - 01 02T00: 00: 00
Prediction error
IPS : Ground trurts
ECMWF
WeatherGPP
Figure 5: Visualization of z500 Predictions.
4.3 Generalizing to Fine-grained Time Scale for Nowcasting
used in medium-range weather forecast- ing, WeatherGIT has the ability to break through time scale limitations of the dataset, making the generalization to fine-grained temporal scales possible. This capability is facilitated by the dynamic progression of our PDE kernel modeling and multiple lead time training. Specifically, we use the second HybridBlock of the total 24 HybridBlocks generate 30-minute generalized forecasts through the lead time conditional decoder, which very important for precipitation nowcasting.
In contrast to conventional black-box AI models 32
29 58
To quantify the accuracy of the model's generalized nowcasting, we utilize the NASA satellite precipitation observation dataset the ground truth, which has time resolution of 30-m 30-minute. We evaluate forecasts at 30, 60, 90, and 120 minutes. It is important to note that data of NASA were not used for training. For other comparison models that cannot directly produce half-hour forecasts, we use the frame interpolation models (i.e., Flavr 28 and UPR (27 to generate 30-minute predictions.
30-min CSI
60-min CSI e1.
90-min CSI 61,5
120-min CST @15
CST
RMSE
CSI @0.
RMSFJ
CSR @0.5
RMSEJ
CSI
RMSEL
@0.5
01.
tpth
tplh
1plb
e0,
tplh
Four
0.26
0.09
0.67
0.61
0.49
0.24
0.25
0.09
0.65
0.37
0.26
0.46
FourCast+UPR
0.20
0.10
0.76
0.61
0.49
0.24
0.11
0.05
1.49
0.37
0.26
0.46
Keisler+Flavr
0.25
0.09
0.66
0.59
0.48
0.23
0.25
0.08
0.66
0.41
0.29
0.35
Keisler+UPR
0.26
0.13
0.69
0.59
0.48
0.23
0.26
0.13
0.68
0.41
0.29
0.35
ClinODE+Flavr
0.26
0.09
0.67
0.62
0.51
0.25
0.66
0.34
0.32
ClIRODEUUPR
0.25
0.12
0.67
0.62
0.49
0.21
0.25
0.11
0.66
0.46
0.32
0.31
WeatherGFTrours)
0.28
0.17
0.72
0.62
0.50
0.21
0.28
0.16
0.71
0.54
0.40
0.27
Table 3: Generalized Nowcast. 60-min and 120-min are trained lead times, while 30-min and 90-min are generalized lead times. Gray represents the results obtained through the frame interpolation model, purple indicates the results obtained through unified model without interpolating. For precipitation nowcasting, CSI (Critical Success Index) is the most important metric.
CSI@th (Critical Success Index) refers to the hit rate the area that reaches the threshold precipita- tion value th. CS1@0.5 can reflect the overall forecast accuracy in rainy areas, and CSI 1.5 reflects the forecast accuracy moderate rainy areas. Table 3 shows that our model surpasses others across different lead times, especially in forecasting regions of moderate rainfall, i..., CSI@1.5.
The visualization in Figure 6 reveals that when using frame interpolation to obtain 30-minute predictions, there is blurring occurring at different scales, resulting in the loss of extreme values, as indicated in the box. Our model, which incorporates physical constraints, provides clearer predictions retaining extreme values without the need for frame interpolation.

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00: 30: 00
Error
01: 00: 00
Error
01:30:00 00
Error
02: 00: 00
Error
Ground truth
FourCast
Flavr
FourCast +
UPR
Keisler
+
Flavr
Keisler
+
UPR
ClimODE
Flavr
ClimODE
UPR
WeTtherGFT
(ours)
Figure 6: Visualization of Precipitation Nowcast. Precipitation in the area ranging from 34N to 50S and 148E to 128W during the time period from 00:00 to 02:00 on July 1, 2017.
4.4 Weather Forecasts can Benefit from Physics and AI via WeatherGFT
As hybrid model combining both physics and AI components, it is crucial to analyze their contribu- tions to the prediction process. We present insights into their respective proportions by visualizing the weight parameter within the learnable router (refer Figure 3). The visualization in Figure reveals that the weights of the 24 HybridBlocks display a similar distribution:
a) The physical weight of the vast majority of HybridBlocks is significantly higher than the weight of AI, which shows that in the process of simulating time evolution, the PDE kernel plays a more important role, while the Attention Block only plays supportive correction role. b) The physical weight gradually decreases while the weight of Al increases throughout each hour (dataset time resolution). This aligns with underlying motivation, which acknowledges that errors may accumulate over time in the physics- based evolution. Consequently, a greater emphasis on AI corrections becomes necessary to compensate for these accumulated errors.
0.7
Datoset resolution (I-hou))
Physical weight Al weight
TimeBlock (15-min)
Figure 7: The Weights in the Router of 24 HybridBlocks.

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By averaging the 4 x 6 HybridBlocks into 4 time steps, the average weight every 15-minute is obtained Figure DI which shows the above two conclusions more clearly. To summarize, physics plays the main evolutionary role in the model, while Al plays an dynamic corrective role.
4.5 Ablation Studies
We use Swin Attention Block 39 the base- line for the ablation studies. For this baseline networks without PDE kernel constraints, as
30-min
RMSE@1-b
RMSE@6-
3
nowcast
12ml] 2500)
(2mi z500
(2m] 25001
Block
a black-box model, it will only learn the map- ping of data pairs corresponding to the lead time. Consequently, its internal information between blocks is unexplainable, which also results in being unable to predict moments without data labels, such as 30-minute nowcasting.
PDE Kernel
0.57 20.43
0.70 21.78
1:22 153.8
Muti Time
0.49 16.66
0.67 21.80
1.14 152.4
Table 4: Ablation Experiment.
Attent Block
Atlent Block Kerne
125
Attent Block + PDF Kerne Muti I end Time
PDE kernel is crucial to the generalization of finer-grained predictions. Instead of simply learning the mapping between data, the model learns the evolution of the corresponding time step according to the physics laws, making in- formation of each neural network layer explain- able, thereby facilitating generalized 30-minute noweasting. addition, we find that the intro- duction of the PDE kernel also improved the prediction accuracy of the model.
100
75
50
10
20
30
40
50
Multiple lead time training accelerates con- vergence and improves the accuracy of model prediction, as shown in Figure g We hypothesize that this phenomenon can be attributed to the loss backward from different lead times, which alleviates the issue of vanishing gradients [23], allowing the parameters of different layers to quickly warm up and improve the expression of the model.
Figure 8: RMSE z500 as Training Epochs.
5
Conclusion
Most existing data-driven weather forecast methods which operated as black-box models via purely performing data mapping unable to generalize finer temporal scale beyond the inherent time resolution of the training datasets due to the absence the fine-grained physics modeling. This paper proposes physics-Al AI hybrid model to solve this problem. Through the exquisitely designed PDE kernel, each block in the networks can simulate the evolution of physical variables at finer- time step, while Al plays the role of adaptive correction, which makes our model capable of generalizing predictions to finer time scale beyond dataset. By employing our proposed multi-lead time training strategy, our model trained forecasts, while maintaining prediction errors that are competitive with those of pure Al and physical models in both medium-range forecast and precipitation nowcast.
hourly dataset exhibits remarkable ability of generalized 30-minute
The main limitation of our model is that only five important atmospheric equations are currently considered, which is still far from fully modeling the atmospheric motion process. Another limitation of this paper is that the experiments have been conducted solely at a spatial resolution of 128 x 256. As part of our future work, plan to extend experiments higher resolutions such 721 x 1440 to assess the model's performance under different settings. Additionally, while the minimum evolution time scale of our model is 15 minutes, were unable to evaluate 15-minute generalized predictions due to the absence of corresponding validation data at that specific time scale. Therefore, we are currently only able to perform evaluations of 30-minute generalized predictions.
For future work, we plan to incorporate additional physical laws into our model and conduct higher- resolution experiments to ascertain the upper limit of its capabilities.
10

---

Acknowledgements
This work is supported by Shanghai Artificial Intelligence Laboratory.
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14

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A PDE Solver
We constrain atmospheric variables, that is, u (Iatitude-direction wind), v (lonitude-directinn wind), or & (geopotential), (humidity). T (temperature), through the following set of five partial differential equations (PDEs) 30]
dp + - FA.
(7)
06
(8)
0p P
V Ow 0
(9)
Op
dT
- 10 Q
(10)
di
p pRT
(11)
The expansion of d is as follows: de
0 01
0 Op
V ()
(12)
dt
The PDE above is in the pressure coordinate system, which is aligned with the input to our model, as the input the model comes from 13 pressure layers. In the air pressure coordinate system, the following equation is also satisfied:
0p
(13)
represents the vertical wind speed and is not directly included of the input variables in model. However, it can be derived from u and using following equation:
Ow
Ou
Ov
0p
dr
Oy
(14)
Ov
w
dp
Oz
Oy
After getting w, get
du
and according Equation 7
dy
Bt
Ou
Ou
du
Du
06
fv
dx
Dy
Op
dr
(15)
Ov
dr
dv
dv
06
+
u
1
+ w
+ fu
Øt
xx
Oy
Op
Oy
Ou
Ou
Ou
du
06
Øt
0r
Oy
w
0p
+ fu
dr
(16)
de
du
dv
06
f
Øt
0x
by
0p
Oy
where / = 7.29e 5 is constant.
According to Equation 10
0T. can get Ør
15

---

0T
ØT
0T
Cp
+
+
1
+
W
Q
0t
dr
0y
0p
p
(17)
0Φ 0p
Q
W
L 2
06 Up
9T
0p
w
w
0T
0T
8T
U
w0
(18)
0t
Cp
Dy
Op
where L 2.5e6 and Cp 1005 are constants.
According to Equations and Equations
we can get
06
1
RT
(19)
Op
p
8 Φ
i P
Opåt
Ot
0T
0p
- R
(20)
01
p2 0t
R&T
06
/
86 dp
01
Epdt
(21)
/
R 0T
p Ot dp
where R= 8.314 is constant.
Finally, according to the water vapor equation 22
8g. we can get 22
dq
8F dø
dt
RT dt
0, dø
< 0 and q 2
dt else
F 4s LR C. R.T
R.T
+ L2-g
(22)
17.671"
6. 6.112 x eap
T' + 243.5
T' = T - 273.115
0.622e,
4. P - 0.378E,
0g
5F
06
+ U dΦ
do
0g
0q
0g
u
+ 0z
w
(23)
Øt
RT
Øt
dr
dy
Dy
dz
where R. 461.5 and R₂
287 constants.
16

---

B Implementation of Integrals and Differentials
Integral p-direction (pressure levels direction) implemented with PyTorch 25 as follows:
def integral_ (inputt (eanputttennorr:......__
Pressure direction integral
B, pressure_ H, W input_tensor shape input_tensor input_tenson reshape (B, pressure_levee
H*w) output to (in.tttiittt.teesor dtype) (input__tnssor device) e input_tenson output output reshape (B, pressure_level.num H, W) return output
Ms obtains the integral through matrix multiplication. Given the input matrix z below, the result of x MMg is:
47
4
1 + 47
x
2
xM,
2
=
2
2 2
(24)
3
6
3
6
3
3
6
Differentials in x-direction (latitude direction) is implemented with PyTorch as follows:
def d_ (input (iipuuttteesoor).
# Latitude direction differential B, C, H, w input_tensor shape conv_kerne
torch. ([1,
device-input. tensor. device,
dtype= inpppeiinpttttesorr dtype, eqquires-grad-aane
conv conv_kernel [0,0,0,000 =
conv_kernel [0,0,0,1] -8
conv_kerne [0,0,0,3]
conv_kernel [0,0,0,4] -1
input_tensor torch. cat ((input_tensol ,,,,,,
input_tenaor,
input_tensor i;,;;; ;2]), dim=3)
H M_ imput_tensor.ssaay
input tensor input_tenne tensor. reshape (B+C, 1, H,, W_) conv_kernel)/12
output_x conv2 (inputtitnn output_x output_ x reshape (B, C, H, W)
dtype) to (output_x, device) output_x = output_ popput_x/pixe____ to
output_x
K2 the convolution kernel. Assume one-dimensional input data -2,0,, 1, It gradually increases from left to right by 1, that is, its gradient is 1. Applying convolution kernel K2 to the result is: Cone(x, K₂) (-2)xI+(-1)x( ( 1000x1118--22(1-11
By using this convolution
kernel, the data gradient can be determined.
Differentials in y-direction (longitude direction) is implemented with PyTorch as follows:
def d_y (iiniitttttesoor
# Longitude direction differential B, C, H, W input_ shape conv conv_kernel torch. zeros [111, device-input, device-input. pensorrttteetroo device, loype=input_tte tensor. dtype, requires_grad=Fale)
conv kernel [0,0,0] -1
conv conv_kernel [0,0,1]
conv_kernee [0,0,3] -8
conv_kerner [0,0,4]
17

---

input_tensor torch. cat ((iputt_eenor( (;
input_tensor
input_tensor [;,;, -2:1), dim=2)
H., W_ Input_tensor.ste
input_tensor input_tensor.. reshape (B+C, 1, H_, W_)
output_y connudd tensor, fov_kernell//
output_y output_y reshape (B, c, H, w) output_y = output_y //iptyy/pixe..
return output_y
Differentials in p-direction (pressure levels direction) is implemented with PyTorch as follows:
def d_z (input_tensorr
Pressure direction differential conv conv_kernel forch.zerow zeros 1,,, 11, device-input.eenson derettteeeor..eccee
dtype- btype-input.o tensor dtype, Soquires-grad-Faa
konv_kernel [0,0,0] -1
conv_kerne [0,0,1]
conv_kerned [0,0,3] -8
conv_kernel [0,0,4]
input_tensor torch. cat ((input_ [2]
input_tensor,
imput_tensor [: -2:1), dim=1)
input_tensor input_tensor unsqueeze (1) B, 1, C, H, W
output_z (copu((innt__tennso output_z output_2 squeeze (1) output_2 output_ 2 //pxxel_2 to (outputt_ dtype) to couuputt_z..co device)
conv_kernel)
return output_2
C Hyperparameter Details
Hyperparameter
Value
Max epoch
50
Batch size
4x8 (GPUs)
Learning rate
5e-4
Learning rate schedule
Cosine
Patch size
4x4
Embedding dimension
1024
MLP ratio
Activation function
GLUE
Input (0-hour)
[4, 69, 128, 256]
Output (1, 3, 6-hour)
[4, 3, 69, 128, 256]
Table Hyperparameters the Model
Datasets
Training
Validation
Test set
Time resolution
Variable
WcatherBench
1980-2014
2015
2017-2018
1h
tp. 12m, u10, v10, z, v. tp
NASA
None
None
2017-2018
30min
Table 6: Datasets Information
18

---

D Additional Experiments
D.1 Prediction Bias Evaluation
Bias 2 8 56] indicates the disparity between the model's predictions and the ground truth. Negative bias indicates underestimation, a prevalent issue in forecasting models. Although the PDE kernel was not specifically designed to address bias underestimation, experimental results indicate that its usage helps ameliorate underestimation.
bias pred gl
(25)
lar
-304
ai leee.
4.15
3.15
377
+2.75
Figure 9: Bias. The closer to the better.
D.2 Prediction Energy Evaluation
This assesses the energy [24] changes in the model's predictions. The experiments reveal that employing the PDE kernel aids in energy preservation.
energy (u² + 2²)
Cp T2
(26)
2T,
1403
FDE we
The weights of router the norm input features
14035
192.0
05
1435
1400
Figure 10: Left. Energy: the more consistent the better. Right. The norms of the outputs from the two networks are similar and stable. This indicates: a) The two networks produce outputs the same scale. b) The router is decoupled and dynamically selects the more crucial features from the two branches without affecting the scale of the two networks.
D.3 Router Weights and Features Norm Change
Figure 10 complements Figure 7 in the paper. It illustrates that physical and AI features comparable scale, with the router dynamically selecting the more effective aspects from each. The router's weight adjustments do not impact the output of the Al or physical branches, highlighting the router's decoupling characteristics.
E
Code Of Ethics and Broader Impacts
Our research is ethical. The physical and Al hybrid model proposed in this paper can be used for global weather forecasting, which can serve many fields such as transportation and agriculture, and bring huge benefits to socicty.
19

---

The dataset used in this paper is public and there are no issues of infringement privacy leakage. The experiments conducted this paper are fair and reproducible. The resource consumption during the experiments is minimal and will not have impact on the environment and society.
The model propose is free of bias and discrimination issues. We open-source the model code and checkpoints on GitHub
Safeguard of Model
This paper presents a hybrid physics-Al model for global weather forecasting. It is important to acknowledge that all models inherently carry certain degree of forecasting error. Hence, the model proposed this paper should not be solely relied upon as the sole basis for predicting significant events. Instead, it is recommended integrate the findings from this model with other models and expert insights to draw comprehensive and informed conclusions.
G Assets
Our study adheres to the licenses governing the usage of existing assets, the data utilized in this paper are publicly available and permitted for academic research purposes.
The model introduced in this paper represents a novel contribution and is considered new asset.
20

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer:
[Yes]
We demonstrate the contribution and scope of the paper in the Abstract and Introduction
Justification:
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals are not attained by the paper.
2. Limitations
Question: Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification:
We show the limitations of our paper in Section
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged judgment and recognize that individual actions in favor of transparency play impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
the paper. The authors should use their best
3. Theory Assumptions and Proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [Yes]
21

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Reviews:
[{"review_id": "XHrmad810Y", "reviewer": "Reviewer_CYo5", "summary": "The paper proposes WeatherGFT, a physics-AI hybrid model designed to generalize weather forecasts to finer temporal scales beyond the training dataset. By integrating PDE kernels for physical simulation and neural networks for adaptive bias correction, the model aims to provide accurate 30-minute forecasts using an hourly dataset. The lead time-aware training framework enhances the model's ability to generalize across multiple lead times, achieving state-of-the-art performance in both medium-range and nowcasting tasks.", "questions": "1. Can you further explain Equation 5 and how you formulate $K_x$ and $M_x$\n2. How does the convolution layer work to align neural network features with physical features?\n3. How do you update the learnable factor $r$? How do you compute the weight to draw Figure 1 since the weight is a vector?\n4. Does tp (hourly precipitation) have a related PDE? How do you deal with variables that are not related to any PDE?\n5. How fine-grained can the model achieve? Doing 30-minute forecasts with hourly data is impressive, but can the model achieve finer resolutions like 15 minutes or even 5 minutes?", "limitations": "Yes.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The framework that fuses AI and PDE is innovative and improves the model's generalibility.\n2. The model demonstrates generalization capabilities across time scales, achieving finer temporal resolutions (e.g., 30-minute forecasts) from coarser data.\n3. Extensive experiments validate the model's state-of-the-art performance across various forecasting tasks and lead times.", "weaknesses": "1. The paper is well-structured but could benefit from clearer explanations and analyses of the novel modules and specific contributions.\n2. The physics model relies on a limited set of PDEs for simulation, which may not fully capture the intricacies of real-world atmospheric dynamics.\n3. The baseline comparisons are limited, as many AI models available for weather forecasting have not been included.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nThank you for your thoughtful review and detailed feedback! We are delighted that you value the innovative fusion of AI and physics in our research. We have revised the paper according to your suggestions and will now respond to the remaining queries you have.\n\n> Q1: The paper is well-structured but could benefit from clearer analyses.\n\nThank you for your affirmation. We will add more quantitative and rigorous analysis to the paper to fully demonstrate our model. In the `global response PDF`, we added the evaluation of two indicators, *`bias`* and *`energy`*. The results show that **using PDE kernel can assist in making the model's prediction field energy more consistent**.\n\n> Q2: The physics model relies on a limited set of PDEs.\n\nWe agree that exploring more PDEs is valuable. Currently, we have employed only a few PDEs which results in performance enhancements and demonstrates the effectiveness of our design in combining AI and PDEs. Moving forward, we intend to augment the number of PDEs to simulate atmospheric dynamics more comprehensively.\n\n> Q3: The baseline comparisons are limited.\n\nWe have included additional model comparisons, as depicted in the table below. Specifically, SphericalCNN is a CNN model designed for spherical data, DMNWP utilizes a diffusion model for weather prediction, and EWMoE employs a mixture of experts (MoE) for weather forecasting.\n\n|RMSE z500|6h|3day|4day|5day|\n|-|-|-|-|-|\n|SphericalCNN|28.40|161.1|239.9|338.4|\n|DMNWP|52.33|272.3|360.7|466.8|\n|EWMoE|23.52|165.3|240.1|341.6|\n|WeatherGFT|**22.08**|**152.3**|**225.8**|**315.7**|\n\n|RMSE t850|6h|3day|4day|5day|\n|-|-|-|-|-|\n|SphericalCNN|0.494|1.183|1.493|1.860|\n|DMNWP|1.073|1.823|2.247|2.551|\n|EWMoE|0.513|1.259|1.593|1.865|\n|WeatherGFT|**0.457**|**1.176**|**1.480**|**1.839**|\n\n> Q4: Can you further explain Equation 5 and $K_x$ and $M_x$?\n\n`Equation 5` shows the differential and integral operators in the model. $K_x$ is the convolution kernel. Assume a one-dimensional data $x=[-2,-1,0,1,2]$. It gradually increases from left to right by 1, that is, its gradient is 1. Applying convolution kernel $K_x$ to $x$, the result is:\n$$Conv(x, K_x)=\\frac{(-2)\\times 1+(-1)\\times (-8)+0\\times 0+1\\times 8+2\\times (-1)}{12} = 1$$\nBy using this convolution kernel, the data gradient can be determined.\n\n$M_x$ obtains the integral through matrix multiplication. Given the matrix $x$ below, the result of $xM_x$ is:\n\n$$x=\\begin{bmatrix}\n1 & 4\\\\\\\\\n2 & 5\\\\\\\\\n3 & 6\n\\end{bmatrix},\\ xM_x=\n\\begin{bmatrix}\n1 & 4\\\\\\\\\n2 & 5\\\\\\\\\n3 & 6\n\\end{bmatrix}\n\\begin{bmatrix}\n1 & 1\\\\\\\\\n0 & 1\n\\end{bmatrix}=\\begin{bmatrix}\n1 & 1+4\\\\\\\\\n2 & 2+5\\\\\\\\\n3 & 3+6\n\\end{bmatrix}$$\n\n> Q5: How does the convolution layer work to align features?\n\nThe size of the tensor of the output of the PDE kernel is different from that of the output of the attention. The size of the output of the PDE kernel is $[8,5,13,32,64]$, where each dimension represents *batch size, (z, q, u, v, t), pressure levels, H-patch, W-patch*. The size of the output of the attention is $[8,1024,32,64]$, where each dimension represents *batch size, embedding dim, H-patch, W-patch*. We first reshape the output of the PDE kernel to $[8,5\\times 13,32,64]$. Then, through a convolution layer `Conv(in_channel=65, out_channel=1024)`, the size of the tensor is converted to $[8,1024,32,64]$, aligning with the output from the attention block.\n\n> Q6: How do you update the learnable factor $r$ and compute the weight to draw Figure 1?\n\nBelow is the PyTorch code for initializing and using the learnable factor $r$.\n\n```python\nimport torch.nn as nn\nr = nn.Parameter(torch.zeros(1, 1, 1, dim), requires_grad=True)\n\ndef router(physics_features, ai_features):\n    # Features size: [B, H, W, dim]\n    r1 = 0.5*torch.ones_like(physics_features) + r\n    r2 = 0.5*torch.ones_like(ai_features) - r\n    mixed_features = r1*physics_features + r2*ai_features\n    return mixed_features\n```\n\nThrough the automatic gradient function of PyTorch, $r$ will be updated through model backward. After the model training is completed, we first average the $r$ of each HybridBlock to obtain 24 values corresponding to 24 blocks. Then we divide them into 4 groups, namely:\n\n```python\n#   1,     2,     3,     4,     5,   ...,   24\n# 00:15, 00:30, 00:45, 01:00, 01:15, ..., 06:00\n[1, 5, 9,  13, 17, 21] # 15min\n[2, 6, 10, 14, 18, 22] # 30min\n[3, 7, 11, 15, 19, 23] # 45min\n[4, 8, 12, 16, 20, 24] # 60min\n```\n\nThen, we can average the $r$ of each group and draw `Figure 1`.\n\n> Q7: Does tp have a related PDE? How do you deal with variables that are not related to any PDE?\n\nThere is no PDE specifically for precipitation, since precipitation is considered as a diagnostic variable in the atmospheric dynamics, meaning that precipitation can be derived from other atmospheric variables. For these diagnostic variables, we utilized neural networks to predict their values. While these variables may not directly influence the PDE kernel, their information will be amalgamated through the router to accomplish implicit modeling.\n\n> Q8: How fine-grained can the model achieve? Doing 30-minute forecasts with hourly data is impressive, but can the model achieve finer resolutions like 15 minutes?\n\nAs given in `Figure 2` and `Section 3.6`, each PDE kernel simulates atmospheric dynamics within 300 seconds, allowing our model to make predictions at various time intervals by stacking these kernels. Theoretically, the smallest time scale for our model to forecast is 15 minutes, this is because one attention block is composed of 3 PDE kernels, equating to 3x300s=15min. Presently, we are unable to evaluate forecasts at finer scales than 30min due to the lack of ground truth at the 15min. Nevertheless, the generalization of 30min predictions showcased in the paper has already proven the efficacy of our physical-hybrid modeling approach.\n\nIf our responses have clarified your inquiries, we kindly ask for an update to your score. Thank you very much for your time and feedback.", "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": "OiIKCd3bcj", "reviewer": "Reviewer_oEsH", "summary": "In this paper, a hybrid model (a model that combines machine learning with physics) is demonstrated for nowcasting and medium-range weather forecasting. WeatherGFT uses machine-learned weightings that combine two successful methods for weather forecasting (machine learning and the traditional numerical, PDE-based method). The combination of both methods allows for significantly increased temporal resolution compared to purely data-driven methods (6-hourly vs 15 minutes). This approach also provides a new framework for combining physics-based methods with data-driven methods, which allows for the ability to differentiate through the hybrid model.", "questions": "In section 4.3, I don’t understand how FourCastNet, ClimODE, or Keisler was used for nowcasting. They all have a time step of 6 hours, how is it possible to get 30-minute forecasts using frame interpolation methods? Are you using interpolation between ERA5 (e.g. the initial conditions) and a 6-hour forecast from these models? \n\nIn section 4.3, how are the forecasts including WeatherGFT initialized? Do they all use ERA5? \n\nI would like to see the effects of the PDE Kernel on other metrics. Does this inclusion of a physics kernel improve spectral bias or allow stability? What about the conservation of energy or momentum? \n\nWhat is the computational cost during inference of having a PDE kernel compared to just the transformer by itself?", "limitations": "Overall the limitations of the paper are well laid out, however, some claims are not supported by the paper. \n\nClaims about being the first to move away from fixed lead times for data-driven weather forecasting are not true. Stormer (https://arxiv.org/abs/2312.03876) and MetNet (https://arxiv.org/abs/2306.06079) have demonstrated this previously. \n\nClaim “In addition, the prediction error of our model at the lead time of 6-hour is significantly smaller than that of the physical dynamic model ECMWF-IFS”. For Z500 this is not supported by Figure 4. \n\nFor Figure 5 the prediction of the subtropical high (it should be a subtropical ridge) isn’t convincing.  That seems to be a function of the contouring in Matplotlib as the difference plots show similar magnitude of errors. \n\nOverall, if some of these claims are toned down and the addition of the appropriate citations the manuscript will be improved.", "rating": 7, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Combining machine learning with physics-based approaches (e.g. hybrid modeling) is an exciting and highly researched topic for weather and climate. As mentioned in the paper, purely data-driven models are black-box systems even if they do perform well. Existing data-driven models also are too temporal coarse for some operational weather forecasting applications. This paper does a good job address both of these existing problems in the field.", "weaknesses": "The novelty of the paper is not as strong as claimed in the paper. Other hybrid models that combine machine learning with physic-based approaches exist (see Arcomano et. al. 2022 and 2023, Clark et. al. 2022, and others). The Arcomano et. al. 2022/2023 even includes a machine-learned weighting for the combination of the ML-based model and the physics-based model. \n\nThe time-embedding and multi-lead times in one model have also been demonstrated before, see Stormer (https://arxiv.org/abs/2312.03876) and MetNet (https://arxiv.org/abs/2306.06079).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nThank you for your insightful review and detailed feedback! We are glad for your recognition of the significance of our hybrid modeling of physics and AI. \n\nWe appreciate your feedback on specific claims in our paper, and will refine specific statements and add citations of relevant papers as necessary. In the following, we will address your remaining questions.\n\n> Q1: Comparisons with other hybrid models that combine machine learning with physics.\n\nWhile there exist various methodologies for integrating AI and physics, **our approach diverges in both methodology and focus from those mentioned**. The referenced paper primarily integrates machine learning techniques into a complete dynamical model (e.g., AGCM) to improve the forecast performance of this physics model. But our method focuses on combining PDE processes with neural networks, rather than improving the existing physical model. In addition, our focus is on finer-scale physical modeling to generalize finer-scale predictions without valid training labels, which differs from focusing on improving forecasting skills.\n\n> Q2: The time-embedding and multi-lead times in one model have also been demonstrated before, see Stormer and MetNet.\n\nThank you for your reminder. It should be noted that time-embedding and multi-lead times are not our core innovations (our core innovation lies in the physical modeling). They represent technical innovations in model implementation. We will increase the citations of the papers related to these concepts.\n\nHowever, we would like to emphasize that these methods differ from previous work. In Stormer and MetNet, there is no direct correlation between the number of network layers and the forecast lead time. In MetNet, the lead time condition serves as input for all network layers, while our network module (HybirdBlock) evolves within a short timeframe without lead time conditions. In our approach, **the lead time condition is solely fed into the decoder to extract predictions from different network layers' outputs**.\n\nFor multi-lead time training, the method of our paper is a completely different technology from the Multi-step finetuning in Stormer. **Stormer's finetuning focuses on autoregressive prediction, and its network itself can only output predictions for the next single step.** In contrast, our method has the capability to generate predictions for multiple steps within a single forward process.\n\n> Q3: How FourCastNet, ClimODE, or Keisler was used for nowcasting?\n\nFor the precipitation nowcasting experiment, we use 1-hour interval ERA5 to train FourCastNet, ClimODE, and Keisler, respectively, to obtain models that can make 1-hour predictions. *(NOTE: In the medium-range forecast experiment, these models have a lead time of 6 hours during training.)* However, as ERA5 lacks half-hour data, these models are unable to directly provide half-hour predictions. Therefore, we utilize additional interpolation models to interpolate the 1-hour predictions of these models to 30-min.\n\n> Q4: In section 4.3, how are the forecasts including WeatherGFT initialized? Do they all use ERA5?\n\nThe initial fields of all models are from ERA5. The only difference is that other models need to interpolate their prediction results to 30-min, while our method can directly get 30-min of prediction from the networks.\n\n> Q5: Does this inclusion of a physics kernel improve bias? What about the conservation of energy?\n\nThis is a good suggestion and helps us to have a more comprehensive understanding of the role of the PDE kernel. We measured *`bias`* and *`energy`* and found that **the PDE kernel plays a positive role in maintaining energy**, as shown in `global response PDF`. We believe that the preservation of energy is intimately linked to enhanced physical modeling, exemplified by the dynamic equations outlined in `Equation 7`. We present the calculation methods and related references of *`bias`* and *`energy`* in this `PDF`.\n\n> Q6: What is the computational cost of having a PDE kernel compared to just the transformer by itself?\n\nAs shown in the `global response PDF`, introducing the PDE kernel slightly increases training time, but the added computational cost is acceptable.\n\n> Q7: Claim “In addition, the prediction error of our model at the lead time of 6-hour is significantly smaller than that of the physical dynamic model ECMWF-IFS”. For Z500 this is not supported by Figure 4.\n\nWe apply the Wilcoxon test to demonstrate that **our model achieves a lower RMSE compared to the ECMWF-IFS at the 95% confidence level (p-value<0.05)**. The table below indicates that the improvement in z500 is relatively smaller than that in other variables when compared to the ECMWF-IFS, which results in the z500 curve in `Figure 4` closely resembling that of the ECMWF-IFS.\n\n|Lead time (h)|6-hour|3-day|4-day|5-day|\n|-|-|-|-|-|\n|p-value t2m|1.42e-14|1.42e-14|1.42e-14|1.42e-14|\n|p-value t850|1.42e-14|1.42e-14|1.42e-14|1.42e-14|\n|p-value u10|1.42e-14|1.42e-14|1.42e-14|2.84e-14|\n|p-value z500|1.42e-14|1.25e-03|9.86e-04|6.26e-04|\n\n> Q8: For Figure 5 seems to be a function of the contouring in Matplotlib as the difference plots show similar magnitude of errors.\n\nThe color bar we employ is consistent and standardized, with its upper and lower bounds derived from the maximum prediction error across all forecasts. By examining the error visualization in `Figure 5`, it is evident that our predictions exhibit reduced errors. The experiment depicted in `Figure 4` quantitatively demonstrates that our model boasts a relatively lower RMSE. Moreover, as detailed in the response to Q7, the results displayed in `Figure 4` hold significance.\n\nWe will introduce additional visualizations in the appendix to illustrate that our model yields comparatively smaller prediction errors.\n\nIf our responses have clarified your inquiries, we kindly ask for an update to your score. Thank you very much for your time.", "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": "t7FY2WbMre", "reviewer": "Reviewer_jiN5", "summary": "The paper proposes a physics-AI hybrid modeling framework for fine-grained weather forecast. They propose to adaptively tune a PDE kernel together with a neural network as the encoder. Following the Euler time stepping, the PDE kernel can perform a fine-grained temporal forecast, which act as a physics-guided modeling part.", "questions": "I encourage the authors to address my concerns listed in the weaknesses.", "limitations": "Limitations are discussed.", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "1. The combination of AI and physics is crucial and novel for weather forecast.\n2. The fine-grained weather forecast is of interest to nowcasting and temporal downscaling.", "weaknesses": "Some opportunities to improve:\n\n1. The experimental details are way from sufficient. What are the hyperparameters you use, except for learning rate? How do you divide validation and test set? How do you divide input and label? What are the inputs and outputs to the model? Their sizes? What are the datasets' statistics, as there is none introduced? What is the time cost or number of parameters for your model compared to other models? How do you obtain the results in the tables, since there are no error bars? Are these all based on a one-time run? Are they statistically significant, and what are the p values, etc.? Table 3 does not demonstrate that the proposed model is the best model. Is there a specific reason why 120-min forecast is especially good? Any pseudo algorithm for understanding? Is there code for understanding and checking?\n\n2. The experimental results are both not good enough and not analyzed well enough. It looks strange from Figure 4 that FourcastNet is much worse than ECMWF-IFS even though FourcastNet should have been better as reported in the original paper. There needs discussion to justify, or maybe it is due to experimental setting. It is hard to know due to lack of experimental details pointed out above. The nowcast results are basically suggesting that the proposed model is no better than previous models, except for 120 min that is a less realistic setting for real life. Figure 6 is revealing very minimum information about the comparison between models. All the errors look alike, and it is hard to know which model's prediction is better without ground truth. I am not convinced by the explanation why the physics weight is decaying within each hour: unless your neural network counterpart's prediction does not change much, the ratio between physics and AI does not directly reflect how much they change. It could be that the AI part is predicting/contributing less, although the weight is surging. It does not mean anything. \n\n3. I want to highlight this problem using a separate paragraph. The ablation study is strange: it seems the physics part is not always helping. This might relate to how you use the PDE kernel and derive the PDEs. Your appendix A needs to cite references and name the equation names you are using. For example, how do you determine the coefficient for those constants in the equations? Shouldn't that be learned, since you cannot tell how big is the friction coefficient, let alone the term, for example? Moreover, Eq. 14 seems incorrect. I cannot recall such an equation in fluid dynamics. It is the continuity equation if you change p to z. However, pressure level slash geopotential slash height are not strictly the same thing. I feel concerned that they are assumed to be the same without any justification.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer,\n\nThank you for your thoughtful review! We are pleased that you appreciate the innovative combination of AI and physics. We will address your remaining questions below.\n\n> Q1: The experimental details are not sufficient.\n\nThank you for your suggestions. The table below outlines the experimental details:\n\n|Hyperparameter|Value|\n|-|-|\n|Max epoch|50|\n|Batch size|4x8(GPUs)|\n|Learning rate|5e-4|\n|Learning rate schedule|Cosine|\n|Patch size|4x4|\n|embedding dimension|1024|\n|MLP ratio|4|\n|Activation function|GLUE|\n|Input (0-hour)|[4,69,128,256]|\n|Output (1,3,6-hour)|[4,3,69,128,256]|\n\n|Datasets|Training set|Validation set|Test set|Time resolution|Variable|\n|-|-|-|-|-|-|\n|WeatherBench|1980-2014|2015|2017-2018|1h|tp,t2m,u10,v10,z,q,u,v,t|\n|NASA|None|None|2017-2018|30min|tp|\n\n> Q2: The time cost for your model.\n\nAs shown in the `global response PDF`, introducing the PDE kernel slightly increases training time, but the added computational cost is acceptable.\n\n> Q3: Error bars and p values.\n\nThe error bars of RMSE is displayed in the `global response PDF`.\n\nWe apply the Wilcoxon test to demonstrate that **our model achieves a lower RMSE compared to the original model at the 95% confidence level (p-value<0.05)**.\n\n|Lead time (h)|6-hour|3-day|4-day|5-day|\n|-|-|-|-|-|\n|p-value t2m|1.42e-14|1.42e-14|1.42e-14|1.42e-14|\n|p-value t850|1.42e-14|1.42e-14|2.84e-14|7.10e-14|\n|p-value u10|1.42e-14|1.42e-14|1.42e-14|2.84e-14|\n|p-value z500|8.04e-04|2.83e-02|5.68e-05|6.39e-08|\n\n> Q4: Table 3 does not demonstrate that the proposed model is the best model. Why 120-min forecast is especially good?\n\nFirstly, it should be pointed out that our model did not fit 30 and 90-min predictions during training, but achieving generalization in a unified model. In other words, our model achieves SOTA on most metrics under a more stringent setting (without interpolation models).\n\nSecondly, precipitation forecasts with a lead time of 120-min or even longer serve as experimental settings `[1]` and play a crucial role in predicting disasters such as mudslides `[2]`.\n\nThirdly, as the lead time increases, the forecasting difficulty increases, and the advantages of our model can be better demonstrated. We added the tp RMSE at 180-min to emphasize that the benefits of our model are more pronounced.\n\n|Model|tp RMSE@180-min ↓|\n|-|-|\n|FourCastNet|0.88|\n|ClimODE|0.39|\n|Keisler|0.43|\n|WeatherGFT (ours)|**0.28**|\n\n> Q5: Pseudo algorithm and code.\n\nPart of the code for the PDE kernel is included in `Appendix B` of the paper. The complete code will be released after the paper is accepted.\n\n> Q6: FourCastNet should have been better than ECMWF-IFS in the original paper.\n\nIn the original paper of FourCastNet `[3]`, `Figure 1` shows that RMSE z500 of FourCastNet are higher than those of ECMWF-IFS, which indicates that its prediction is relatively poor. Even in the case of FourCastNet v2 `[4]`, its forecast skill for z500 does not surpass that of ECMWF-IFS.\n\n> Q7: All the errors in Figure 6 look alike.\n\nThe red boxes in `Figure 6` show that the predictions obtained by other AI models often have problems with smoothness and lack of extreme values. On the contrary, the predictions of our model have more details and more accurate extreme values.\n\n> Q8: Explanation on router weight.\n\nTo ease your concerns, we calculate the input norm for each router as outlined in the global response PDF. The AI and physics feature norms remain consistent and similar (0.5:0.5), indicating **the router's independence from both PDE kernels and attention blocks**. This decoupling ensures that weight variations do not influence input. **Both AI and physics branches outputs of similar magnitude**, while the router dynamically selects the better part from them.\n\n> Q9: Physics is not always helping.\n\nUsing PDE kernel without changing the training method may not necessarily lead to improved RMSE. If we only use lead time=6h labels for training supervision, the relatively front PDE kernels of this deep networks will not be effectively trained due to gradient disappearance. After adding 1h and 3h supervision to the middle part of the network, the role of the PDE kernel can be better reflected.\n\nIn addition, the *`bias`* and *`energy`* metrics in the `global response PDF` indicate that **integrating the PDE kernel effectively addresses the issue of energy decay with increasing lead time**.\n\n> Q10：How do you determine the coefficient for constants.\n\n`Equation 7` is the kinematic equation in the basic atmospheric equations. $F_h$ in `Equation 7` represents the friction force, which has a very small dimension (10e-12), so it can be omitted in the calculation refer to the `Section 2.4` of `[5]`. $fv$ in `Equation 15` represents the Coriolis force, where $f$ is the geostrophic parameter setting as a constant refer to the `Section 4.6.3` of `[5]`. Other constants are also referenced from `[5]`.\n\n> Q11: Pressure level and geopotential height are not the same thing.\n\nWhile we appreciate the reminder, we would like to clarify that we do not confuse the p-coordinate and the z-coordinate. Our PDEs are all converted to the p-coordinate refer to `Sections 1.6.1-1.6.2` of `[5]`. We will give a more detailed proof in later versions.\n\n> References\n- `[1]` *Sønderby C K, Espeholt L, Heek J, et al. Metnet: A neural weather model for precipitation forecasting.*\n- `[2]` *Brunetti M T, Melillo M, Peruccacci S, et al. How far are we from the use of satellite rainfall products in landslide forecasting?*\n- `[3]` *Kurth T, Subramanian S, Harrington P, et al. Fourcastnet: Accelerating global high-resolution weather forecasting using adaptive fourier neural operators.*\n- `[4]` *Bonev B, Kurth T, Hundt C, et al. Spherical fourier neural operators: Learning stable dynamics on the sphere.*\n- `[5]` *Holton J R. An Introduction to Dynamic Meteorology. Forth edition.*\n\nIf our responses have clarified your inquiries, we kindly ask for an update to your score. Thank you very much for your time.", "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:
Dear Reviewers,

We thank all reviewers for their efforts in reviewing our submission and their recognition of our work, e.g., *'fuse AI and PDE is innovative'* and *'demonstrate generalization capabilities'* from **Reviewer CYo5**, *'the paper does a good job address both of these existing problems'* from **Reviewer oEsH**, and *'crucial and novel for weather forecast'* from **Reviewer jiN5**.

In the following, we offer general responses to the common questions and concerns raised by the reviewers. More detailed responses to each specific comment can be found in our rebuttal to each review.

- In response to questions from **Reviewer oEsH** and **Reviewer jiN5** about the significance of our model's performance improvement, we included the Wilcoxon test to demonstrate that **our model's enhancement is statistically significant at a 95% confidence level (p-value < 0.05)**.
- **Reviewer CYo5** has expressed concerns regarding the adequacy of our explanation of integral and differential operators, while **Reviewer oEsH** and **Reviewer jiN5** have raised questions about certain experimental details. Owing to the constraints of the paper's page limit, our coverage of these specifics in the main text is incomplete. We have responded to these queries in each rebuttal and will incorporate them into the appendix for the upcoming version. Thanks for your suggestions.
- **Reviewer oEsH** has noted similarities between some of the methods in our model and those found in previous papers. Thanks for your reminder, we will include references to relevant work where appropriate. However, we contend that although these works may share some similarities, they diverge in both motivations and methods. Our research emphasizes achieving generalization on a smaller scale through physical modeling, contrasting with studies that leverage AI techniques to enhance the forecasting capabilities of a physical dynamic system.

In the `global response PDF`, we present visualizations of five new results: 
- Bias: Bias indicates the disparity between the model's predictions and the ground truth. Negative bias indicates underestimation, a prevalent issue in forecasting models. Although the PDE kernel was not specifically designed to address bias underestimation, experimental results indicate that its usage **helps ameliorate underestimation**. 
- Energy: This assesses the energy changes in the model's predictions. The experiments reveal that **employing the PDE kernel aids in energy preservation**.
- Comparison of Time Consumption: Introducing the PDE kernel slightly increases training time, but the added computational cost is acceptable.
- RMSE Error Bars: Error bars of RMSE values for different variables across various lead times.
- Router Weights and Features Norm Change: This figure complements `Figure 7` in the paper. It illustrates that **physical and AI features are on a comparable scale**, with the router dynamically selecting the more effective aspects from each. The router's weight adjustments do not impact the output of the AI or physical branches, **highlighting the router's decoupling characteristics**. 

We have provided tailored responses to each reviewer's queries. Should you have further questions, we are eager to engage in discussion and address them promptly. If our responses have clarified your inquiries, we kindly ask for an update to your score. Thank you very much for your time and feedback.

Sincerely,

Authors

Meta-review: This paper introduces WeatherGFT, a physics-AI hybrid model designed to extend weather forecasts to finer-grained temporal scales beyond the limits of the training dataset. The proposed model demonstrates state-of-the-art performance across multiple lead times and shows a notable ability to generalize to 30-minute forecasts, addressing a significant challenge in the field.

The authors have provided additional results during the rebuttal phase, including statistical tests and visualizations, which further support the robustness of their approach. Following the rebuttal, the paper received three positive scores (7, 7, 5), reflecting the overall strength of the contributions. Based on the reviewers' assessments and the authors' responses, this paper offers a valuable contribution to the field of weather forecasting.
Decision: Accept (poster)