DeepPrim: a Physics-Driven 3D Short-term Weather Forecaster via Primitive Equation Learning
Authors: Jiawei Chen
Weiqi Chen
Rong Hu
Peiyuan Liu
Haifan Zhang
Liang Sun
Venue: ICLR 2026 Poster (2026)
Keywords: Weather forecasting
Physics-informed neural networks
Primitive equations
Earth atmospheric dynamics
Deep learning.
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
TL;DR: We propose DeepPrim, a physics-informed 3D deep weather forecaster designed to learn primitive equations of the Earth’s atmospheric dynamics.

Abstract: Solving primitive equations is essential for accurate weather forecasting. However, traditional numerical weather prediction (NWP) methods often incorporate various simplifications that limit their effectiveness in parameterizing unresolved physical processes. Meanwhile, existing deep learning-based models mostly focus on pure data-driven paradigms, overlooking the fundamental physical principles that govern atmospheric dynamics. To address these challenges, we present DeepPrim, a novel 3D \underline{deep} weather forecaster designed to learn \underline{prim}itive equations of the Earth’s atmosphere. Specifically, DeepPrim aims at accurately modeling 3D atmospheric motion through Navier-Stokes equation in pressure coordinates, and effectively capturing the interactions between the solved advection and key weather variables (e.g., temperature and water vapor) through corresponding equations. By seamlessly integrating fundamental atmospheric physics with advanced data-driven techniques, our model effectively approximates complicated physical processes without relying on empirical simplifications. Experimentally, DeepPrim achieves impressive performance in both short-term global and regional weather forecasting tasks, and exhibits the superior capacity to capture 3D atmospheric dynamics. It is now deployed as part of the Baguan weather forecasting system, especially specializing in short-term forecasting. The code is available at https://github.com/DAMO-DI-ML/DeepPrim.

Published as a conference paper at ICLR 2026
DEEPPRIM: A PHYSICS-DRIVEN 3D SHORT-TERM WEATHER FORECASTER VIA PRIMITIVE EQUATION LEARNING
Jiawei 1.2 Weiqi 2. Rong Hu, Peiyuan Liu Haifan Zhang Liang Sun *Zhejiang University DAMO Academy, Alibaba Group 3 Hupan Laboratory 4 Tsinghua University
ABSTRACT
Solving primitive equations is essential for accurate weather forecasting. How- ever, traditional numerical weather prediction (NWP) methods often incorporate various simplifications that limit their effectiveness in parameterizing unresolved physical processes. Meanwhile, existing deep learning-basee models mostly fo- cus on pure data-driven paradigms, overlooking the fundamental physical princi- ples that govern atmospheric dynamics. To address these challenges, we present DeepPrim, novel 3D deep weather forecaster designed to learn primitive equa- tions of the Earth's 's atmosphere. Specifically, DeepPrim aims at accurately mod- eling 3D atmospheric motion through Navier-Stokes equation in pressure coordi- nates and effectively capturing the interactions between the solved advection and key weather variables (c.g., temperature and water vapor) through corresponding equations. By seamlessly integrating fundamental atmospheric physics with ad- vanced data-driven techniques, our model effectively approximates complicated physical processes without relying empirical simplifications. Experimentally. DeepPrim achieves impressive performance in short-term global and regional weather forecasting tasks and exhibits the superior capacity to capture 3D atmo- spheric dynamics. It is now deployed as part of the Baguan weather forecasting system, especially specializing in short-term forecasting. The code is available https: httes://githuae github. com
INTRODUCTION
Weather forecasting is an essential tool for planning, decision-making and risk mitigation in many industries (Coiffier 201T). Traditionally, atmospheric researchers developed numerical weather pre- diction (NWP) systems by solving primitive equations and other equations related to the Earth at- mosphere, such as the water vapor equation (Hurrell et al., 2013 Bouallègue et al. 2024). As a widespread and prevalent paradigm in recent decades, NWP-based forecasting systems offer no- table interpretability and accuracy, but may suffer from parameterization errors of unresolved phys- ical processes and high complexity in solving partial differential equations (PDEs) (Kochkov al. 2024]. To address these limitations, deep learning-base based weather forecasting methods have emerged, leveraging deep neural networks such as Vision Transformers (ViT's (Nguyen et al. 2023 Bi al., 2023). Graph Neural Networks (GNNs) [Lam et al.) 2023) Keisler 2022), or neural ordinary dif- Terential equation (ODEs) (Verma al. 2024
data-driven fashion based on comprehensive datasets such as ECMWF Reanalysis v5 (ERA5) (Hersbach et al. 2020).
Despite demonstrating promising performance, existing deep weather forecasters are still facing three key challenges: Pressure-tevel-wware 3D atmospheric dynamics. As illustrated in Fig. [ the weather state evolutions are influenced by the ougitude--atuuu---ressur level 3D atmospheric dynamics, which can often be described by Navier-Stokes equations (Bauer et al., 2015). While cur- rent deep forecasters achieve strong performance by modeling atmospheric variables across pressure levels as spatial-temppral videos or 2D fluid fields, these methods may not explicitly model verti-
Equul contribution. This work was done when Jiawei Chen was research intern at DAMO Academy. Corresponding author.

---

Published as a conference paper at ICLR 2026
cally coupled dynamics governed by physical principles (e.g., pressure aware advection, inter-level energy transfer). Thus, the exploration of 3D interaction modeling could serve as a complementary inductive bias to further enhance the learning of weather representation. 2 Critical physics priors of the Earth & atmosphere. While recent advances in data-driven weather forecasting demonstrate certain capabilities in capturing atmospheric physics (Hakim & Masanam 2024) with end-to-end training, most deep forecasters predominantly rely on statistical correlations from historical data without explicit incorporation of fundamental physical constraints such mass, momentum, and energy conservation. 3 Short-term forecasting for real-time decision-makine casting, especially sub-24-hour prediction, crucial for real-time actionable planning in agricul- ture, disaster mitigation,, and energy management (Weckwerth et al. 2005 Klein et al., 2015). More importantly, physical law adherence (e.g., abrupt convective processes, localized storm evolution) (Verma et al. 2024) and 3D atmospheric dynamic modeling (e.g., convective updrafts and inter-level energy transfer) are effective and crucial modeling short-term weather evolutions.
Short-term fore-
In this paper, we reconeile the intrinsic tension between these challenges and address them by incorporating physical priors to model 3D at- mospheric dynamics in a coherent framework, DeepPrim, which integrates pressure-level- aware 3D neural networks with primitive equa- tions of atmospheric physics for continuous- time short-term weather forecasting in a Neu- ral ODE system. DeepPrim first models at- mospheric motion by learning force terms in the Stvier-Stokes equation with 3-dimensional bicomponent Vision Transformer (3D-BiViT), which can explicitly model the atmospheric dy- namics in upper-air and at surface differentially yet collaboratively. Followingly, the model uses the learned advection to model key weather variables through corresponding equations, with a source- sink network to characterize the resolved and unresolved physical processes. In summary, tributions include:
Upper Variebles
Inter- Presoure-leeee Intoractions
100+Po
200NPo
Upperc--Uppe--ii
900mPs
1000kPa
Latrfate
Surface Varinbles
Intra Prese-re---eeeee Advection
Surfoce
Longitade
Figure 1: 3D atmospheric dynamics in weather evolutions with the coordinates of longitude- latitdee-resssure levels (z-g-p).
We introduce DeepPrim for short-term weather forecasting, pioneering the explicit learn- ing of primitive equations using deep learning and enhancing deep weather forecasting by integrating critical physical principles of the Earth's 's atmospheric dynamics.
We propose 3D-BiViT to learn the force terms in the Navier-Stokes equation, effectively modeling the 3D atmospheric dynamics by capturing both inter- and peraaee interactions for upper-air and surface variables differentially yet collaboratively.
DeepPrim outperforms most recent counterpart methods, reducing RMSE by 38.7% and 35.8% in short-term global and regional weather forecasting tasks, respectively. It also demonstrates superior abilities to capture pressure-level-wwarr 3D atmospheric dynamics, reliably characterizing continuous-time weather evolutions. DeepPrim has been deployed as part of the Baguan weather forecasting system (Niu et al. 2025). ), specializing in short- term forecasting.
2 RELATED WORK
Numerical Weather Prediction As a predominant paradigm, NWP systems mostly formulate the atmospheric physical rules via PDEs and numerical simulations. Typical methods include Earth System Models (ESM) (Hurrell et al. 2013) and the operational Integrated Forecast (IFS) of ECMWF (Bouallègue et al.] 2024). By integrating physics laws, NWP approaches have achieved remarkable success, demonstrating high accuracy, robust stability, and strong interpretability. On the other hand, it suffers from sensitivity to initial conditions, errors of parameterization, and high computational cost (oockoov et al. 2024).
Deep Learning for Weather Forecasting Recently, deep learning methods have emerged as pow- erful weather forecasters due to their strong expressive ability. Built upon vision backbones, e.g., VIT and its variants, ClimaX (Nguyen et al., 2023), FourCastNet (Pathak et al., 2022), Pangu (Bi eet al.] 2023), FengWu (Chen et al. (023a). and FuXi (Chen et al.) (023b) treat regional grids as patch tokens to forecast future weather states via certain pre/post training strategies. Alternatively,
2

---

Published as a conference paper at ICLR 2026
Table I: Overview and comparison of typical data-driven weather forecasters. z-y-P denote Longitde-latitud--pee levels. More detailed analysis can be found in Table Table Section B.5
Method
Communcs
Hime Sifemnalllinnn
Physks dup 3D
Key techniques for 3D atmospheric modeling
GraphCast (20233)
X
X
NIA
Clima ( (7
N/A
WeatheraF (2044)
NIA
ClimODE (21244
NIA
Pange (2023)
Unified attention blocks for s-y-p coordinates by treating pressure levels geometric heights Local ncural networks in space on indiv idual vertical columns the stmospheer that oulput tendencies.
Neura M (2024)
DeepPrin (Ours)
V /
Physics- informed dual artention for explicit and customized intra- und inter- pressure level modeling.
GraphCast (Lam et al. 2023) and Artificial Intelligence Forecasting System (AIFS) (Lagg et al., 2024) employ GNNs and transform earth systems as graph representations for weather predictions. Compared with NWP methods, deep learning-based models present higher inference efficiency and scalability, but may overlook the fundamental physical laws and often focus on forecasting at pre- defined fixed lead time.
Physics-informed Deep Weather Forecasting To integrate both physics priors and deep learn- ing's expressive capacity, physics-informed deep weather forecasting Luo et al. 22255) has been attracted increasing attention recently. For example, DeepPhysiNet (Li et al. 2024). PINNs (Cai ct al. 202I), and PINO (Li al. 202T) incorporate physics law-embedded PDEs into loss functions to guide the deep forecasters. ClimODE (Verma et al.) 2024), as a pioneering work, first intro- duces continuity-equation -insaiire neural advection within Neural ODE framework for weather forecasting. NeuralGCM (Kochkov et al. (2024) combines general circulation models with machine learning for weather and climate. WeatherGFT (Xu al. 2024) incorporates physical PDEs into ViTs in a parallel manner and leverages the latter as the correction module. Recently, (Karlbauer al. 2024) provides a systematic comparison of deep learning backbones for Navier-Stokes and atmospheric dynamics, offering a useful guideline for backbone choices.
Deep Learning for PDEs Beyond weather-specific applications. there is rapidly growing literature on deep learning methods for partial differential equations (PDEs) that explicitly incorporate phys- ical constraints. Hard-constrained approaches learn differentiable solvers that enforce constraints during training and inference (Négiar et al., (022). and have recently been scaled using mixture- architectures (Hansen et al.)
of-experts designs (Chalapathi et al.) and conservation-lav-resseett 2023). Other works leverage uncertainty quantification to improve out-of-domain generalization for PDEs (Mouli et al.)) or formulate end-to-end probabilistic frameworks for learning with hard constraints (Utkarsh al. 2025).
Research Gap In general, existing weather forecasters mostly focus on ppatial-temporal variations by treating weather states videos or 2D fluids. The pressure-level information and interactions, de- spite being fundamental and critical, are seldom investigated and not explicitly exploited, as shown in Table Pangu (Bi et al. 2023) height information, which may not explicitly distinguish heterogeneous horizontal-vertion tions. NeuralGCM (Kochkov al. (024) employs individual networks to model physical tendencies in the corresponding single vertical column, which may omit the inference of intra- petra-pressure-lvee in- fluences. Inspired by the primitive equations, our DeepPrim bridges the gap between pressure-level. aware explicit and flexible 3D dynamics modeling with physics-informed continuous-time weather forecasting in the neural ODE system, offering more realistic and reliable short-term weather forecasting paradigm.
unified attention blocks and positional embedding to encode interac-
3 LEARNING PRIMITIVE EQUATIONS VIA DEEPPRIM
We model the evolution of weather states as a spatial-temporal process u(c,t) (u (c, (x(ctt) E IR of K variables u (c t) E R over continuous time E IR and latitude- x [-180P, 180 E RR?. The total number of variables is represented as K Ds + P Du. where Ds denotes the surface variables, include Uy (latitude-directian wind), U, (longitude-diree direction wind), and T (temperature). Additionally, Du refers to the upper-air variables measured at P pressure levels, with added & (geopotential) and q
longitude coordinates c = (a. y) E Ω
[-900, 909 [
3

---

Published as a conference paper at ICLR 2026
(humidity). The weather forecasting problem can be formulated as:
2 :t)
ú(t : :xN),
(I)
where f(-) denotes the (learndd) weather dynamics. The model takes as input three weather states, which correspond to the current time, lo, and predicts weather states at the next N time steps. The complete notations and explanations are summarized in Table 5 in Section B.
3.1 PHYSICAL MODEL
Solving the primitive equations (along with the water vapor equation) of the Earth's atmosphere, which comprise a set of equations related to wind, temperature, pressure, geopotential, and water vapor, provides the foundation for NWP (Kalnay) 2003). Although various variables are intercon- nected, the momentum equation that describes atmospheric motion serves as the cornerstone (Bauer) itt al. 2015),, as the variations of other weather variables (such as temperature and water vapor) are primarily driven by horizontal advection within the atmospheric circulation scale (Emanuel) [994]. Therefore, accurate modeling of atmospheric motion is crucial for forecasting these prognostic vari- ables. To achieve this, we first consider the Navier-Stokes equation in pressure coordinates (z-yyp), which enables us to account for detailed viscous friction as
fkxv Corialis force
dt
Op?
(2)
presouuee gradiest force
Viscons friction
where v = -(2, vy)) is the horizontal wind, & is the geopotential, f 7.29e-5, V, =
8 8
Ge by
denotes horizontal gradients, and A are the viscosity coefficient and turbulent exchange coefficient, respectively. The expansion d(-)
dr is as follows:
(-) a()
8(-)
di de +v +v. (.
Op
where is the vertical wind, which dynamically evolves over time and can be derived from the continuity equation:
1 @v Dr
Ouy Dy
w
dr
dp.
(4)
The coefficients and A are typically estimated using empirical formulas in practice, which may lead to a certain degree of imprecision. Moreover, other physical processes are modeled under moderate simplifications when solving primitive equations. For example, in most general circulation models, temperature is constrained by a pseudo-adiabatic process considering moist processes such condensation and evaporation, while other processes, including heat originating from friction, radiative heating, and cooling, remain not considered (Randall et al. 2007 Stocker 2014).
To describe these processes, parameterization methods are introduced to approximate the interac- tions between resolved and unresolved processes. It is important to note that these parameterization methods are not unique and be customized or adjusted within different numerical schemes based on experiments and observations. However, since experiments and observations data are geograph- ically limited, this can result in considerable variations in the calculated outcomes of parameteri- zation schemes for the same process. Thus, the selection of specific parameterization schemes and configurations can lead to significantly different simulation results (Banks al., 2016]. Conversely, deep neural networks offer a way to bypass these simplifications by directly learning resolved and unresolved processes through a data-driven approach.
Takeaways. The aforementioned mathematical models provide three key physical insights guiding the design of our learning framework As estimating the horizontal advection is the key in solv- ing the primitive equations, the modeling of horizontal motion should be carefully designed. 2 The viscous friction term in the Navier-Stolfe equations highlights the appropriateness of incorporating higher-order gradients and 3D interactions to accurately model atmospheric motion. The pressure gradient force term suggests that additional variables should also be incorporated.
The use of
We provide detailed formulations of primitive equations and analyze their relation to our model designs in Section
4

---

Published as a conference paper at ICLR 2026
(a). Overall Pipeline of DeepPrim
(b). Force Network (3D-BIVIT)
Dattal Weather Stutes
Burface feetures Re. Vuy, Y5 Var
Toleevartion
Selfurmantin
Self Altection
pl tempers
Upper festures Bey Vuay Vo
Patchify Taken aollinn
Entre-praare Call-AmTation
embedd uo Op
Airem woot
Serfucn
1995
Projector
Protacten
800 e
C
000
000
000
Uul
Londin Spmi enbed
Presure- unbedding
(c). Initialization Network
(d). Source-sink Network
Repert
Net
Adrccionn'
(PuGe)
OCE Schvr
(x100)
Trahstartion
Sourc sink Nej
130 000
1d0 end
(800)
009
600
Future Piedictiees
3D ocendinotes
U6J
empesding Boy
Pay
WGA
Figure 2: DeepPrim Framework and key modules, including (a) overall pipeline and architectures of (b) force network based on 3D-BiViT (Sec. 3.2.2), (c) initialization network (Sec. 3.2.10 and (d)
source-sink network (Sec. (.2.3) h. More figure details can be found Appendix B.. and B.4
parameterization methods encourages us to learn a source-sink term in the advection equation to simulate both resolved schemes with learnable source-sink network that minimizes the discrepancy between simulations and observations.
and unresolved processes. By doing so, we can replace the parameterization
3.2 DEEPPRIM AS PHYSICS-DRIVES WEATHER FORECASTER
Model Overview Building upon the insights derived from the physics discussed in Section 3.1 we model the force term and the source-sink term using specially designed neural networks. Treating these networks as universal approximators, the derived equations are expressed as follows:
Ov
Ov"
+ Force(u, v")
0t
0p
du
Source-Sink(u,, v"), Op
at
where w can be calculated by Eq. (4). The force model Force(-) encapsulates external forces includ- ing pressure gradients, Coriolis forces, and frictional effects. Mcanwhile, the source-sink model Source-Sink(.) embodies the contributions from both resolved and unresolved processes, which may include phenomena such as turbulent mixing and energy exchanges that are not explicitly mod- eled. important note that in the formula for the primitive equations, denotes horizontal wind, and u typically represents other prognostic variables (i.e.. T, q, Φ). To enhance the model's capability and flexibility, our learning framework treats u by including all prognostic variables, en- compassing horizontal wind, with v denoting the intermediate atmospheric motion. Consequently, applying the source-sing sink network the wind component can be viewed as correction mechanism.
Given the initial values u (to) and v* (ta), the numerical solution can be accurately approximated with ODE solvers such as Euler (Biswas al. 2013) and Kunge-Kutte (Runge) 1895) by discretizing the system into multiple discrete time steps as in Eq. (6) as follows:
u
u (to)
dr,
(to)
(T(T)
(6)
(1)) (Ux, My (to) + Initialization(u) (fo)).
Backpropagation of ODEs is compatible with standard autodiff, while also admitting tractable ad- joint form (Chen et al. 2018). Since the introduced intermediate motion v" is not entirely equivalent to the wind v, utilize residual initialization network estimate the initial values of v*.
To sum up. we propose DeepPrim with its overall pipeline depicted in Fig. (a). It consists of three key modules motivated by Eqs. (5 6): Initialization network that estimates the initial values of the
5

---

Published as a conference paper at ICLR 2026
intermediate motion fields considering the temporal variations and 3-dimensional spatial gradients of weather states; 2) Force network that leverages 3-dimensional bicomponent Vision Transformer (3D-BiViT) to model the time deviation of intermediate motion for upper-air and surface separately inspired by the force term of the Navier-Stokes equation; 3) Source-sink network that captures the value gains losses of each variable caused by resolved and unsolved processes for final predic- tions. Note that we do not simulate the source and sink rate at each discrete ODE step in Eq. (5), but directly model the total amount of the source and sink after integration from start time to fore- cast time, which can ensure the stability of ODE integration and avoid overfitting Details of each module are elaborated below.
3.2.1 INITIALIZATION NETWORK
Estimating initial values is crucial for ensuring the precision and stability of ODE solutions. In the numerical solutions of primitive equations in NWP, the initial values are typically derived from assimilated weather states. As we incorporate the intermediate motion into our learning framework, we introduce an initialization network that estimates the initial values of this intermediate motion more coherently within the ODE learning system.
Specifically, we model the initial intermediate motion by modulating the horizontal wind using con- volutional networks. This approach accounts for both temporal variations and 3D spatial gradients of the initial weather conditions, represented mathematically as:
v² (to) = [Vas vy/o + Conv Conv(u(t-2 to), ú(to):
Vu(to)
(5T),
temporal varlations spacial gradients
0. denotes 3D spatial gradients. The initialization network integrates vari- where Bar By 8p
ous additional features to capture essential information relevant to atmospheric conditions. These features include spatial-temporal derivatives of various weather variables, land-sea mask, incoming solar radiation, orography, and spatial-temporal embeddings (denoted (SST) that convey both tem- poral information (such as day and season) and spatial location information (longitude and latitude). Detailed implementation of these embeddings can be found in Section B.3
3.2.2 FORCE NETWORK
The force network is introduced to learn the force term in the Navier-Stokes equation for modeling atmospheric dynamics. As previously discussed, the structure of viscous friction implies 3D interac- tions within the neural network. Thus, design a 3 3-dimensional bicomponent Vision Transformer (3D-BiViT) to learn the force term for upper-air and surface, differentially yet collaboratively. This network utilizes features consistent with the initialization network as input along with v (t), en- compassing three core modules.
Tokenization and Embedding Technically, we adopt ViT (Dosovitskiy et al. 2021 as the back- bone. As depicted in Fig (b), the input channels are first tokenized into a sequence of N (H/p) x (W/p) h w patches, where p x p the patch size and N is the number of patches. These input tokens are then embedded into vectors of dimension embed_dim with convolution net- works. Moreover, due the distinct characteristics of surface and upper-air dynamics, we set two distinct embedding layers for upper-air and atmospheric varibles.
Pressure-Level Embeddings Subsequently, these token embeddings are added with adaptive pressure-level embeddings. The pressure-level embeddings are designed to inform the model of pressure information. We implemented the pressure-level embeddings t/p as learnable parameters with shape [1, 1, P embed dim], where P+1 1 is the total pressure levels (with upper-air variables of P and surface variables of 1). The pressure-level embeddings are added to the token embeddings with shape N, P 1, embed_dim)
Level Attentions To characterize 3D atmospheric dynamics, we pro- pose a two-stage self-attention mechanism for 3D dynamics learning. For the concatenated repre- sentations of surface and upper-air variables [Zg. Zul with shape (B x (P+1) x N x embed .dim), the intra- pressure self-attention operates the tokens within each pressure level (i.e.. N) to model the inter anter-pressure-evw dynamics. Afterwards, the cross- pross-rressur level self-attention layer operates on the tokens across different pressure levels (i.e., P + 1) model the vertical (nnra-pressuue-levee)
Intra- and

---

Published as a conference paper at ICLR 2026
atmospheric dynamics. Similar with embedding layers, projections of query, key, and value, as well as the forward networks in the intra-pressure attention blocks, are learned separately for upper-air and surface tokens.
3.2.3 SOURCE-SINK NETWORK
The weather state evolutions may undergo value loss gain processes due to the energy or matter exchange within the atmosphere and between the atmosphere and the external system, such as the radiative heating and cooling, day-night cycle, as well as condensation and evaporation. To model such impacts, we introduce source model to further account for these resolved and unresolved processes. Following the aforementioned discussion, in Eq. (5), but model the total amount of source and sink from to to {t4)i=]. Specifically, the source-sink network receives the initial weather states u (fv) and initial intermediate motion v* (to),
do not model the source and sink rate
and ODE solutions (u' (t,))== N along with spatiotemporal embeddings ØST input correct the output of ODE solutions, which is formulated as
(á (++) = Conv (fu v" (u v) + fu' (t.)
(8)
where (á (t+)) the final prediction of the model.
3.3 OPTIMIZATION OBJECTIVE
We provide the overall implementations of DeepPrim in Algorithm 1 in Section B.2 In the training stage, we minimize the latitude-weightt MSE loss between the predicted values ñ and the ground truth the target lead time tN P
H ΣΣΣ a(h) (0,,h,, (tN) (t
K
C
KHW
where (h) is the latitude weighting factor that accounts for the varying grid cell areas on spherical Earth, and more details on the weighting factor can be found in Section E
4 EXPERIMENTS
Dataset We use the ERA5 dataset (Rasp et al. 2020) with 5.625 (64 x 32 latitude-longitude grids), 1.40625 resolutions (256 x 128) and 0.25 resolutions (1440 720). Following ClimaX (Nguyen et al. 2023),
the data from 1979 to 2015 the training set, 2016 the validation set, and 2017 to 2018 as the test set. The performance is evaluated on five key variables: geopotential at 500 hPa (z500), temperature 850 hPa (t850), temperature at 2 meters (f2m), and wind speeds at 10 meters (u10 and v10). More details of the dataset and experimental settings can be found in Section D
Baselines We extensively compare DeepPrims with several counterparts on Resolution-Matched
"Apples-to-Apples" comparison scenarios: FourCastNet (FCN) (2022) ClimaX (2023) Pangu (2023), GraphCast (2023), ClimODE (2024), NeuralGCM (2024) and WeatherGFT (2024) We also add Integrated Forecasting System (IFS) (2023) as a gold-stane gold-standard baseline, which is recog- nized as amongest the most skillful NWP in the world. Also, we evaluate the performance using latinudewweighted root mean squared error (RMSE). More details of baselines and the performance results can be found in Section Faad Table [6 More details of metrics can be found in Section H
The backbones of the initialization and the source-sink models are ResNet (He et al., 20166), and the backbone of the force network is based on ViT (Dosovitskiy et al.) 2021). We report the results of two variants of DeepPrim: 1) "DeepPrim" takes all surface and upper-air variables as output variables for training; "eeeeeprrm only outputs five key variables, in line with ClimODE and ClimaX. More implementation and hyperparameter details are in Section
Implementations
Tasks We consider the tasks of global weather forecasting (Section 4.1 and regional weather fore- casting (Section of Appendix). More detailed task settings are provided Section D.3
"Alternatively, one can also add supervision signals to the sequence of weather states {ú (f:) In our experiments, we chose the form in Eq. ( for computational efficiency.

---

Published as a conference paper at ICLR 2026
Table 2: Global weather forecasting results on 5.625 and 1.40625" ERA5 dataset. The best results are bolded and the second-best results are underlined, except for IFS. The RMSE curves along lead time are shown in Fig. 9
The result sources are summarized in Table 16
Latinud-wwiighted RMSE (5.625" ERA5)
Latiudo-weighted RMSE (1 (1.40225 ERA5)
Lead-Time At (h)
Variable
MODIE [200
FCN
CliraX' [2225]]
ClmuX [+00
ClarOSE [+114]
Degphin Ozs
DespPrin! Okes
NeurlOCM 12:0
ClinsX 20023
WashorFF [2024
Desphin
[2523]]
Oun
6
26.9
300.6
149.4
247.5
62.7
102.9
50.1
53.6
26,96
28.43
49.67
22.10
27.44
12
(NIA)
460.2
217.8
265.3
81.9
134,8
71.0
77.4
(NIA)
35.01
(N/A)
36.71
40.15
2500
18
(N(A)
627.6
275.0
319.8
88.9
162.7
94.1
102.2
(N/A)
40.52
(NIA)
41.41
46.82
24
51.0
877.8
333.0
364.9
96.2
193.4
121.0
126.7
50,96
48.21
72.76
54.13
59.72
6
0.69
1.82
1.18
1.64
0.88
1.16
0,76
0.72
0.69
0.52
0.84
0.46
0.44
12
(NIA)
2.32
1.47
1.77
1.09
1.32
0.93
0.87
(N/A)
0.58
(N/A)
0.59
0.53
1850
(NIA)
2.93
1.65
1.93
1.04
0.98
(NIA)
0.62
(NIA)
0.63
0.55
24
0.87
3.35
1.83
2.17
1.11
1,55
1.13
1.06
0.87
0,76
1.02
0,71
0.69
0.97
2.72
1.28
2.02
0.95
1.21
0.94
0.73
0.97
(NIA)
1.11
0.67
0.63
12
(N/A)
3.16
1.48
2.26
1.24
1.45
1.12
0.88
(N/A)
(N/A)
(NIA)
0.72
0.70
12m
18
(N/A)
3.45
1.61
2.45
1.19
1.43
1.17
0.97
(N/A)
(NIA)
(NIA)
0.76
0.73
1.02
3.86
1.68
2.37
1.19
0.99
(NIA)
1.19
0.81
0.76
0.80
2.30
1.47
1.58
1.08
1.41
0.89
0.82
0.79
(NIA)
1,04
0.54
0.62
12
(N/A)
3.13
1.89
1.96
1.23
1.81
1.07
1.00
(N/A)
(NIA)
(NIA)
0.71
0.71
w10
18
(NIA)
3.41
2.05
2.24
1.27
1.97
1.24
1.15
(N(A)
(NIA)
(N/A)
0.77
0.79
4.10
2.33
2.49
2.01
1.39
1.25
1.11
(N/A)
1.31
0.90
6
0.94
2.58
1.54
1.60
(N/A)
1.53
0.92
0.85
(NiA)
(N/A)
(N/A)
(N/A)
(NIA)
12
(NIA)
3.19
1.81
1.97
(NIA)
1.10
1.03
(N/A)
(N/A)
(NIA)
(NIA)
v10
18
(NIA)
3.58
2.11
2.26
(N/A)
1.96
1.27
1.19
(N/A)
(N/A)
(NIA)
(NIA)
(NIA)
24
1.33
4.07
2.39
2.48
(N/A)
2.04
1.43
1.24
(N/A)
(N/A)
(N/A)
(N/A)
(NIA)
(I), "DoeeDDoPri"" " takes all surfeee and upper-air variables as cotput varishles for tminnig, whose full reulls are summorfeed in Table 19 "DeepPimm only ouppus
live key variables: (2), "ClimaX"* ** denotes ClimaX that NeuraGGUM model; The results 1tS cited from Climax (220m))2222555)ll IFS WoaterBeack2 Section
trainedi CMIS6 (Eyning al. (016) (3), "NeuraGGCM" denotas the official 1.4" deieministic
those ClimDDE and WeatherGFF. We provide discussion
4.1 GLOBAL WEATHER FORECASTING
We summarize the results of DeepPrim and baselines in Table 2 (5.625 and 1.40625 ERA5 datasets) and Table 3 (0.25% ERA5 datasets) at At {6. 12, 18, 24} hours. We also present and discuss the forecasting at longer lead time, i.e., At = (36, 72, 144) hours, in Section G.2
Results on 5.625 Our DeepPrim/DeepPri'' generally achieves the best RMSE, outperforming ClimODE by 38.6%, and suppressing the pre-trained ClimaX by 10.7% on average. DeepPrim is compara- ble with IFS and outperforms IFS in predicting t2m and v10. Also, the ceeooutputvariabble DeepPrim generally outperforms the cotouuutttvriibbbt Deep- Prim, full-variable optimization may require bal- ance and compromise for all variables.
Table 3: Global forecast results on 0.25%.
Variable
Load Time
IFS
Pangu
GraphCastt
DeepPrinn
(houns)
(2223)
(2023)
Outs
27.6
16.9
15.9
24.3
12
38.6
30.2
28.5
31.8
:500
18
39.7
35.6
31.2
36.3
24
50.8
45.2
41.0
44.8
0.70
0.42
0.27
0.24
12
0.77
0.58
0.45
0.44
185
0.78
0.64
24
0.86
0.72
0.59
0.58
2 Results on 1.40625 Our full-output-variabe DeepPrim surpasses IFS in 1850, t2m, and u10 pre- dictions and outperforms the pre-trained ClimaX by 36.11% on average. With only 1/20 of the param- eters of WeatherGFT as shown in Table 10] Deep- Prim achieves comparable performance while ex- celling in forecasting 1850 and 12m. DeepPrim also achieves competitive performance with Neural- GCM.
6
1.05
0.62
0.51
0.53
1,10
0.68
0.54
0.56
18
1.07
0.75
0.58
0.56
24
1.14
0,72
0.63
0.62
0.95
0.47
0.38
0.34
w10
1.08
0.70
0.61
18
1.11
0.79
0.68
0.67
1.23
0.91
0.81
0.98
0.38
12
1.1
0.72
0.63
0.58
110
1.1
0.82
0.70
0.66
24
1.26
0.94
0.84
0.82
4 Results 0.25 In comparison with IFS, Pangu(0.255) and GraphCast(0.255
DeepPrim has
achieved the best overall performance, especially in the prediction of U &V wind speed, as Deep- Prim comprehensively models the force term and horizontal advection in physically-informed manner. GraphCast excels at z500, as leverages a graph-based structure that explicitly encodes both local and global interactions for modeling large-scale global circulation patterns.
4.2 REGIONAL WEATHER FORECASTING
In practice, making regional weather forecasting based on accessible local data is also critical. Tra- ditional NWP methods require the given boundary conditions for regional prediction. In contrast,
8

---

Published as a conference paper at ICLR 2026
2500 [m/1s")
1850 [K]
12m [K]
610 [mus]]
VLO [mis]
Source-drt
150
1.05
3.5
15
will triiniztton
1.55
15
23
weld iniiication Rot
win modiles
30
23
24
24
DrepPrin
Lood Time (hours)
Load Time (houss)
Liad Time trours)
Load Time thours)
Load Time (hours)
Figure 3: Ablation studies of DeepPrim based on the global weather forecasting task. results can be found in Table 20
The full
component
Ground Truth
Grouns Tur
Ground Trush
Irdidiinns medim
(surface)
heltiia ol Desshim
Red cio el Dazshiim
Pretition ol Desshin
Temperature 850hPa
-10
Geopotential
Temporature 700hPu
Indicion of we modu
helitiain of 30
Prodition of 30 medu
-10
36
Temporature at 600hPu
tner m
Eeer Desphrr
Ener DeepPrin
-335
Temperature
Temporature at 500hPs
-1.55
-3.2
Ever mobules
trar mod
Crte ct 33 modulss
Temparature at 250hPs
Pressure Levels
wind 10m (Surface)
wind at 700hPa
U wind at 2500h
(a) Corr. maps
(b) Continuous-time preds.
(c) Global predictions and errors (Ar = 12h)
Figure 4: Necessity and Effect of 3D atomphric dynamic modeling. (a) Correlation maps of typical weather variables at different pressure levels based on ERA5 dataset; (b) Continuous-timg predictions for temperature at different pressure levels of a specific location based on DeepPrim and its w/o 3D module version; (3) Global weather forecasting for U component of wind at different pressure levels.
DeepPrim adaptively learn the boundary conditions in data-driven manner and is flexi- ble. Following previous work (Verma et af. 2024), we focus on regional forecasting in three typical areas: North America, South America, and Australia. We summarize the comparison in Table on Appendix A It can be observed that DeepPrim consistently achieves the best forecasting per- formance for five variables across three regions, suppressing the second-best baseline ClimODE or ClimaX by 35.8% on average in different tasks, which validates the superiority of DeepPrim in effectively capturing regional weather dynamies via 3D pressure-level aware modeling.
4.3 INVESTIGATION OF 3D ATMOSPHERIC DYNAMIC MODELING
As presented in Fig. (a), some weather variables (e.g., wind speed, temperature) at different pres- sure levels show typical correlation patterns, with variables more adjacent pressure levels showing higher correlations. As mentioned in Section 3.1. the correlation maps further emphasize the neces- sity of 3D atmospheric dynamic modeling.
Ablation evidence We conduct ablation studies (denoted as w/o 3D modules) by removing the 3D- related modules of DeepPrim, i.e., cross-pressure self-attention and the pressure level derivatives V uu and equally model the surface and the upper variables in the 2D coordinates. As shown in Fig. removing the 3D modules would bring an average performance degradation of 12.3%.
Visualization showcase Fig. (b) presents the continuous time predictions based DeepPrim and its w/o 3D module version for local grid with a lead time At = 12h. Remarkably. the temperature at different pressure levels presents typical correlated variations and generally shows downward trends due to the dig-night cycle. Due to pressure-leved -vae 3D modeling designs, our DeepPrim successfully captures these correlations and models the variation trends of weather evolutions

---

Published as a conference paper at ICLR 2026
more precisely in comparison with its w/o 3D module version. Furthermore, Fig. 4 (c) highlights the improvement of predictions and errors on a global scale led by the 3D modeling strategy.
4.4 ABLATION STUDIES
Investigation of initialization network We adopt both the temporal variations ú and 3D spatial derivations Vu of initial conditions to estimate the initial intermediate motion fields v² (to). Ac- cording Fig. 3 both the two terms can facilitate the estimation and bring performance improve- ments, and the inclusion of Vu enjoys better improvement.
Effeet of source-sink network The results Fig. alss emphasize the importance of the source- sink network, as it is vital to characterize the value loss gain processes of weather states, e.g., day-night cycles, and can also alleviate the error accumulation problem of the ODE systems.
5 CONCLUSION
In this paper, we introduce the 3D atmospheric modeling in a physics-driven deep-learning paradigm, and propose DeepPrim for short-term weather forecasting. Through learning primitive equations, DeepPrim effectively models atmospheric motion by portraying the force term in Navier- Stokes equation with 3D-BiViT and captures weather state evolution by leveraging the learned ad- vection and source-sink sink model. In the future, we plan to extend DeepPrim with customized rolling training and inference strategies to improve long-term predictions and explore more applications by incorporating additional physical priors with rigorous stability and conservation analysis and advanced time-stepping methods.
10

---

Published as a conference paper at ICLR 2026
ACKNOWLEDGEMENT
This work was supported by DAMO Academy through DAMO Academy Research Intern Program.
7 ETHICS STATEMENT
Our work focuses solely on scientific problems and does not involve human subjects, animals, or environmentally sensitive materials. We foresee no ethical risks conflicts of interest.
REPRODUCIBILITY STATEMENT
We have rigorously formalized the model architecture through equations and illustrations in the main text. We involve the reproducibility details in Appendix, including algorithm implementa- tions and hyperparameters (Appendix dataset descriptions (Appendix D). and evaluation met- com/ DI-ML/
rics (Appendix E
We provide our source code in:
DeepPrim
USE OF LLMS
The authors used LLM solely as general- formatting suggestions. LLM did not contribute research ideation, experimental design, data analysis, or interpretation, and all content was reviewed and is the full responsibility of the authors.
general-purpose purpose assistive tool for grammar refinement and minor
REFERENCES
Robert F Banks, Jordi Tiana-Alsina. José María Baldasano, Francese Rocadenbosch, Alexandros Papayannis, Stavros Solomos, and Chris Tzanis. Sensitivity of boundary-layes variables to phl schemes in the wrf model based surface meteorological observations, lidar, and radiosondes during the hygra-cd campaign. Atmospheric Research, 176:185-201, 2016.
Peter Bauer, Alan Thorpe, and Gilbert Brunet. The quiet revolution of numerical weather prediction. Nature, 525(7567):47-55,2)) 2015.
Kaifeng Bi, Lingxi Xie, Hengheng Zhang, Xin Chen, Xiaotao Gu, and Qi Tian. Accurate medium- range global weather forecasting with 3d neural networks. Nature, 619:533-5388 2023.
BN Biswas, Somnath Chatterjee, SP Mukherjee, and Subhradeep Pal. A discussion euler method: A review. Electronic Journal of Mathematical Analysis and Applications. 1(2):2090-22) 2792, 2013.
Zied Ben Bouallègue, Mariana C. A. Clare, Linus Magmusson, Estibaliz Gascón, Michael Maier- Gerber, Martin Janoušek, Mark Rodwell, Florian Pinault, Jesper S. Dramsch, Simon K. Lang, Baudouin Raoult, Florence Rabier, Matthieu Chevallier, Irina Sandu, Peter Dueben, Matthew Chantry, and Florian Pappenberger. The rise of data-driven weather forecasting: A first statistical assessment of machine learning-ban based weather forecasts in operational-like context. Bulletin of the American Meteorological Society, 105(6):E864 F883, 2024. doi: 10.1175/BAMS --2AMS-D-22-012..1.
Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Kariiadakis. Physics- informed neural networks (pinns) for fluid mechanics: A review. Acta Mechanica Sinica, 37(12): 1727-1738, 2021.
Nithin Chalapathi, Yiheng Du, and Aditi S Krishnapriyan. Scaling mhysics-informec hard con- straints with mixture-off-eppets. The Twelfth International Conference on Learning Represen- tations.
Kang Chen, Tao Han, Junchao Gong, Lei Bai, Fenghua Ling, Jing-Jia Luo, Xi Chen, Leiming Ma, Tianning Zhang, Rui Su, et al. Fengwu: Pushing the skillful global medium-range weather forecast beyond 10 days lead. arXiv preprint 2xXi:23344002455 2023a.

---

Published as a conference paper at ICLR 2026
Lei Chen, Xiaohui Zhong, Feng Zhang, Yuan Cheng, Yinghui Xu, Yuan Qi, and Hao Li. Fuxi: A cascade machine learning forecasting system for 15-day global weather forecast. npj Climate and Atmospheric Science, 6(1):190, 2023b.
Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K. Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018.
Jean Coiffier. Fundamentals of numerical weather prediction. Cambridge University Press, Cam- bridge; New York, 2011.
Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, Jakob Uszko- reit, and Neil Houlsby. An image is worth 16x 16 words: Transformers for image recognition at scale. In International Conference on Learning Representations, 2021.
ECMWE. IFS Documentation CY48RI. ECMWF, 2023.
Kerry A Emanuel. Atmospheric convection. Oxford University Press, USA, 1994.
Veronika Eyring, Sandrine Bony, Gerald A Meehl, Catherine A Senior, Bjom Stevens, Ronald J Stouffer, and Karl E Taylor. Overview of the coupled model intercomparison project phase (cmip6) experimental design and organization. Geoscientífic Model Development, 9(5):1937- 1958, 2016.
Yuan Gao, Hao Wu, Ruiqi Shu, Fan Xu, Rui Ray Chen, Yibo Yan. Qingsong Wen, Xuming Hu, Kun Wang, Jiahao Wu, et al. Oneforecast: A universal framework for global and regional weather forecasting. In Forly-second International Conference on Machine Learning.
Gregory Hakim and Sanjit Masanam. Dynamical tests of a deep learning weather prediction model. Artificial Intelligence for the Earth Systems, 333):2 2024.
Derek Hansen, Danielle C Maddix, Shima Alizadeh, Gaurav Gupta, and Michael W Mahoney. Learning physical models that can respect conservation laws. In International Conference on Machine Learming, pp. 12469-12510. PMLR, 2023.
Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recog- nition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp- 770-778, 2016.
Hans Hersbach, Bill Bell, Paul Berrisford, Shoji Hirahara, András Horányi, Joaquín Muñoz-Sabater, Julien Nicolas, Carole Peubey, Raluca Radu, Dinand Schepers, et al. The era5 global reanalysis. Quarterly Journal of the Royal Meteorological Society. (6(7300):199-2000 2020.
James R. Holton and Gregory J. Hakim. An Introduction to Dynamic Meteorology. Academic Press, 5th edition, 2013.
James Hurrell, Marika Holland, Peter Gent, Steven Ghan, Jennifer Kay, Paul Kushner, J-F Lamar- que, William Large, D Lawrence. Keith Lindsay, et al. The community earth system model: framework for collaborative research. Bulletin of the American Meteorological Society, 94(9): 1339-1360 2013.
E Kalnay. Atmospheric Modeling, Data Assimilation and Predictability, volume 341. Cambridge University Press, 2003.
Matthias Karlbauer, Danielle C Maddix, Abdul Fatir Ansari, Boran Han, Gaurav Gupta, Yuyang Wang, Andrew Stuart, and Michael W Mahoney. Comparing and contrasting deep learn- ing weather prediction backbones on navier-stokes and atmospheric dynamics. arXiv preprint arXiv:2407.14122 2024.
Ryan Keisler. Forecasting global weather with graph neural networks. arXiv preprint arXiv:2202.07575. 2022.
Benjamin Klein, Lior Wolf, and Yehuda Afek. A dynamic convolutional layer for short range weather prediction. In Proceedings of the IEEE Conference on Computer Vision and Pattenn Recognition, pp. 4840- 2015.
12

---

Published as a conference paper at ICLR 2026
Dmitrii Kochkov, Janni Yuval, Ian Langmore, Peter Norgaard, Jamie Smith, Griffin Mooers, Milan Klöwer, James Lottes, Stephan Rasp, Peter Düben, et al. Neural general circulation models for weather and climate. Nature, 632(8027):1060-1066 2024.
Remi Lam, Alvaro Sanchez-Gonzalez, Matthew Willson, Peter Wirnsberger, Meire Fortunato, Fer- ran Alet, Suman Ravuri, Timo Ewalds, Zach Eaton-Rosen, Weihua Hu, et al. Learning skillful medium-range global weather forecasting. Science, 382(6677):1416-144 2023.
Simon Lang, Mihai Alexe, Matthew Chantry, Jesper Dramsch, Florian Pinault, Baudouin Raoult, Mariana CA Clare, Christian Lessig, Michael Maier-Gerber, Linus Magnusson, et al. Aifs- ecmwf's data-driven forecasting system. arXiv preprint arXiv:2406.01440 2024.
Wenyuan Li, Zili Liu, Keyan Chen, Hao Chen, Shunlin Liang. Zhengxia Zou, and Zhenwei Shi. Deepphysinet: Bridging deep learning and atmospheric physics for accurate and continuous weather modeling. arXiv preprint arXiv:2401.04125 2024.
Zongyi Li, Hongkai Zheng, Nikola Kovachki, David Jin, Haoxuan Chen, Burigede Liu, Kamyar Azizzadenesheli, and Anima Anandkumar. Physics-informed neural operator for learning partial differential equations. ACM/MMS Journal of Data Science, 2021.
Yingtao Luo, Shikai Fang, Binqing Wu, Qingsong Wen, and Liang Sun. thysics-guided learning of metcorological dynamics for weather downscaling and forecasting. In Proceedings of the 31 ACM SIGKDD Conference on Knowledge Discovery and Data Mining V.2, KDD '25, pp- 2010 2020. Association for Computing Machinery, 2025.
S Chandra Mouli, Danielle C Maddix, Shima Alizadeh, Gaurav Gupta, Andrew Stuart, Michael W Mahoney, and Bernie Wang. Using uncertainty quantification to characterize and improve out-of- domain learning for pdes. In Forty-first International Conference on Machine Learning.
Geoffrey Négiar, Michael W Mahoney, and Aditi S Krishnapriyan, Learning differentiable solvers for systems with hard constraints. arXiv preprint arXiv:220.008675 2022.
Tung Nguyen, Johannes Brandstetter, Ashish Kapoor, Jayesh K Gupta, and Aditya Grover. ClimaX: A foundation model for weather and climate. In ICML, 2023.
Peisong Niu, Ziqing Ma, Tian Zhou, Weiqi Chen, Lefei Shen, Rong Jin, and Liang Sun. Utilizing strategic pre-training to reduce overfitting: Baguan pre-trained weather forecasting model. In Proceedings of the 31 ACM SIGKDD Conference on Knowledge Discovery and Data Mining Association for Computing Machinery, 2025.
V.2, KDD 225 pp. 2186-2197
Jaideep Pathak. Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chattopadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzadenesheli, et al. Four- CastNet: A global data-driven high-resolution weather model using adaptive fourier neural oper- ators. arXiv, 2022.
D. A. Randall et al. Climate Models and Their Evaluation. Cambridge University Press, 2007.
Stephan Rasp, Peter D Dueben, Sebastian Scher, Jonathan A Weyn, Soukayna Mouatadid, and Nils Thuerey. Weatherbench: a benchmark data set for data-driven weather forecasting. Journal of Advances in Modeling Earth Systems, 12(1 202200000000,,,,,,, 2020.
Carl Runge. Über die numerische auflösung von differentialgleichunge. Mathematische Annalen, 46(2):167-1788 1895.
ed. Stocker, Thomas. Climate change 2013: the physical science basis: Working Group contribu- tion to the Fifth assessment report of the Intergovernmentall Panel on Climate Change. Cambridge University Press, 2014.
Utkarsh Utkarsh, Danielle C Maddix, Ruijun Ma, Michael W Mahoney, and Yuyang Wang. End-to- end probabilistic framework for learning with hard constraints. arXiv preprint arXiv:2506.077 07003, 2025.
Geoffrey K. Vallis. Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 2nd edition, 2017.
13

---

Published as a conference paper at ICLR 2026
Yogesh Verma, Markus Heinonen, and Vikas Garg. Climode: Climate and weather forecasting with physics-informet neural odes. arXiv preprint TrXiv:2404.10024 2024.
Tammy M Weckwerth, Crystalyne R Pettet, Frédéric Fabry, Shin Ju Park, Margaret A LeMone, and James W Wilson. Radar refractivity retrieval: Validation and application short-term forecasting. Journal of Applied Meteorology, 44(3):285-300, 2005.
Wanghan Xu, Fenghua Ling, Wenlong Zhang, Tao Han, Hao Chen, Wanli Ouyang, and LEI BAL Generalizing weather forecast fine-grat ene-grained temporal scales via physics-AI hybrid modeling. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URL attpps://ppeneevew net/ wtt.://ooummmiiiiammmmm zcELTf
14

---

Published as a conference paper at ICLR 2026
CONTENTS
A EXPERIMENT RESULTS ON REGIONAL WEATHER FORECASTING
Due to the spece limitation, here we summarize the experimental results of regional forecasting in Table 4
Table 4: Regional weather forecasting results based on 5.625 ERA5 dataset with Latitude- weighted RMSE. "A" denotes the relative error reduction of DeepPrim in comparison with the best baseline. The RMSE curve plots along lead time are presented in Fig. Fig. 12 and Fig. 13
North-Americs
South-Americe
Australia
Ar
Nonn
Cliex
Clincii::
Deighte!
ARMSSB
NOIE:
Clinx
Clienite
Deephtel
ARMSS
None:
Clinx
Clivnite
Deephin!
ARMSE
232.8
273.4
134.5
69.1
148.0%
225.6
205.4
107.7
59.9
144.4%
251.4
190.2
103.8
53.4
143,
x500
12
469.2
129.5
225.0
121.5
146.9%
365.6
220.2
199.4
85.6
149.5%
144.6
184.7
170.7
70.3
158
18
667.2
545
307.7
194.5
269.2
222.2
211.2
292.0
324.9
6
1.96
1.62
1.25
0.50
57.5%
1.58
1.38
0.97
0.72
255 85
1.57
1.19
1.05
0.60
44..%
12
3.34
1,86
1.81
1.25
1.30
1,20
2:75
1.79
24
2,27
2.25
33.0%9
1:97
1.65
22
1.92
1.63
2,65
1,75
1,85
1.35
1,57
11889
188
2,08
1.57
1,10
18
3.53
2.27
1,96
2.60
2.15
3.51
1.72
1.23
3%5
24
3.39
1,93
2.15
0,85
5500%
2.55
2.25
1.17
1.12
14.3%
2.46
2.15
1,25
0.89
L2X M98
12
2.24
28.9%
1.57
1.49
1,77
18
3.4
3.24
2.17
1,66
123.5%
3.24
1,83
1,31
1.13
137.0%
3.44
2,03
1,96
1.33
132, 1%6
24
3.96
3.14
2.34
2.08
111.1%
3.77
2.04
2/08
1.33
134.8%
3.91
2.64
2.53
1.52
(34.8%
1.67
140.1%
1.31
1,30
1,47
1.44
12
3.2
2,43
2,03
1.54
124.1%
3,42
1.64
1.71
1.03
137.2%
3.60
1.79
1.57
1.30
133 333.0%%
+10
18
3.96
3.52
2.51
1.91
117.3%
4.16
1,90
2.07
1.20
156.8%
4.51
2.53
2.23
1.35
199.59
24
4.57
3.39
2.50
2.44
12.4%
4.76
2.14
2.43
1.42
133.05
33.0%%
4.88
2,58
2.53
1.53
199,5%
B IMPLEMENTATION DETAILS OF DEEPPRIM
B. NOTATIONS
To clearly describe the task of weather forecasting and introduce the proposed approach, we first introduce some necessary notations in Table Specifically, we use subscript s to denote surface variables, use subscript to denote upper-air variables, and use subscript sta to denote static vari- ables.
B.2 ALGORITHM IMPLEMENTATIONS OF DEEPPRIM
We summarize the model feed-forward procedures of DeepPrim in Algorithm | where K Ds + PDv. ú du denotes time derivatives, C' denotes the channels of spatiotemporal embeddings. Also, Algorithm Algorithm2ssummarizee summarizes the implementations of the force network based on 3D-BiViT, which is used for estimating the time derivatives of intermediate motion fields v
de
For the input initial weather states u(t-2 to) with time window length of 3, we first compute the 3D spatial derivatives Vu(to) along the longitude-latitude- pressere level coordinates. The initial states ull-2 : (o) and both their time derivatives ii(t- (o) and spatial derivatives Vu(la) along with the spatialtemporal embeddings ves are fed into the initialization network to estimate the initial values of intermediate motion fields v* (to). Then, both u(fo) and (to) are fed into the force-network- based ODE system, where ú(to) is calculated via advection equation and V (to) is calculated based on 3D-BiVIT. By solving the ODE system with numerical solvers, we can obtain the future weather predictions {u' (+- Furthermore, the source model takes the inputs of initial state (t-2 tt), initial intermediate motion fields v (lo), ODE predictions {u' [4:) and spatiotemporal embed- dings &6 to characterize the external source-sink effects and correct the ODE predictions, output the final predictions (ú (t+))
B.3 SPATIAL--TMPORAL EMBEDDINGS
The spatial- embeddings vsr E Xxx are designed to inform the model of both longitue-llaiiude spatial location information and day-season time information. The spatial-
15

---

Published as a conference paper at ICLR 2026
Algorithm DeepPrim as weather forecaster (Kindly note: for notational convenience and to be consistent with prior work, flatten the variables across all pressure levels into a single dimension in our notation. Actually, K = 1 + x Du)
Input: Initial weather states u(t-2 to) {3x(K+3)xHXX (include surface variables ug(t_2 to) E B3xIxDsxHXW upper-air variables up to) E R3xPxDy xHxW and static vari- ables Uste = (Istt E R3xHxW), discrete time steps N, the length of time step A
spatialtemporal embeddings tSsT
Output: Future weather states {ú (t:))= E RN
1: # Intermedia ate Motion Field Estimation
2: Obtain 3-dimensional spatial derivatives Vu(to) E RIхKжиx
3: Estimate the initial intermediate motion fields:
4: v* (to) (Vas vy)ts + Conv(u(t=22 to), ú(to),
Vu(to)
#ST) E R2
temporal variations spatial gradients 5: # Force Network and ODE System
6: for in range(N) time steps do
v" (1) Vpu (1:) + w au(t)) 8p
a(4))
7:
v(4))
(v- (4) v,v" (4) (v'((.))
Force(u, v*)
Op
u (t++1)
u (t.) Atú (t:) (t.) + ANV (t,)
8:
v" (t++1)
9: end for
10: Solve the ODE system and obtain the predictions {u' (4)) E BRNXXHH
11: # Source-s sink Estimation and Correction
12: Obtain the source terms (ugource-sink (4 = Conv ((o))) v" (to), (u' (t))) (SST)
Correct and obtain the final predictions (ú (44)) {uoorrc-sink (1)) + [u' (4))
16

---

Published as a conference paper at ICLR 2026
Algorithm Force Network for learning time derivative of intermediate motion field via 3D-BiViT
Input: Current weather states and their spatial derivatives, current intermediate mo-
tion fields, spatiotemporal embeddings and pressure-level embeddings Ω (u (4:), (44,, (44,, sSST, vr).
Output: Time derivatives of intermediate motion fields (4))
I: # Patch Embedding
2: Get the inputs surface variables 2s = [us (t:), Ustas Vus (4:) v5(44, tST]
3: Get the inputs of upper-air variables Sy = [uo (lx), Usta, Vuy (44) vt WSTl E 4(Dr+
4: Patch-wise token embedding of surface variables Zs + Embeds (Rt) E R IxNxxd
5: Patch-wise token embedding of upper-air variables Zu + Fmbedy (Qt) E PxNpxd
6: Add pressure-level embeddings Zs + Zs 4p(0 1,;, Zu Zu + 1ppl1 P.:.
7: # Peera-pressure--ie level AAteettinn
8: Zs Self-Attns (Zs|:;,,))))))
9: Zy <- Self-Attny [2
10: # Cross Cross-pressee presstre-eeeeel Self-Attention
11: Concat the embeddings Z + [Zs, Zu] ((P+))xxpxd
12: Z Z.transpose (0,1) Nex
13: Z + Self-Attn (Z),), P+1
# Projecti on Head
15: Obtain the representations of surface variables Zs RRIXNxxd
16: Obtain the representations of upper-air variables Zy E PRNxxd
17: Compute the intermediate motion fields' time derivatives of surface variables and upper-air variables.
(t,) Projector,(Zgg) (t,),) Projector,(Zz))
17

---

Published as a conference paper at ICLR 2026
Table Notations and their meanings.
Notations
Meanings
temperature
latitude-direction wind
By
longitude -erettuue-drection wind geopotential
humidity
w
vertical wind speeed
viscosity coefficient
turbulent exchange coefficient
f
Coriolis force coefficient
rypp
longitude-latitud--rrse pressure level coordinates
H
number of latitude grids
W
number of longitude grids
Ds
number surface variables
Du
number of upper-air variables each pressure levels
P
number of pressure levels
K
number of total variables at all pressure levels K = Ds + PDu
u(t)
weather states time
(t)
intermediate motion fields at time time derivatives of weather states du dr time derivatives of intermediate motion fields dv" dt horizontal gradients of weather states
Puu
Vu
3D (horizontal and vertical/pressurelleee) spatial gradients weather states divergence of intermediate motion fields
V v
N
number of discrete time steps the ODE system
Al
the length of time interval/step in the ODE system spatialtemporal embeddings
VST
UP
Pressure-level embeddings
C
the number of spatiotemporal embeddings channels number of patches
Nr
d/Embed dim
embedded dimensions
temporal embeddings Vsr are encoded as additional input channels to the model and consist of the following parts:
Spatial Embedding Consider the spherical properties of the Earth, we encode latitude h and lon- gitude w values with trigonometric and spherical coordinates as
[sin (h), cos(h) (in(w), cos(w), sin(h) cos(w), sin(h) sin(w))
Temporal Embedding Consider that daily and seasonal periodic patterns (e.g., day-night, winter- summer) are important for weather forecasting, we encode daily and seasonal cycles are using trigonometric functions as:
2nt
2ht
1/ sin (2Tt),, (2s(( sin
cos
365
365
where denotes the global time of the corresponding samples.
18

---

Published as a conference paper at ICLR 2026
Initial Weather States
{u (t
(ta)
(t_.))
(t.2))
Spatial-tempore embeddings Wir
3D Gradients
Initialization Net
Season Day Time embed.
Initial Intermediate Motion
i Lat-Lon Spatial embed.
(fa)
Repeat N times
Force Net
Advection v' v()) w 0p
8()
G
ODE Solver
{v (t))
(ū()))
Source-sink Net
(tx)
(fz)
(1))
Future Predictions
Figure 5: Overall pipeline of DeepPrim Framework
Spatial-temporal Joint Embeddings To capture the cyclical spatial location and temporal peri- odic patterns jointly, design spatial-temporal joint embeddings
Vest v/a t/c.
where denote the Hadamard product.
Finally, the overall spatial-temporal embeddings integrates all the aforementioned terms as:
UST W&. the, vas]]
B.4 MODEL ARCHITECTURES AND PYPER-PARAMETERS
In our DeepPrim, the backbones of the initialization network and the source-sink network based on ResNet (He et al. 2016), and the backbone of the force network is based on ViT (Dosovitskiy al. 2021p. The hyperparameter configurations of the initialization model, the force network, and the source-sink model are summarized in Table 6 Table 7 and Table respectively.
B.5 DIFFERENCE BETWEEN PANGU, NEURALGCM, AND DEEPPRIM IN 3D ATMOSPHERIC DYNAMICS MODELING
As pioneering methods, both Pangu and NeuralGCM emphasize the importance of 3D atmospheric modeling with customized techniques. In this section, we'll discuss the difference between Pangu, NeuralGCM, and DeepPrim in 3D atmospheric dynamics modeling summarized in Table
19

---

Published as a conference paper at ICLR 2026
Surface features Vus, Vs Ver
Patchify
Untro-pressure Self Self-Attention
Gross pressure Self- Attentionn
Tokenization
Concat&
Self Asto, (12gl:12
Upper-oir features Up Vug, Ve VSr
Patchify & Tokenization
Untra-preesure Self-Atrention
2 [2,,
2+ 2trongpeen Seif Astos ((8) (Mja))))
Ze
Self Atta (...))
Surfoce
925hPa
800hPa
50hPa
Projector
Projector
000
D
Surface
Upper-air
Pressure-level embeddings
Figure 6: Architecture of Force Network that composes of 3D-BiViT. The force network encap- sulates external forces (including pressure gradients, Coriolis forces, and frictional effects) by mod- eling the time deviation of intermediate motion for upper-air and surface inspired by the force term of the Navier-Stokes equation. The feature forward steps involve (1). tokenization and embedding, (2). pressure-Level embedding, and (3). Intra- and cross-pressure level attentions.
(u(L_zito)
)
(ta))
Vta(ts) V.u (ta) V (t,)
u(t.)
(t.,))
Initialization Net
Pressure P
Latitude y
10
000
-00
Longitude
Spatial-temport embeddings Ver
coordinates
(to)
Figure 7: Architecture of Initilization Network Model. The initialization network integrates vari- additional features for initial intermediate motion estimation. The input features include spatial- temporal derivatives of various weather variables, land-sea mask, incoming solar radiation, orogra- phy, and spatial-temporaa temporal embeddings tSsr.
Specifically, Pangu leverages Unified attention blocks and positional embedding model pressure information, which may not explicitly distinguish heterogeneous Morizontal-vertin interactions. NeuralGCM employs individual neural networks model physical tendencies in the corresponding single vertical column, which may omit the inference of intra-pressure-lett influences. In compar- ison, our DeepPrim explicitly couples pressure-level dynamies with primitive equations and uses separate attentions for horizontal and cross pressppresuure level interactions rather than solely relying
Table 6: Default hyperparameters of the initialization network.
Hyperparameter
Description
Value
Kernel size
Size of each convolutional kernels
3
Padding size
Size of padding of each convolutional layer
1
Stride
Step size of each convolutional layer
1
Dropout
Dropout prohability
0.1
Leakage Coefficient
Slope of LeakyReL.U U for negative inputs
0.3
ResBlock List
(Number of ResBlocks, Hidden dimensions)
[5, (5,256), (3, 64),, (2, out channels)
20

---

Published as a conference paper at ICLR 2026
(t(t
(t))
Source-sink Net
$00 1000
100
@
temporll
embeddings Ver
û ((,)
(L))
Figure 8: Architecture of Source-Sink Network Model that captures the value gains losses of each variable caused by resolved and unsolved processes for final predictions. The source-sink network receives the initial weather states u (to) and initial intermediate motion v (to). and ODE solutions (u' (4+)) along with spatiotemporal embeddings Uss as input to correct the output of ODE solutions.
Table 7: Default hyperparameters of the VIT in 3D force network. Kindly note the patch size Psize set to 2 for 5.625 model, is set for 1.40625 model, and set to 24 for 0.25 model.
Hyperparameter
Description
Value
Paiza
Patch size
2/4/24
D
Dimension of hidden layers
128
Number of Transformer blocks inter-pressure level self-attentions for surface variables
LUpper inter
Number of Transformer blocks of inter-pressure level self-attentions for upper-air variables
Liutrsprssurr
Number of Transformer blocks intra-pressure level self-attentions
Heads
Number of attention heads
MLP ratio
Expansion factor for MLP
Decoder Depth
Number layers the final prediction head
Drop path
Stochastic depth rate
0.1
Dropout
Dropout rate
0.1
positional biases. Thus, DeepPrim enables more flexible and physically consistent 3D atmospheric dynamics modeling capacity, as we claborately demonstrated in the Section 4.3
B.6 NUMBER OF MODEL PARAMETERS AND COMPARISON WITH BASELINES
We summarize the parameter number of DeepPrim and other advanced deep weather forecasting approaches in Table 10 It can be seen that DeepPrim generally maintains great lightweight de- signs with a relatively small number of parameters in comparison with most baselines, and also
Table 8: Default hyperparameters of the source-sing sink model.
Hyperparametter
Description
Value
Kernel size
Size of each 3D convolutional kernels
3
Padding size
Size of padding of each 3D convolutional layer Step size of each 3D convolutional layer
1
Stride
Dropout
Dropout probability
0.1
Leakage Coefficient
Slope of LeakyReLUU for negative inputs
0.3
BesBlock List
(Number ResBlocks, Hidden dimensions)
[55 256), (3, 64), (2, out channels)
21

---

Published as a conference paper at ICLR 2026
Table Comparison with Pangu, NeuralGCM, and DeepPrim in 3D atmospheric dynamies model- ing in the perspectives of key ideas and technical designs.
Aspect
Pangu (2023)
NeuralGCM (2024)
DeepPrim (Ours)
Key ideas
Treats pressure levels as ge-
Feeds data for individual Leverages intra-pressure-leved
ometriic height equivalents
columns
the atmosphere
tention for horizontal interactions
in positional embeddings, may not explicitly distin- guish heterogeneous interac- tions
into neural network used to and cross-pressure-leve attention
produce corresponding physics tendencies
for vertical coupling separately yet collahoratively
Technical Designs
Unified attention blocks for Neural networks that act locally lat-lon-pressure coordinates in space, on individual vertical (implicit horizontal/vertical columns of the atmosphere. interaction blending)
Physics-informed modular design explicitly separates intra-pressure- layer horizontal dynamics and inter-pressure-layer vertical inter- actions
presents superior performance in both global and regional weather forecasting tasks. Compared with ClimODE, our DeepPrim further exploits 3D atmospheric dynamics and integrates a parametrized initialization network for motion velocity estimation. The resulting performance gains and 3D at- mospheric modeling capabilities over ClimODE require acceptable additional parameters.
Table 10: Parameter number of DeepPrim and advanced deep weather forecasting approaches. "Dataset Resolution" denotes the resolution of the ERA5 dataset used in experiments.
PawesWWrtter
FlimaX
WeatweCFT
NerariM
Derphtin Dapphin Dugffin
Lan a 0023
Bidd. 2633
Nasyen a 0023
Nasyen
2023
2024
Xus al 2024
Keckkow e 2024
Ons
Ous
Ozo
Parastees (Milion)
27M
250M
1053
10981
2.85
470M
18.3M
23M
23M
45M
B.7 OPTIMIZATION CONFIGURATIONS
We conduct all the experiments with PyTorch In the training stage, choose the batch size of 8, and run experiments on 8 NVIDIA 3800-SXM4-80GB GPUs. In the ODE system, we set time interval A/ 1h. We the AdamW optimizer with 31 0.9, B2 0.999. The learning rate is set to le-5 for the ODE model components and 5e-4 for the rest. weight decay of le-5 is applied to all parameters except for the positional embeddings. The learning rate follows linear warmup schedule starting from 1e-8 for the first 20, 000 steps (approximately epoch), transitioning cosine-ance annealing schedule for the remaining 90, 000 steps (approximately 9 epochs), with minimum value of 1e-8. The maximum epoch is set as 50, and we use an early-stopping strategy with patience value 3. 3.
B.8 TRAINING AND INFERENCE TIME
We summarize the Inference time of DeepPrim in Table 11 and the training time cost of DeepPrim in Table [2 In general, the training time is affordable. Also, the inference speed of DeepPrim is appreciable due to its lightweight design.
C PRIMITIVE EQUATIONS THE ATMOSPHERE
The primitive equations, which form the foundation numerical weather prediction (NWP), consist of set of nonlinear partial differential equations governing atmospheric dynamics and thermody- namics. These equations describe the conservation of momentum, mass, energy, and water vapor in the atmosphere. Below, we present the complete set of primitive equations in pressure coordinates (x-y-p), along with detailed explanations each term.
htpss//pytorch.org/
22

---

Published as a conference paper at ICLR 2026
Table 11: Inference time of DeepPrim for each sample different forecasting lead times based on single NVIDIA A800-SXM4880GB GPU. The evaluation metric is seconds (s).
Lead time (hours)
Resolution
At 6
At
12
At
18
At 24
DeepPrim(5.6259)
0.16s
0.33s
0.47s
0.66s
DeepPrim(1.49)
0.64s
1.23s
1.81s
2.31s
DeepPrim(0.259)
2.48s
4.98s
7.47s
9.48s
Table 12: Training time of DeepPrim different forecasting lead times. The maximum epoch is set as 50, and we use early-stopping strategy with a patience of 3. Please refer to Section B. 7 and Section B.4 for more optimization configurations.
Lead time (hours)
Resolution
Hardware
=t 12
= 18
A 24
DeepPrim(5.625%) 4 NVIDIA A800-SXM4800GB GPUs
13hours
1day2hours
Iday7hours
Iday 19hurs
DeepPrim(1.44
4 NVIDIA A800-SXM4-80GB GPUs
5days
6days
6days
7days
DeepPrim(0.259) NVIDIA A800-SXM4-80GB GPUs
8days
15days
Kindly Note: Due to computational resource limitations, did not directly train DeepPrim(0.25% models with At 12 and At 18. Instead, we obtained the results by rolling the DeepPrimm0.25"" model of At = 6. For example, obtained the results of At 12 by rolling the DeepPrim(0.255 (0.25) model of At 6 twice.
C.1 MOMENTUM EQUATION
The horizontal momentum equation (Navier-Stokes equation) describes the evolution of the hori- zontal wind vector (Uz, Uy) under the influence of the Coriolis force, pressure gradient force, and horizontal force (we) ):
dv dt + fk x v = = V +
FR
pv
F3 =u=Vay +
Op?'
where:
v = (Uz, vy): Horizontal wind velocity vector (0g: latitude-direction wind, ey: lontitude- direction wind).
f 7.29 x 10-5 s s: Coriolis force coefficient.
k: Unit vector in the vertical direction.
Φ ge: Geopotential (g: gravitational acceleration, æ: geometric height).
Vp Oz by
Horizontal gradient operator in pressure coordinates.
FA: Horizontal Force.
Horizontal viscosity coefficient.
A: Vertical turbulent exchange coefficient.
The material derivative is expanded as:
di
i
dt at +v. V
+ Op'
23

---

Published as a conference paper at ICLR 2026
where dp is the vertical velocity in pressure coordinates.
C.2 CONTINUITY EQUATION
The continuity equation expresses the conservation mass in the atmosphere:
dw
Oy V = 0.
+
Integrating this equation vertically yields the vertical velocity wt
/ Vp ddp,
w
Pup
where Ptop the pressure at the model top.
C.3 THERMODYNAMIC EQUATION
The thermodynamic equation describes the evolution of temperature T due to adiabatic processes and diabatic heating:
dT
RT Q Cp
0T Op?
V K
Kr
dt
cpp
where:
T: Temperature.
R 287 J -g-- Gas constant for dry air.
= 1004 kg -K-1 Specific heat capacity at constant pressure.
Q: Diabatic heating rate (e.g., radiative heating, latent heat release).
Ky Horizontal thermal diffusivity.
K2 Vertical thermal diffusivity.
C.4 WATER VAPOR EQUATION
The conservation of water vapor is governed by the following equation:
dg S + K p + K 09
dt
Op2
where:
g: Specific humidity (water vapor mixing ratio).
Sy Sources and sinks of water vapor (e.g., evaporation, condensation). Under saturated pscudo-adiabatic conditions, S, can be computed as:
Sg
6F de
0, di
de
0 and y 4s
8 =
1, else
F T.T LR-C,
cpR.T*+LAq
e, 6.112 X exp
17.677
243.5
T' T - 273.15 0.622er gs 0-.3Tee
Kgi Horizontal diffuusivtty for water vapor.
K₂; Vertical diffusivity for water vapor.
24

---

Published as a conference paper at ICLR 2026
C.5 HYDROSTATIC BALANCE EQUATION
The hydrostatic balance equation relates the vertical gradient of geopotential Φ to the temperature T:
RT
0p
C.6
IDEAL GAS LAW
The ideal gas law provides the relationship between pressure, density, and temperature:
= PRT,
where p the air density.
C.7
SUMMARY OF SYMBOLS
Coordinates: x (longitude), y (latitude), p (pressure).
Time: t.
Wind: v= (y) (horizontal), (vertical in pressure coordinates).
Thermodynamic Variables: T (temperature), (specific humidity), e (geopotential).
Physical Constants: f (Coriolis parameter), R (gas constant), Cp (specific heat capacity).
Diffusivity Coefficients: (horizontal viscosity), (vertical viscosity), Kr (thermal dif- fusivity), Kg (water vapor diffusivity).
C.8 THE DERIVED EQUATIONS AND ANALYSIS
This set of equations forms the basis for atmospheric modeling and numerical weather prediction, as described in (Kalnay| 2003), (Bauer et al. 2015). (Vallis 2017). and (Holton & Hakim 2013).
The resulting equations T. q. and are:
L 06 w tw 0p
0T
OT
OT
OT
0t
=
Cp
U2 dr
By Oy
w
0p"
0q
8F
OΦ
00
0Φ
06
0q
0q
8q
+ 0r
+ Byy Oy
+ 0p/
Us dr
vy dy
w
Øt
RT
Öt
0p"
/
OT
dp.
Øt
Öt
Although these variables are interconnected, the momentum equation that describes atmospheric motion serves the cornerstone (Bauer et al. 2015). the variations of other weather variables (such as temperature and water vapor) are primarily driven by horizontal advection within the at- mospheric circulation scale (i.e., the advection term has larger magnitude than other terms in thermodynamic equation and water vapor equation) (Emanuel 1994). Therefore, accurately model- ing atmospheric motion is crucial for forecasting these prognostic variables. As result, in Deep- Prim, we first introduce a 3D force network together with an initialization network to simulate the Navier-Stokes Equation. Moreover, in NWP, physical processes are modeled under moderate simplifications. For example, in most general circulation models, temperature is constrained by pseudo-adiale aeeabataiiaatti system considering moist processes such as condensation and evaporation, while other processes, including heat originating from friction, radiative heating, and cooling, remain un- solved (Randall et al. 2007] Stocker 2014). To describe these processes, parameterization methods introduced to approximate the interactions between resolved and unresolved processes. It is important to note that these parameterization methods are not unique; they can be customized and adjusted within different numerical schemes based human heuristics. The choice of parameter types and configurations can lead to significantly different simulation results. Conversely, deep neu- ral networks offer way bypass these simplifications by directly learning unresolved processes
25

---

Published as a conference paper at ICLR 2026
through data-driven approach. Thus, in DeepPrim, we additionally apply a source-sink network to parameterize both the resolved and unresolved processes.
D DATASET DETAILS AND EXPERIMENTAL SETTINGS
In our experiments, we use the preprocessed ERAS data from WeatherBench (Rasp et al. 2020 EARS a well-acknowledged weather forecasting benchmark dataset and it widely used in data- driven weather forecasting methods.
WeatherBench regridded the raw ERA5 datase from its 0.25% resolution to 5.625", 2.8125%, and 1.40625 resolutions. To comprehensively evaluate our model, we choose the ERA datasets with 5.625 resolution, the datasets with 1.40625', and the datasets with 0.25 resolution for per- formance evaluation. The processed dataset includes 8 atmospheric variables across 13 pressure levels, surface variables, and 5 static variables. We normalize all the inputs via z-score normal- ization for each variable each pressure level. Also, we apply the inverse normalization for the predictions of future states for performance evaluation.
D.1 ERA5 DATASET WITH 5.625 RESOLUTION
In line with (Nguyen et al. 2023), we selected atmospheric variables at 7 pressure levels, surface variables, and static variables for the ERA5 dataset with 5.625 resolution, detailed in Table T4] In our model training, we choose all variables as input variables, and all variables except three static variables as output variables that are used for loss calculation. For performance evaluation, we focus on five key target variables in line with previous work (Verma et al. 2024). including geopotential 500 hPa (2500), temperature at 850 hPa (1850), temperature meters (f2m), and zonal wind speeds at 10 meters (w10 and v10).
Table 13: Summary of ECMWF variables utilized in the ERA5 dataset with 5.625 resolution. The variables Ism and oro are constant and invariant with time. "Abbrev." denotes abbreviation.
Type
Variable Name
Description
Pressure Levels
Land-sea mask
Ism
Binary mask distinguishing land (1) from sea (0)
NIA
Static Variable
Orography
oro
Height of Earth's surface
NIA
Latitude
lal
Laiitud of each grid point
N/A
metre temperature
Temgerature measured 2 meters above the surface
Single level
Surface Variahle
metre U wind component
ulO
East west wind speed 10 meters above the surface
Single level
10 metre V wind component
c10
North-sou wind speed 10 meters above the surface
Single level
Geopotential
Height relative to pressure level
50, 250, 500, 800, 700, 850, 92. hPa
U wind componcnt
Wind speed the cast-west direction
50, 250, 500, 600, 700, 850, 925 hPa
Upper-air Variable
component
speed the north-south
50, 250, 600, 700, 850, 925
Temperature
Atmospheric temperature
50, 250, 500, 600; 700, 850, 925 hPa
Specific humidity
Mixing ratio of water vapor air mass
50, 250,
Relative humidity
Humidity relative
50,250,
D.2 ERAS DATASET WITH 1.40625 AND 0.25 RESOLUTION
In line with (uu et al. 2024). we selected 6 atmospheric variables at all 13 pressure levels, 3 surface variables, and static variables for the ERA5 dataset with 1.40625 resolution, as detailed in Table 14. In our model training, we choose all variables as input variables, and all variables except three static variables as output variables that are used for loss calculation. For performance evaluation, focus on five key target variables in line with previous work (Verma et al. 2024), including geopotential at 500 hPa (z500), temperature at 850 hPa (t850), temperature at 2 meters (t2m), and zonal wind speeds at 10 meters (u10 and v10).
'More details of FRA5 data can be found in https://conflueree www.:u....ee.e..........m int/dispp... http://iaddapa/c....... ww..533Arcataarrcccueetttaa
26

---

Published as a conference paper at ICLR 2026
Table 14: Summary of ECMWF variables utilized in the ERA5 dataset with 1.40625 and 0.25 resolution. The variables ism and oro are constant and invariant with time.
Type
Variable Name
Description
Pressure Levels
Land mask
Loma
Binasy mask distinguishing land (1) from sea (0)
NIA
Static Variable
Otography
ore
Height of Eartl's surface
N/A
Latitude
lot
Laiituo of each grid point
N/A
metre tomperature
12on
Tempertture measured 2 meters above the surface
Single level
Surface Variable
metre U wind component
u10
East west wind speed 10 meters above the surface
Single level
10 metre V wind component
c10
North-south wind speed 10 meters above the surface
Single level
Geopotential
Height relative to pressure level
50, 100.150, 200, 250,300, 400, 500,
600, 700, 550, 125,1000 lPPa
Upper-air Variable
U wind component
Wind speed in the cast direction
50, 100,150, 200, 250,300, 400, 500,
800, 850, 925, &Ps
V wind component
Wind speed the north-south direction
50, 100,150, 2000, 250,3000 400, 500,
925,1000
Temperature
Atrospheric temperature
60,
2000, 2500,000 5000
600, 850, 925,
Specific humidity
Mixing ratio water vapor air mass
50, 100,150, 200,
400, 500,
600, 700, 850, 925,1000
Relative humidity
Humidity relative to satunation
50, 100.150, 200, 250,300, 400, 500.
200, 850, 925,1000 lPa
Table 15: Latitudinal and longitudinal boundaries with grid size for each region based on the ERAS dataset with 5.625 resolution.
Region
Latitude Range
Longitude Range
Grid Size (lat x lon)
North America
(15, 65)
(220,
8 14
South America
(-55, 20)
(270, 330)
14 10
Australia
(-50, 10)
(100, 180)
10 x 14
Global
(-90, 90, 90)
(0,
32 64
D.3 EXPERIMENTAL SETTING OF REGIONAL FORECASTING
In practice, it is not always necessary or possible to make global weather forecasting, especially when we cannot have access global data. Instead, making regional weather forecasting based on specific local regional data is also critical. In line with the experimental setting of ClimaX (Nguyen jet al. 2023) and ClimODE (Verma et al. (002)), we consider regional weather forecasting for three typical regions, including North America, South America, and Australia, based on ERA datasets with 5.625 resolution. The detailed information is summarized Table 15
E EVALUATION METRICS FOR WEATHER FORECASTING
This section provides detailed explanations of all the evaluation metrics for weather forecasting used in the main experiments. For each metric, u and û represent the predicted and ground truth values, respectively, both shaped as K И W, where K is the number of total weather factors (K = (P + 1) x D, where P is the number of pressure levels, 1) is the number of weather factors), and H W is the spatial resolution of latitude (H) and longitude (W). To account for the non- uniform grid cell areas, the latitude weighting term a(.) is introduced.
Latitude-weighted Root Mean Square Error (RMSE) assesses model accuracy while consider- ing the Earth's '' curvature. The latitude weighting adjusts for the varying grid cell areas different latitudes, ensuring that errors are appropriately measured. Lower RMSE values indicate better model performance.
27

---

Published as a conference paper at ICLR 2026
H HW
cos(lat(h)) Σ h== cos (lat (h")
RMSE
((h) (ūk,h. u.s,,o)) a(h)
K
H
Anomaly Correlation Coefficient (ACC) measures model's ability to predict deviations from the mean. Higher ACC values indicate better accuracy in capturing anomalies, which is crucial in meteorology and climate science.
Enhhw na A
ACC =
V (T/ Eks,, a(h)(u/,,
Ex un representing the temporal mean of the where u' = u and = û - C, with C= = 1 R ground truth over the test set.
F IMPLEMENTATIONS OF BASELINES AND IFS
To make fair and comprehensive experimental validations, experiments strictly adhere to Resolution-matched "Apples-to-Apples" comparison principle. Specifically, different methods may focus different tasks (global or regional forecasting) and dataset with different resolutions (ranging from 0.25% to 5.625%). We first report the results from the original paper. Furthermore, we also replicated the Pangu and GraphCast models based on their official codebase to conduct fair comparisons at resolution of 5.625 degrees in Table
F.1 DETAILS OF BASELINES AND THE RESULT SOURCES
We summarize the details of baselines and the result sources of Table 2 and Table 3 in Table [6 including the original resolution setting of each method and the detailed result sources.
Table 16: Details of Baselines and the Result Sources of Table and Tablle. 3 "Resolution" denotes the resolutions adopted in experimental setting. "Original Resolution" denotes the res- olution used in the original paper of the corresponding method.
Resolutio
Baselines
Paper
Original Resolution
Resuli Scunces
ItS
(3023)
5,025" (downsampled)
Clted from the paper of ClimaX
Nguyes el
2024
NODE
(018)
5
Cited from the
ClimODE
Verma at.
2024
(003)
5663
Citel the paper ClimDDE
Verma
5.625
ClimaX
(2033)
5A25*
Citel from the of ClioDDE Verma et al.
2004
(without pre-training with CMIP6)
(2023)
Ciedd the paper
Nguyes
(pee trained CMIPS)
GraphCast (5.625%)
(2023)
0.25
Reproduced the othical
pie // hhus, (goog le-deepni wtwpegniin//ggppeeee.. ne/grapacest
Pungu (5.625")
(2023)
0.23
actps: / thubb con/ htt.//9688Bee 9889988800/// BeteDeeee/aaooeee Paaee/aaooouaaeee Reather
Retmeliced from the coticaal coles
ClimODE
(0044)
5,625
Cited the paper ClimODE Verma al.
2024
(0033)
1.40025
(dwnnsmplol) Cited from the paper ClimuX NNuyyen et
1, 1
(2024)
officaa hedocs. cocccs.oo//// 1o/en/ www.tt//e/ee.... wwwst//checkooiteeee......o..ete
checkpoint:
(Tible
sttpe: //neurs igon Lig.nnneee readt rww.e-eeeoo.c.oo
ClimuX
(2023)
1.40625*
Cited from the paper ClimX Nguyee el
2023
WestherGFF
(2044)
1,40625
Cited from the paper WeatherGFF Xu
2024
0.25e
IFS
(2033)
0.25"
Dt. Lgg / //wll
Obiaaned from WeatherDench2
(aalle
Pangu (0 (5.)5))
(0033)
0.25
Chinined Waather
3ttps: //sttos....s..ooo.o..... wtw.eesss..scc......eee.o.eeseee www.orch.ccccco.o.oo..ooo.... wtt.://aa...ooo////eaaoooooeeeee
GraphCast
(003)
0.25"
Obtuaeed Wenther
tttpss//ss
28

---

Published as a conference paper at ICLR 2026
Table 17: Global weather forecasting results (ACC only) based on 5.625 ERA5 dataset, The metric reported is weiihh- ACC (t).
Variable
Lead Lome-Time
Al (b)
ClimtDt:
DeepPrmm
DeepPhtm
[206
6
1.00
0.96
0.99
0.97
1.00
0.99
1.00
1,00
12
(N/A)
0.88
0.99
0.96
1.00
0.99
1.00
1.00
=500
(NIA)
0.79
0.99
0.95
1.00
0.98
1.00
1,00
24
1.00
0.70
0.99
0.93
1.00
0.98
0.99
1.00
6
0.99
0.94
0.99
0.94
0.98
0.97
0.98
0.99
12
(N/A)
0.85
0..99
0.93
0.98
0.96
0.98
0.99
1850
18
(N/A)
0.77
0.99
0.92
0.98
0.96
0.98
0.98
24
0.99
0.72
0.99
0.90
0.98
0.95
0,97
0.99
0.98
0.97
12
(NIA)
0.68
0.99
0.90
0.97
0.96
0.98
0.99
12m
18
(N/A)
0.69
0.99
0.88
0.97
0.96
0.97
0.98
24
0.99
0.79
0.99
0.89
0.98
0.96
0.97
0.98
0.98
0.85
0,95
0.92
0.97
0.91
0,97
0.98
(N/A)
0.70
0.93
0.88
0.95
0.89
0.96
=10
18
(NIA)
0.58
0.91
0.84
0.95
0.88
0.95
0.96
24
0.97
0.50
0.89
0.80
0.94
0.87
0.94
0.95
0.98
0.81
0.94
0.92
(N/A)
0.92
0.97
0.98
(N/A)
0.61
0.91
0.88
(N/A)
0.89
0,96
0.97
e10
18
(N/A)
0.46
0.86
0.83
(N/A)
0.88
0.95
0.96
24
0.97
0.35
0.83
0.80
(N/A)
0.86
0.94
0.95
"DeepPrim" takes all surface and upper-air variables as output variables for training, whose full results are
summarized Table 19 "DeepPrim" " only outputs five key variables. "ClimaX** denotes ClimaX that not pre-trained on CMIP6 Eyring et al. (2016).
F.2 CLARIFICATION OF THE IFS RESULTS IN TABLE 2 AND TABLE 3
In Table 2 we use the same IFS baselines ClimaX (i.e., IFS HRES vs. ERA5), and we also adopt the same experimental settings. Therefore, we cited the results from the paper of ClimaX. Also, the results of IFS are also consistent with those ClimODE and WeatherGFT. The results may be inconsistent with those reported in WeatherBench2 (Rasp et al. (020). The main reason lies in different regridding methods. As the results of IFS on resolution down-sampled from higher resolutions, the inconsistent results are mainly due to discrepancies of downsampling strategies (i.e., the ways of regridding from 0.25 to 5.6%). Following ClimaX, we use bilinear interpolation, which considers adjacent grids for regridding, while linear conservative regridding used in WeatherBench2 is more like avg. pooling. the resolution reduction relatively large, the latter would significantly reduce the data noise and the prediction error.
5.625 and 1.4 are
In Table 2 all the comparison results IFS, Pangu (0.259) and GraphCast (0.25%) are obtained from Be
G EXTERNAL EXPERIMENTS AND FULL QUANTITATIVE RESULTS
G.1 GLOBAL WEATHER FORECASTING RESULTS EVALUATED BY ACC
Anomaly Correlation Coefficient (ACC) is a standard metric in meteorology and climate science to assess a model's skill in capturing anomalies (deviations from the climatological mean). Here summarize the weather forecasting results based on ACC in Table 17 The results based on RMSE are in Table 2
G.2 GLOBAL WEATHER FORECASTING AT LONGER LEAD TIME HORIZONS AND DISCUSSIONS
Table 18 summarizes the comparison of our model with ClimaX (both the one that is pre-trained and the one that is trained from scratch) and ClimODE for longer lead time, i.c., At {36h, 72h, 144h) corresponding to (1 (1.5 days, days, 6 days). Empowered by pre-training on the CMIP6 dataset, Cli- maX shows relatively stable performance for forecasting longer lead time, pre-training lays
https::/siites reweeesseeerrcce www.eesssssssarcrgggg.go....oe
n st scores/
29

---

Published as a conference paper at ICLR 2026
great foundation for post-training and degrades the difficulty of long-lead time forecasting. Specifi- cally, our DeepPrim, despite not being pre- are-trained on any external data, presents comparable results in forecasting t850 t2m, and u10 in comparison with pre-trained ClimaX. Moreover, DeepPrim consistently achieves better performance than the non-pre-trained ClimaX* and ClimODE.
Actually, pointed out in recent work (Gao al. Chen al. 2023b), accurate long-term fore- casting often requires specialized training techniques, model ensemble strategies (e.g., OneForecast (Gao et al.)), and autoregressive refinement (e.g., FuXi (Chen et al., 20236)). These aspects are beyond the primary scope of our current work. In the future, it is of interest to incorporate such tech- niques to further enhance medium- and long-range forecasting of DeepPrim. Additionally, adopting more udvanced ODE solvers (e.g., RK4 instead of Euler) may further improve long-term prediction stability and accuracy.
Table 18: Long lead time prediction results of global weather forecasting based on 5.625 ERA5 dataset. The evaluation metrics are Latitude-weighted RMSE and ACC.
Lead Time
ACC
Variahles
A/ (h)
IFS
ClimaX* ClimaX ClimODE DeepPrim DeepPrin"*
IFS
ClimaX* ClimaX CliRODE DeepPrim DeepPrin"
36
66.7
455.0
126.4
259.6
154.3
160.7
1.00
0.89
1,00
0.96
0.99
0.99
<500
147.0
687.0
244.1
478.7
319.4
331.9
0.98
0.73
0.97
0.88
0.95
0.94
144
430.3
801.9
523.5
783.6
558.7
$74.9
0.86
0.58
0.86
0.61
0.81
0.80
0.83
1.25
1.75
1.27
1.19
0.94
0.96
0.97
1850
72
1.19
3.17
1.59
2.58
1.88
1.78
0.94
0.76
0.98
0.85
0.94
0.94
144
2,3
3.97
2.54
3.62
2.75
2.64
0.77
0.69
0.84
0.77
0.82
0.84
36
0.95
2.87
1.33
1.70
1.49
1.21
0.93
0.83
0.97
0.94
0.96
0.97
12m
72
1.15
3.97
1.43
2.75
1.88
1.57
0.90
0.83
0.98
0.85
0.95
0.95
144
1.83
2.01
3.30
2.78
36
1.07
2.98
1.57
2.25
1.70
1.60
0.96
0.69
0.93
0.83
0.93
0.93
72
1.69
3.70
2.18
3.19
2.55
2.44
0.90
0.30
0.94
0.66
0.81
0.82
144
3.17
4.24
3.24
4.02
3.65
3,55
0.66
0.30
0.63
0.35
0.55
0.57
36
1.12
2.98
N/A
2.29
1.75
1.64
0.96
0.69
N/A
0.83
0.92
0.92
v10
72
1.75
3.80
N/A
3.30
2.58
2.43
0.90
0.39
N/A
0.63
0.79
0.80
144
3.3
N/A
4.24
3.70
3.56
0.65
0.25
N/A
0.32
0.52
0.54
Kindly Note:
1. The results of IFS are obtained from WeatherBench:fi
2. "DeepPrim"" denotes our model that takes all surface and upper-air variables as output variables
for training;
3. "DeepPrim" denotes our model that only takes five key variables as output variables.
4. "ClimaX"" denotes ClimaX that is trained from scratch and "ClimaX" denotes ClimaX that is
pre-trained with CMIP6 dataset.
G.3 FULL GLOBAL WEATHER FORECASTING RESULTS
As mentioned in the main text, our DeepPrim receives all surface variables and upper-air variables as input variables to model the 3D atmospheric dynamics, and can also predict the future states of all surface and upper-air variables (K Ds + P x Dr 45, where Ds 3, P 7, and Dr = 6). In Table the forecasting performance of five key weather variables is reported in line with previous mainstream work (Nguyen et al., 2023) Veerma et al.] 2024). Here we summarize the forecasting performance of all surface and upper-air variables in Table
G.4 QUANTITATIVE RESULTS OF ABLATION STUDIES
The numerical quantitative results of ablation studies summarized in Table 20 corresponding Fig. 3 in the main text.
30

---

Published as a conference paper at ICLR 2026
Table 19: Full results of DeepPrim" for global weather forecasting based on ERA5 dataset with 5.625 resolution.
RMSE
ACC
Variable
Pressure lovel
A4
Ar
Át 24
31=
Ar
18
Ar
U wind s.10(mm//a) (a) wind (0(mma)))
Surface
0.89420
107571
1.24577
1,39002
0.9466
0.9780
09534
0.4117
Surface
0.92482
1,10300
1.27185
1,42997
0.9745
0.9637
099188
0.9390
2m Temperature (2m(K)
Surface
0.94463
1,12992
L,17411
1,19324
0.9737
0.9626
09738
0.9299
90.86801
122.57426
152. 14494
172.36325
0.9755
2506PPa
75.98683
108 (03359375
138.29961
174,1012
0.9988
0.9976
0.9962
0.9939
71.07771
91.17221
0.9557
Geopotential (mn )
6006Pa
41,7012
62.57886
82.39021
105
0.9985
0.9972
0.9951
0.9919
42,91045
59.26258
77.16673
97,76039
0.9399
8506Pa
43.85304
59.68118
76.36211
95.31515
0.9972
0.9948
0.9115
0.9667
900hPa
46.10273
62.33511
79.26835
98.23368
0.9968
0.9942
0.9966
0.8855
09782
2506PPa
13.33168
15:45208
16.65293
17.70896
0.9092
0,8758
0.8541
0.8331
Humidity (%6)
6006JP
10.12321
12.19817
13,1334
11.99717
0.9283
0.8910
0.8896
0.8141
12.75557
13,78103
8506PP
9.51776
11. 11.3608
12.21194
12,94124
0.9034
0,8592
0,8353
0.8127
0,8412
6.686F-08
7459F-08
28118408
8.771
0.9419
09318
2508PP
2.924F-05
3,378F455
3.5944.05
3.768F-05
0.8943
0,8557
08348
01666
3.166R-04
4:2998-04
0.9460
0.9156
0.8999
Specific Humidity
kg)
6006PP
4.442B-04
5.556F.04
6.067.004
6479
0.9524
0.9245
0,9092
0.8958
0.9026
8506PP
8.001E-04
9.851E444
LO08E.00
L116E-03
0.9393
0.9064
0.8911
0.8780
6.6298-044
0.9214
50hPa
0.84614
0.97900
L.01100
L.15405
0.9850
0.9801
0.9759
0.9727
250hPs
0.62926
0,80314
0,93336
1.0305
0.9678
0.9801
0
0.9655
5006PP
0.59038
0.72667
0,82702
0.93176
0.9908
0.9861
0.9820
0.9771
Temperature A(K)
600hPPs
0,57139
0,70115
0,80529
0.90681
0.9914
0.9570
0,9300
0.9783
7006Ps
0.60335
0.74169
0,84022
0.93270
0.9909
0.9862
0.9823
0.9781
850hPs
0,76288
0.92355
1,03605
1,12888
0.9882
0.9823
0.781
0.9739
9006PP
0.74482
0.92997
L,03406
1,12530
0.9886
0.9824
0.979
0.9737
50hPa
1,41168
1.77854
1 1,99226
2.16724
0.9898
0.9839
0.9999
0.9763
2500PP
2,38616
2.66888
3.41439
3,85471
0.9782
09711
5006PP
1,75884
2.14611
2.44368
2.74203
0.9812
0.9718
0.9333
0.9536
wind component
alm//4)
600hPa
1,56071
1.90175
2.14444
2.39050
0.9794
0.9677
0.9509
7006Ps
1,18509
1.79435
2,00816
2.21522
0.9758
0.9615
0.9554
0.915
850bPs
1,39223
1.67430
1.87418
2,07397
0.9733
0.9612
0.9112
0.9399
9006PP
1,27750
1.53114
1.73867
1,95376
0.9767
0.9664
0.9564
0.9446
50hPP
1,42118
1.72542
1.91538
2.07149
0.9597
0.94088
09268
0.9143
2500Pa
2.36407
299685
3.44890
3,94055
0.9731
016333
0.5177
5000PP
1,74155
2.13588
2.45400
2.79292
0.9780
0.9667
09558
0.9427
wind component
(mm/s)
600hPa
1,54916
1.88425
2,14326
2,41167
0.9757
09529
0.339
7006PP
1,88933
1.759188
1 1.97805
2.20431
0.9718
0.9588
0.9476
0.345
850bPa
1.36526
1,64698
1,85200
2.06879
0.9701
0.9563
0.9444
0.303
9006P2
1,27886
1.54310
1.76354
1.99765
0.9756
0.9643
0.9531
0.9393
31

---

Published as a conference paper at ICLR 2026
Table 20: Full ablation results of DeepPrim' for global weather forecasting based on ERA5 dataset with 5.625 resolution. The evaluation metric is Latitude- weighted RMSE.
Variablee
Lead time
w/o Sourcee Sink
k/o Vu in Initialization Net w/o in Initialieation Net w/o 3D modules
DerpPrim
1363
68.3
52.4
65.3
50.1
=500
12
149.2
84.2
75.4
63.2
71.0
198.3
1194
94.1
258.5
155.7
130.4
121.0
6
1.34
0.88
0.77
0.84
0.76
1.59
0.99
0.93
6850
1.1
24
1.98
1.28
1,16
1.26
1.13
1.01
0.89
1.1
1,07
18
2.51
1.29
1.18
1.32
1.17
24
2.58
1.34
1.30
1:38
1.19
1.68
0.58
0.92
12
1.81
1.25
1.13
1.19
1.10
=10
18
1.94
1.38
1.28
1.20
1.24
1.52
6
1.70
1.30
1.0
1.05
0.94
12
1.79
1266
1.1
1.19
L.12
v10
1.38
24
2.28
1.5%
1,42
G.5 ADDITIONAL ABLATION ANALYSIS OF BACKBONE ARCHITECTURES
As discussed in Appendix Section B.4, the backbones of the initialization network and the source- sink network based on CNN (ResNet), and the backbone of the force network is based on the carefully designed 3D-BiViT. We further carry out backbone ablation experiments with graph net- works modules using message passing and summarize the results in Table 21 From Table 21 can observe that:
(1). Replacing the CNN(Resnet) in the Initialization Network and Source-Sink Network with GNNs yields comparable performance, as both architectures excel at modeling local node interactions for initial state estimation and residual corrections.
(2). Substituting the 3D-BiViT in Force Network with GNNs caused significant performance degra- dation, our 3D-BiViT leverages two-stage attention to resolve anisotropic interactions (i.e., ver- tical and horizontal advection), GNNs, while effective for graph-structured systems, struggle to pa- rameterize continuous 3D spatial relationships and pressure-level-awan dynamics without explicit inductive biases. The results further demonstrate the effectiveness of the designed 3D-BiViT.
(3). ForceNet (3D-BiViT) is designed learn the force terms in the Navier-Stoker equations, thereby effectively modeling 3D atmospheric dynamics by capturing both inter- and intra- citra-pressure- level interactions for upper-air and surface variables replacing 3D-BiViT with vanilla ViT or CNN modules resulted in varying degrees of performance degradation.
coordinated way. our ablation study,
because neither vanilla VIT, GNN, nor CNN explicitly models 3D interactions between pressure levels, which may lead to bias in estimating the velocity term and thus degrade the forecasting performance. The additional ablations further demonstrate the necessity of our 3D design and its coherence with the overall model architecture
G.6 MORE EXPERIMENTS D'"APPLE-TO-APPLEE" COMPARISON WITH CLIMODE
is noted that ClimODE (Verma et al. 2024) experiments are conducted under an experi- mental setup that differs notably from ours: (i) only five variables are used in total, namely (2500, t2m, u10, v10, t850), both as inputs and outputs, and (ii) the model is trained on only 10 years of data. In our main results, report the results from the original ClimODE paper under this setting for completeness. aligning the data usage and variable configuration closely as possible.
this subsection, we perform a more "apples-to-apples comparison by
32

---

Published as a conference paper at ICLR 2026
Table 21: Additional ablation study results of DeepPrim and its variants. In these variants, each architecture of DeepPrim is replaced with graph neural networks (GNNs) using message passing. The evaluation metric Latitude weightted RMSE, and the forecasting lead time is At 6h.
Model
Initilization Net
Force Net
Sourco-Sink Not
RMSE
6/
2500 (sn"
(8)
12m (K) u10 (m/s))
v10
DeepPrim
CNN (ResNet)
3D-BiVIT
CNN (ReeNet)
50.1
0.76
0.89
0.92
0.94
Varianth1
GNN
3D-BiVIT
CNN(ResNet)
50.9
0.77
0.88
0.95
0.96
Varian082
CNN(ResNet)
3D-BiViT
GNN
0.79
0.92
Variani83
CNN (ResNeU)
GNN
CNN (ResNet)
69.3
0.91
0.98
1.01
1.02
Variand44
CNN (ResNot)
Vanilia VIT
CNN (ResNet)
64.4
0,94
0.99
1.04
1.06
Variances
CNN(RsaNo)
CNN (RosNet)
CNN (ResNen)
63.3
0.87
0.96
0.99
1.03
Our main ERAS-5.625' experiments follow the protocol used in ClimaX and are described in Sec- tion D To disentangle the effects of data volume and variable selection from the architectural dif- ferences between DeepPrim and ClimODE, consider the following controlled settings:
DeepPrim (full configuration), Our default setting described above: 1979-2015 for train- ing, 2016 for validation, 2017-2018 for testing; all selected variables and levels as inputs, all non-static variables as outputs.
DeepPrim (5-variable configuration). We restrict DeepPrim to use exactly the same five variables as ClimODE, both as inputs and outputs, while keeping the same temporal split (1979-20) (1979-2015 train, 2016 validation, 2017-2018 test)
ClimODE (retrained with long-span data). We reimplement ClimODE and retrain it using the same temporal split as DeepPrim (1979-2011 train, 2016 validation, 2017-2018 test), under its original five-variable configuration. This isolates the effect of increasing the training period.
First, when ClimODE is retrained on the longer 1979-2015 period, its performance improves compared to the originally reported 10-year set- ting, which is consistent with scaling-law behavior in data-driven forecasting. Second, when Deep- Prim is restricted to the same five variables as ClimODE, its performance degrades relative to our full nulti-variable, multi-pressuure-leell configuration, reflecting the loss of pross-pressure-leeel and cross-variable information that our model is designed to exploit. Third, and most importantly, under the same five-variable configuration and the same long-span data split, DeepPrim still outperforms the retrained ClimODE baseline across lead times and metrics.
The corresponding results are summarized in Table 22
These results indicate that: (i) both DeepPrim and ClimODE benefit from longer training records and richer input variables; and (ii) even under matched conditions in terms of variables and data span, DeepPrim achieves stronger performance. Therefore, the gains report ClimODE cannot be attributed solely to using more data or more variables, but also arise from our primitive-quaaion-innpeee Neural ODE design, explicit 3D multi-level representation, and force- network-basee treatment of source terms.
H LIMITATION AND FUTURE WORK
Probabilistic and Long-term Forecast. A limitation of DeepPrim is that it only supports determin- istic weather forecasting its current form. Extending DeepPrim to probabilistic forecasting is a promising future direction, and we will explore it in subsequent research. Feasible techniques could be reparameterizing the Source-Sink Network to output distribution parameters (e-g., mean and vari- ance), enabling sampling- based uncertainty quantification while retaining the physics-guide ODE framework. Also, as pointed out in recent work (Gao et al.) Chen et al., (022b), accurate long-term forecasting often requires specialized training techniques, model ensemble strategies (e.g.. OneFore- cast (Gao et al.)), autoregressive refinement (c.g., FuXi (Chen et al.) 2023b)), rolling initialization, and explicit error correction. These aspects are beyond the primary scope of current work. In the future, it is of interest to incorporate such techniques to further enhance medium- and rangerange forecasting of DeepPrim.
33

---

Published as a conference paper at ICLR 2026
Table 22: Global weather forecasting results on 5.625 ERA5 dataset (ClimODE Deep- Prim).
Lead Time At
ClimODE
DeepPrim Suniaall
Vasiable
year for traning
1979-2015 fo tniting
Gull variable ippai
6
102.9
79.1
69.5
53.6
12
134.8
111.2
98.9
77.4
=500
162.7
148.9
129.8
102.2
24
193.4
192.8
159.6
126.7
1.16
1.01
0.81
0.72
12
1.32
1.23
0.98
0.87
1850
1.47
24
1,55
1.52
1.16
1.06
1,21
1.13
0.79
0.73
12
1.45
1.36
0.97
0.88
12m
1.43
1.54
0.97
24
1.40
1.75
1,08
0.99
1.41
12
1.81
1.41
1.13
1.00
197
1.68
1.33
1.15
24
2.01
1.83
1.51
1,25
1,53
0.85
12
1.81
1.47
1.19
1.03
1966
1.65
1.19
2.04
1.86
1.24
Advanced Time-stepping Method. In our implementation, we support Euler, Runge- Kutta (RK4), D for discretizing the ODEs, and we previously used the Fuler method to solve the ODE system, as it is simple, efficient, and adequate for short-term forecasting. Compared to the simple Fuler method, RK4 may require more computational effort but offers higher-order accuracy, improved stability, and better control of local truncation error, which can help capture the complex temporal dynamics in atmospheric modeling, thereby helping mitigate error accumulation, especially for medium- and long-range forecasts. Exploring these advanced time-stepping methods for longer-term forecasting remains a promising direction for future work.
and other ODE solvers (through forchdiffeq
Data Seperation Pipeline. Our experimental setting aligns with ClimaX, i.c., the separation of train (1979~2011 /)//aaiiaiiooo22111 (0ess)2)(0(((11)))) which may also have some limitations and biases. The dataset separation could be more interesting and fair by employing more expensive but more robust scheme by keeping a rolling train/val/tesst set, e.g... 5 consecutive days for train, for val, 1 for test, than for train etc, for all the data. could allow for avoiding biases global changes, potentially from global warming between 1980 and 2018.
VISUALIZATION OF RMSE CURVES FOR GLOBAL AND REGIONAL WEATHER FORECASTING
To provide clear understanding of the forecasting performance comparison, provide the RMSE curves at different lead times for global and regional weather forecasting tasks in Fig. Fig. 10 Fig.
Fig.
12
Fig. 13
2500 [mnP/sP1]
1850 [K]
22m [K]
1/0 [mis]
v20 [mis]
800
3.0
2
32
18
28
12
18
Lead Time (hours)
Lead Time (bmAr))
Time (hours)
Lead Time (hours)
Lead Time (hour))
IFS
NODE
FCN
Climax
ClimaX
ClimODE
DeepPrim
Figure 9: Global weather forecasting results based on 5.625 ERAS dataset evaluated by latitude- weighted RMSE. corresponds to Table 2 our paper.
https://giittuub.co//////////// /ttps///tt/////ttooo/o/////////r ffeg
34

---

Published as a conference paper at ICLR 2026
12m
800
RW0
400
DeepPrim
Figure 10: Long lead time prediction results of global weather forecasting based on 5.625 ERA5 dataset with latinude-weighted RMSE. corresponds to Table 18
in our paper.
2500
1850
12m
WID
V10
400
200
12
18
24
12
1.0
24
12
18
24
12
18
24
12
1.0
24
Lead Time (hours)
Lead Time (bnum))
Lead Time [houm)
Lead Time (houss)
Lead Time (hours)
NODE
ClimaX
ClimODE
DeepPrim
Figure 11: North America regional weather forecasting results based on 5.625 ERA5 dataset with latitude-weighted RMSE. It corresponds to Table 4 in our paper.
J VISUALIZATION OF GLOBAL FORECASTING MAPS
To intuitively demonstrate the forecasting capacity of our method, we present the showcases of weather forecasting results and bias maps with lead time At {6, 12, 18, 24} in Fig. 14 Fig. 15 Fig. 16 and Fig. respectively. Each figure includes the initial conditions, the DeepPrim's predictions of future states, the ground truth of further states, and the bias map of five key weather variables, i.e., 12m, w10, v10, g500, and 1850. Furthermore, we provide showcases to demonstrate the forecasting accuracy of DeepPrim in comparison with ClimaX as in Fig. 18
19 20 21 with lead
time Al (6, 12, 18, 24). respectively. As presented, our DeepPrim can provide more accurate predictions via physics informed 3D atmospheric modeling in comparison with ClimaX.
physics-informed
K BROADER IMPACTS
This research focuses weather forecasting, which has an essential influence on relevant fields such as energy, transportation, and agriculture. As an Al application for social good, our model boosts predictions for various weather factors such as temperature, wind speed, and geopotential in both global and regional perspectives. It is essential to note that our work focuses solely on scientific issues, and we also ensure that ethical considerations are carefully taken into account.
2.5
400
1.5
NODE
ClimODE
DeepPrim
Figure 12: South America regional weather forecasting results based on 5.625 ERA5 dataset with latitude-weighted RMSE. It corresponds Table 4 in paper.
35

---

Published as a conference paper at ICLR 2026
600
3.0
2.5
2.0
200
1.5
2
1.0
0.5
DeepPrim
Figure 13: Australia regional weather forecasting results based on 5.625 ERA5 dataset with latitude-weightt RMSE. It corresponds Table in our paper.
Condition
Preciction
12m
u10
v10
g500
1850
Figure 14: Visualization showcase of DeepPrim' for global weather forecasting with leat time At 6h based on ERAS dataset with 5.625 resolution.
the weather datasets are publicly available for scientific research. Thus, we believe that there is no ethical risk associated with research.
L DECLARATION OF LLM USAGE
The authors used LLM solely purrroe- assistive tool for grammar refinement and minor formatting suggestions. LLM did not contribute to research ideation, experimental design, data analysis, interpretation, and all content reviewed and is the full responsibility of the authors.
36

---

Published as a conference paper at ICLR 2026
v10
g500
1850
Figure 15: Visualization showcase of DeepPrim 1 for global weather forecasting with leat time At 12h based on ERA5 dataset with 5.625 resolution.
Initia Condition
Prediction
Ground Truth
Blas
v10
g500
Figure 16: Visualization showcase of DeepPrim" for global weather forecasting with leat time At 18h based on ERAS dataset with 5.625 resolution.
37

---

Published as a conference paper at ICLR 2026
inttal Condition
Prediction
Ground Truth
22m
u10
ETR0 Bol
Vilee
t850
Figure 17: Visualization showcase of DeepPrim' for global weather forecasting with leat time At 24h based on ERA5 dataset with 5.625 resolution.
ClimaX
DeepPrim
Ground Truth
300
t2m
200
10
u10
v10
51383
STREE
37500
55002
g500
41583
42009
47500
200
290
t850
280
250
Figure 18: Prediction comparison of DeepPrim and ClimaX for global weather forecasting with lead time At = 6h based on ERA5 dataset with 5.625 resolution.
38

---

Published as a conference paper at ICLR 2026
ClimaX
DeepPrim
Ground Truth
t2m
u10
10
10
v10
31
30
50099
SC081
20403
g500
50099
41993
41523
t850
280
266
240
Figure 19; Prediction comparison of DeepPrim' and ClimaX for global weather forecasting with lead time At 12h based on ERA5 dataset with 5.625 resolution.
ClimaX
DeepPrim
Ground Truth
t2m
12
u10
v10
$3900
15823
300es
g500
20823
283
268
20
Figure 20: Prediction comparison of DeepPrim and ClimaX for global weather forecasting with lead time At 18h based on ERA5 dataset with 5.625 resolution.
39

---

Published as a conference paper at ICLR 2026
ClimaX
DeepPrim
Ground Truth
300
330
t2m
280
200
u10
v10
g500
380
30C
330
t850
360
280
Figure 21: Prediction comparison of DeepPrim' and ClimaX for global weather forecasting with lead time At 24h based on ERA5 dataset with 5.625 resolution.
40


Reviews:
[{"review_id": "3Yrn8u9hCX", "reviewer": "Reviewer_axZ9", "summary": "The authors highlight limitations of both numerical weather prediction (NWP) methods and deep learning weather prediction (DLWP) models. They propose DeepPrim that learns primitive equations of the Earth's atmosphere. DeepPRIM models 3D atmospheric motion using the Navier-Stokes equation for the pressure and captures the interactions between the advective and other variables.\n\nThis is a good applied paper but is lacking some algorithmic novelty since it is using a standard Vision Transformer.", "questions": "1. The authors mention that the proposed method does well on short-term prediction. How would it extend to longer term weather prediction?\n2. What made the authors choose a Vision Transformer backbone?\n4. The ODE in Eqn, 6 looks like it is just solved with Neural ODE?\n5. Do the authors know why some of the other baselines do better on the z500 variable?\n6. Do the authors know why GraphCast has better performance in several cases in Table 3?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Nice combination of physical/numerical methods within deep learning methods.\n- Authors clearly identify 3 main challenges with current DLWP methods\n- Nice emphasis on the importance of physical constraints\n- Strong results of RMSE improvement by ~35-38%\n- Nice method overview in Table 1.\n- Nice use of RK4 instead of Euler time-stepping\n- Nice use of ERA dataset at various resolutions over various predictive variables, e.g., geopotential, air temperature, wind speeds.\n- Strong results in most cases.\n- Good comparisons to IFS, Pangu-Weather and GraphCast in Table 3.\n- Nice that the method can learn the BCs\n- Nice ablation studies", "weaknesses": "- Missing reference to DLWP benchmarking paper across the various SOTA methods Karlbauer et al., \"Comparing and contrasting deep learning weather prediction backbones on navier-stokes and atmospheric dynamics\", 2024.\n- Missing references on physical constraints in deep learning models for PDEs especially to hard-constrained methods: \n  -  Negiar et al., \"Learning differentiable solvers for systems with hard constraints\", ICLR, 2023\n  - Chalapathi et al., \"Scaling physics-informed hard constraints with mixture-of-experts\", ICLR 2024\n   - Hansen et al., \"Learning Physical Models that Can Respect Conservation Laws\", ICML 2024\n    - Mouli et al., \"Using uncertainty quantification to characterize and improve out-of-domain learning for pdes\", ICML, 2024 \n  - Saad et al., \"Guiding continuous operator learning through Physics-based boundary constraints\", ICLR, 2023\n   - Utkarsh, U., \"End-to-End Probabilistic Framework for Learning with Hard Constraints\", https://arxiv.org/pdf/2506.07003?, 2025.\n- Metrics other than point-wise ones should be considered, e.g., probabilistic ones like CRPS or a constraint/physics-based metric", "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": "fncIgesUFL", "reviewer": "Reviewer_k1Xf", "summary": "The paper presents DeepPrim, a physics-driven 3D deep learning framework for short-term weather forecasting. DeepPrim combines the learning of primitive equations—including the 3D Navier-Stokes equations for atmospheric dynamics—with data-driven neural modules, specifically a 3D bicomponent Vision Transformer (3D-BiViT) to capture both horizontal and vertical (pressure-level) interactions, source-sink networks for unresolved processes, and explicit ODE solvers for continuous-time forecasting. Extensive experiments on global and regional ERA5 datasets show that DeepPrim outperforms recent deep learning and NWP baselines in RMSE and demonstrates improved physical fidelity in 3D temporal weather modeling", "questions": "1. Out-of-Sample Evaluation: Have the authors tested DeepPrim on years/geographies not included in the training/discussed regions, or on different reanalysis datasets (e.g., JRA-55, MERRA-2), to further demonstrate generalization capabilities?\n2. Physical Interpretability of the Source-Sink Module: Please elaborate on how the source-sink network is parameterized. Can its outputs be mapped back to interpretable physical processes? Are particular source/sink terms discoverable in the learned model, and how does this inform trust in the system for operational forecasters?\n3. Long-range Forecasting and Accumulated Error: Can the authors provide quantitative results for forecast horizons >24h, using the full DeepPrim model, to assess whether short-term gains remain stable at longer lead times? If degradation is significant, what additional mitigations (e.g., rolling initialization, explicit error correction) might be needed?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. Integration of Physics and Deep Learning: DeepPrim thoughtfully incorporates the full set of primitive atmospheric equations—including the 3D Navier-Stokes equation in pressure coordinates—into a neural architecture\n2. Comprehensive Ablations and Interpretability: The authors ablate every core module—initialization, source-sink, and especially 3D pressure-coupling\n3. Architectural Innovation for 3D Coupling: The force network introduces explicit intra-pressure and cross-pressure attention mechanisms, directly modeling vertical and horizontal dynamics.", "weaknesses": "1. Limited Experimental Scope for Longer-term Forecasting: Despite strong short-term (sub-24h) results, the paper’s focus on short-range forecasting leaves a gap in evaluating DeepPrim’s stability and error accumulation at multi-day or weekly horizons.\n2. Insufficient Comparison on Precipitation and Rare-event Metrics: Although DeepPrim is evaluated on standard atmospheric state variables (z500, t850, t2m, u10, v10), it lacks explicit performance metrics for short-term precipitation, extreme wind/storm events, or severe weather proxies restricts insight into the practical significance for domains that are most sensitive to rare or impactful events.\n3. Absence of Explicit Uncertainty Quantification: Modern operational forecasting demands not just pointwise predictions, but calibrated uncertainty—either through ensembles or explicit probabilistic modeling. DeepPrim delivers deterministic outputs, and there is no discussion of uncertainty, confidence intervals, or systematic error analysis.", "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": "8gRWjHtRM9", "reviewer": "Reviewer_ZswK", "summary": "Physics-informed modeling of weather has recently gained attention. Building on the existing work on neural advection, this work advocates modeling 3D dynamics of weather evolution with both inter- and intra-pressure-level interactions pertaining to upper-air and surface variables, drawing inspiration from the Navier-Stokes equation. A bicomponent vision transformer architecture is invoked for this purpose. Experimental focusing on short-term global and regional weather forecasting tasks are also provided.", "questions": "Could the authors please address my concerns mentioned in the Weaknesses section?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 1, "contribution": 2, "strengths": "--- Weather modeling has clearly emerged as an important area of research within the ML community, so the topic is of broad interest.\n\n--- Incorporating additional physics-inspired priors (e.g., using pressure gradients), to augment data-driven pipelines can potentially provide a useful inductive bias.", "weaknesses": "--- Novelty of the proposed approach is limited. Essentially, with the exception of  a component called 'force network' that models external forces such as pressure gradients, the entire approach is directly adopted from ClimODE. Surprisingly, in the entire paper, the authors fail to acknowledge ClimODE for proposing and implementing neural advection as a fundamental principle for weather modeling.\n\n--- The implementation is not consistent with the formulation. For instance, equation 5 describes a source-sink network operating on  \\dot{v} (the gradient of intermediate atmospheric motion), but in practice, a ClimODE-style approach is adopted, wherein the network is applied on ODE predictions instead of ˙\\dot{v}˙.\n\n--- It is unclear how the pressure-based information is handled in practice. The dataset contains a few variables pertaining to pressure, and estimating the gradient with respect to pressure (equation 5) would necessitate segregating the equations for each variable. Absent this, information leakage due to correlations between pressure gradients across different variables cannot be ruled out.    \n\n--- The evaluation and reporting of results (e.g., in Table 2) is also unsatisfactory, and oftentimes, even misleading. For instance, it is mentioned in Appendix D that \"In line with (Nguyen et al., 2023), we selected 6 atmospheric variables at 7 pressure levels, 3 surface variables, and 3 static variables for the ERA5 dataset with 5.625° resolution, as detailed in Table 14. In our model training, we choose all variables as input variables, and all variables except three static variables as output variables that are used for loss calculation.\" This is in contrast to ClimODE that reported use of only five variables in total, namely {z, t2m, u10, v10, t at 500hPa}, for training. \n\n--- Comparisons are also unfair in terms of size of data used for training. DeepPrim used data from 1979–2015 for training, while ClimODE was trained with only 10 years of data.  \n\nOverall, the methodology (including evaluation) falls considerably short of expected standards.", "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": "U9EywMUrTL", "reviewer": "Reviewer_Rdm2", "summary": "A novel 3D deep weather forecaster that explicitly integrates physical principles for short-term, continuous-time prediction. It addresses the trade-off between traditional Numerical Weather Prediction (NWP) models and purely data-driven deep learning models.", "questions": "1.Can the authors provide a targeted ablation study comparing the current Euler solver against a higher-order numerical integrator (e.g., RK4) for lead times of 36h and 72h? This is crucial for determining if the model's reduced performance at longer ranges is due to the learned dynamics $f_{\\theta}$ or the numerical instability of the integration method.\n\n2.What is the specific physical interpretation and necessity for introducing the \"intermediate atmospheric motion\" $v^*$ separate from the horizontal wind components $(v_x, v_y)$ within the prognostic variable $\\mathbf{u}$? Is this required for numerical stability or does it represent a specific component of the flow?\n\n3.The paper emphasizes capturing interactions with the solved advection. How is the advection term itself computed? Is it handled entirely implicitly by the neural network, or is it computed explicitly using a traditional numerical scheme (like semi-Lagrangian or finite differencing) and then corrected/fused by the network?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.The model pioneers the explicit learning of the Primitive Equations (Navier-Stokes Force Terms) via a Neural ODE system. This design is highly original, ensuring physical consistency and enabling continuous-time prediction, which is superior to fixed-step forecasting methods.\n\n2.It introduces the novel 3D Bi-Component Vision Transformer, which provides a physics-motivated inductive bias for modeling the atmosphere. This architecture is specifically designed to accurately capture heterogeneous inter- and intra-pressure-level interactions, which are crucial for 3D atmospheric motion.", "weaknesses": "1.While the Learnable Source-Sink Network is an innovative structural component, it essentially replaces complex, semi-empirical parameterization schemes used in NWP with a data-driven black-box parameterization. The network lacks explicit physical constraints or built-in interpretability, limiting the scientific novelty. It represents a missed opportunity to provide causal or physically restricted learned parameterizations that could advance our understanding of unresolved atmospheric processes.\n\n2.The empirical validation is heavily concentrated on short-term forecasting (sub-24h). While the paper provides some longer-term results (up to 5-6 days) in the Appendix, the core analysis and demonstrated dominance are not extended to medium-to-long-range prediction. This limitation leaves the model's robustness and scalability in capturing cumulative errors over longer time horizons largely unexplored in the main text.\n\n3.The paper lacks theoretical guarantees (e.g., stability analysis, energy conservation) that typically accompany physics-driven models. Without proof that the learned dynamics $f_{\\theta}$ adhere to fundamental laws beyond empirical fitting, the claimed \"physics-driven\" novelty is limited to an architectural constraint rather than a proven theoretical foundation.\n\n4.The paper omits a crucial analysis of inference time/computational complexity (FLOPs). Given the complexity of the 3D-BiViT architecture combined with the iterative nature of the Neural ODE solver, DeepPrim is likely more computationally expensive for real-time deployment than direct mapping models, a trade-off that is not quantified.", "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)