VeinCast: Physics-Guided Dynamic Field Graphs with Graph-Conditioned Fusion for Global Medium-Range Weather Forecasting

VeinCast: Physics-Guided Dynamic Field Graphs with Graph-Conditioned Fusion for Global Medium-Range Weather Forecasting

Abstract

Global medium-range weather forecasting requires modeling structured yet state-dependent interactions among heterogeneous atmospheric fields. Existing data-driven models largely learn these interactions implicitly, whereas equation-level physical constraints may inherit approximation and model-form biases. We present VeinCast, a physics-guided dynamic field graph and graph-conditioned fusion framework that jointly forecasts 69 surface and upper-air fields. Within each local window, its Physics-Guided Dynamic Field Graph combines predefined atmospheric relations with state-dependent Top- K residual edges and adapts Earth-window attention using the resulting graph context. Graph-Conditioned Latent Fusion further employs graph context and source-node centrality to guide field-to-latent aggregation, while bounded feedback preserves field-specific information. On the 1.5◦ ERA5 benchmark, VeinCast demonstrates competitive forecasting performance across all 69 meteorological fields at lead times of up to 14 days, compared with representative global weather forecasting models including FuXi, Pangu-Weather, GraphCast, FengWu, and ARROW. Ablations confirm that the two modules provide complementary gains, demonstrating the effectiveness of relational-level physical guidance for data-driven weather forecasting.

Introduction

Medium-range weather forecasting predicts atmospheric evolution over lead times from several days to approximately two weeks and supports weather-sensitive decisions in disaster preparedness, agriculture, energy, transportation, and water-resource management. Reliable global forecasting requires jointly modeling heterogeneous surface and upper-air fields across multiple vertical levels. Their interactions exhibit persistent physical structure, but their relevance and strength vary across locations, pressure levels, and atmospheric states. Capturing these structured yet state-dependent cross-field couplings is therefore essential for skillful and computationally efficient medium-range forecasting.

Operational forecasting traditionally relies on Numerical Weather Prediction (NWP), which uses data assimilation to initialize the atmosphere and numerically integrates discretized governing equations. Although physically grounded, high-resolution integration and ensemble forecasting are computationally demanding, while errors from finite resolution, unresolved-process parameterizations, numerical discretization, and imperfect initial conditions accumulate with lead time. These trade-offs have motivated data-driven approaches that learn atmospheric state transitions from historical data.

Recent AI forecasters have established a competitive data-driven paradigm for global weather forecasting while substantially reducing forecast-generation costs. Nevertheless, many architectures represent meteorological fields as channels or tokens and infer their interactions through generic convolution, attention, or message-passing operators. Graph-based forecasters likewise construct graphs primarily over spatial grid or mesh nodes, leaving cross-field dependencies implicit in feature representations. Such designs can learn statistical couplings but do not explicitly distinguish persistent atmospheric relations from interactions whose relevance changes with the local state. Vertical dependencies and couplings among wind, geopotential, temperature, and moisture have recognizable physical structure, yet their effective strengths vary across atmospheric conditions.

Physics–AI hybrid models offer another route by embedding governing equations, differentiable solvers, or equation-based constraints into neural forecasting. Although these approaches provide strong physical guidance, the incorporated formulations may still reflect approximations, closure assumptions, parameterizations, and numerical discretization choices. Imposing such formulations too rigidly may transfer model-form biases to the learning process and restrict data-driven correction, while remaining errors can be amplified during chaotic forecast rollout. This motivates incorporating physical knowledge at the relational rather than equation level: selected relationships provide a stable inductive bias without prescribing exact field evolution. However, a fixed relation graph cannot describe all regime-dependent interactions, whereas learning all relations without guidance may overlook robust atmospheric knowledge. We therefore combine a physics-defined backbone with data-dependent connections that capture couplings emerging under specific local states. The resulting graph should also guide multiscale spatial modeling and cross-field aggregation rather than remain an isolated auxiliary representation, while avoiding excessive fusion that obscures field-specific characteristics.

To address these challenges, we propose VeinCast, a physics-guided dynamic field graph and graph-conditioned fusion framework for global medium-range weather forecasting. VeinCast learns a 6-hour state transition and generates medium-range forecasts through autoregressive rollout. We consider a closed-set setting with 69 input and output fields, including 65 upper-air fields from five variables at 13 predefined pressure levels and four surface fields. Within each local spatial window, these fields form the nodes of a directed graph. Physics-defined edges provide a stable relational scaffold, while state-dependent Top- K residual edges capture additional couplings absent from the predefined relations. The resulting graph context adapts Earth-window spatial attention and conditions field-to-latent aggregation through source-node centrality. A Latent U-Backbone models multiscale spatial dependencies and interactions among fusion latents, after which bounded feedback returns the fused context without overwhelming field-specific features. Finally, the Forecast Decoder reconstructs the fixed forecast fields as residual atmospheric updates.

Main Contributions

• We introduce a Physics-Guided Dynamic Field Graph that incorporates physical knowledge at the relational rather than equation level. It combines a stable physics-defined scaffold to distinguish persistent atmospheric relations from locally varying couplings, while using the graph context to adapt Earth-window spatial attention.

• We develop Graph-Conditioned Latent Fusion, which uses dynamic graph context and source-node centrality to guide field-to-latent aggregation. A multiscale latent backbone captures shared interactions, while bounded feedback returns the fused context without obscuring field-specific information.

• Systematic experiments on global medium-range forecasting demonstrate competitive accuracy across representative surface and upper-air fields. Ablation studies further verify the complementary contributions of the dynamic field graph and graph-conditioned fusion.

Method

Given the normalized global atmospheric state Xt ∈ R B × M × H ×W at time t, VeinCast learns a state-transition model with a base interval of delta = 6 h:

bt+delta = Ftheta Xt, S, Pt, ecal

X

t +delta,

where S denotes static geographic features, Pt is a field-availability mask used to exclude invalid source fields, and ecal

represents calendar conditions at the target valid time. We use M = 69 meteorological fields, consisting of 65 upper-air fields from five variables at 13 pressure levels and four surface fields. Long-range forecasts are obtained by recursively applying the 6-hour transition model.

Field Encoding

Surface and upper-air fields have different vertical semantics and are therefore processed by two lightweight stems with independent parameters. Each scalar field is projected into spatial patch tokens and augmented with field metadata:

em = Evar (vm) + Epres (e

pm, sm) + Esurf (sm),

where vm is the variable identity, sm ∈ 0, 1 is the surface indicator, and e

pm = log(pm /1000) for upper-air fields while e

pm = 0 for surface fields. The land–sea mask, orography, and spherical positional features are projected separately and added to all field representations. The two stems are then merged in the canonical 69-field order and processed by a shared multi-scale Earth-aware encoder.

Physics-Guided Dynamic Field Graph

Regional Descriptors: VeinCast constructs a directed graph within each local spatial window, where nodes represent meteorological fields rather than latitude–longitude locations. Given X ∈ R B × M × h×w× d, the grid is partitioned into R windows, and the T spatial tokens of each field are averaged into a regional descriptor:

Db,r,m = (1/T) Σ tau=1..T Xwin

b,m,r,tau.

Physical Relations: For target field i and source field j, the relation score combines state-dependent similarity, a relation-type bias, and a physics-derived prior:

ub,r,i, j = (Wq Db,r,i)⊤(Wk Db,r, j)/√d + erho¯ij + lambda Pphy

ij.

Here, Pphy

ij ∈ 0, 1 indicates whether a predefined rule permits information flow from j to i, and rho¯ij is its relation type. The prior contains self, within-variable vertical, wind–geopotential, temperature–moisture, moisture-transport, and surface–atmosphere relations. Pairs without a predefined relation share a learnable generic residual type. Thus, the rules specify a stable relational support while the edge strengths remain conditioned on the local atmospheric state.

Dynamic Residual Edges: For each sample, window, and target field, the source set Ωb,r,i contains all physics-defined edges and the K highest-scoring residual edges among the remaining candidates; sources masked by Pt are excluded. The normalized relation weights and relation-specific context are:

Ab,r,i, j = Softmax j ∈ Ωb,r,i (ub,r,i, j),

Cb,r,i = Σ l=1..Nrho Σ j ∈ Ωb,r,i Ab,r,i, j 1[rho¯ij = l] Wvl Db,r, j,

where each relation type has an independent value projection Wvl. The physics-defined edges form the relational backbone, whereas the top- K residual edges capture state-dependent dependencies not covered by the rule set.

Graph-Adapted Earth Attention: The graph context is broadcast to the tokens of its spatial window and combined with field and static features to generate an adapted key and a per-head gate:

K ada = Wada fada ([X, C, S]),

g key = sigma fgate ([X, C, S]),

K = g key ⊙ K base + (1 − g key) ⊙ K ada.

The dynamic graph therefore modulates Earth-window spatial attention without replacing its native spatial representation.

Graph-Conditioned Latent Fusion

Graph Context and Source-Node Centrality: At both spatial resolutions, the encoder alternates dynamic graph modules with Earth-window attention. The final low-resolution graph is also reused to condition multivariate fusion. Since Ab,r,i, j measures information flow from source j to target i, the source centrality of field j and its latent-specific bias are defined as:

cb,r, j = (1/M) Σ i=1..M Ab,r,i, j,

bcent

b,r,n, j = fcent(log max(cb,r, j, epsilon)), n,

where n indexes a fusion latent. The bias is broadcast to the corresponding spatial window. After enriching low-resolution fields with graph context, the fusion latents aggregate the available fields through:

el = Wx [X l, C l],

X

Z1 = Z0 + MHA(Z0, Xel, Xel; bcent, Pt).

The graph context modifies field values, while source-node centrality biases field-to-latent aggregation toward locally influential source fields.

Latent U-Backbone: The fusion latents are initialized by learnable embeddings, static geography, and calendar conditions. They are processed by a Latent U-Backbone whose blocks alternate Earth-aware spatial attention within each latent slot and self-attention across latent slots at each location. Downsampling, a high-capacity bottleneck, and skip-connected upsampling capture multi-scale spatial dependencies while preserving interactions among latent semantics.

Bounded Fusion Feedback: The fused memory is written back to the low-resolution field representations by cross-attention. Let ΔX l = MHA(X l, Z, Z). A field- and location-dependent gate is computed from the field representation, fusion response, and field metadata:

alpha = alphamin + (alphamax − alphamin) sigma(falpha[X l, ΔX l, e]),

X l = X l + alpha ⊙ ΔX l,

where (alphamin, alphamax) = (0, 0.25). The bounded feedback preserves field-specific information while allowing each field to recover context from the shared multivariate memory.

Relation-Aware Forecast Decoder

For each of the registered 69 output fields, the decoder constructs a forecast query from its variable identity, pressure metadata and surface indicator:

qi = Evar (vi) + Epres (e

pi, si) + Esurf (si).

The query is combined with the local mean of available field features and performs relation-aware cross-attention over the low-resolution field memory:

Q field

l = MHA(q + x, X l, X l; eta, Pt),

where eta is a fixed bias derived from the same registered field relations, favoring the identical field, vertically related levels, and physically related variables. The field-derived query then reads the fused latent memory and combines the two sources through a bounded gate:

Q fusion

l = MHA(Q field

l, Z, Z),

beta = betamin + (betamax − betamin) · sigma(fbeta[Q field

l, Q fusion

l, q]),

Q l = (1 − beta) ⊙ Q field

l + beta ⊙ Q fusion

l,

where (betamin, betamax) = (0, 0.5). Low-resolution queries are upsampled and fused with high-resolution field-attention outputs, followed by Earth-aware spatial blocks and patch recovery. For the standard 69-field forecast, the model predicts a residual update in normalized space:

bt+delta = SCL (Xt + gammaRtheta),

X

SCL (x) = L tanh(x / L),

where Rtheta is the decoded residual and gamma controls its magnitude.

Training Objective

VeinCast is trained only with a latitude-weighted Huber loss in normalized space. Let e(s)

b,m,h,w = b

X(s)

b,m,h,w − X(s)

b,m,h,w denote the error at rollout step s, and let omegah = cos(phih)/ Σ hH′ =1 cos(phih′) be the normalized weight at latitude phih. The loss for one forecast step is:

L(s) = Σ b,m,h,w omegah Hkappa(e(s)

b,m,h,w) / Σ b,m,h,w omegah,

Hkappa(e) = 1/2 e2, e ≤ kappa; kappa(e − 1/2 kappa), e > kappa,

Ltrain = (1/S) Σ s=1..S L(s).

Here, S is the number of supervised rollout steps. No auxiliary reconstruction term or physical regularizer is used in the reported experiments.

Experiments

Dataset and Implementation Details

We conduct experiments on the ERA5 reanalysis dataset at 6-hour intervals and a spatial resolution of 1.5◦ (121 × 240 grids) provided via WeatherBench2. Data from 1979–2017, 2018, and 2020 are used for training, validation, and testing, respectively. Each atmospheric state contains 69 input and output fields: geopotential (z), temperature (t), specific humidity (q), and zonal and meridional winds (u and v) at 13 pressure levels, (300,400,500,600,700,850,925,1000 hPa), together with four surface variables: T2M, U10, V10, and mean sea-level pressure (MSL). Per-field normalization statistics are computed only from the training set. VeinCast and baseline models predict a 6-hour state transition and generate longer-range forecasts autoregressively.

VeinCast and baseline models are trained on 16 NVIDIA A100 GPUs. VeinCast uses AdamW with a weight decay of 10−3 and a linear warm-up followed by cosine learning-rate decay. We optimize a latitude-weighted Huber loss with threshold kappa = 2.0. Training follows a three-stage rollout curriculum, with each stage initialized from the best checkpoint of the preceding stage. All baseline models are trained under a unified experimental framework, and their hyperparameters are systematically optimized to obtain the best-performing checkpoints for a fair comparison with VeinCast.

Model Performance and Analysis

Table 2 compares VeinCast with five representative global weather forecasting models, including FengWu, GraphCast, Pangu-Weather, ARROW, and FuXi. We report results for three surface variables (T2M, U10, and V10) and three upper-air fields (Q500, T850, and Z700) at lead times from 3 to 14 days. Performance is evaluated using latitude-weighted root mean square error (RMSE) and anomaly correlation coefficient (ACC), where lower RMSE and higher ACC indicate better forecast accuracy.

VeinCast achieves the lowest RMSE for most variable–lead-time combinations while maintaining consistently high ACC at extended forecast horizons. Across all five lead times, it delivers the best overall performance for T2M, Q500, and T850, with particularly clear advantages for temperature and moisture prediction. It also performs strongly on near-surface winds, achieving the highest V10 ACC at every lead time and the highest U10 ACC from 5 to 14 days. These results highlight the benefit of explicitly modeling cross-field relationships and their joint temporal evolution. Moreover, the performance gains remain stable as the forecast horizon increases, indicating improved robustness against error accumulation during recursive forecasting. Although Pangu-Weather retains a modest advantage in Z700 RMSE at 10 and 14 days, VeinCast provides stronger and more consistent performance across heterogeneous surface and upper-air variables.

Fig. 3 provides a rank-based summary of VeinCast’s performance on normalized RMSE and ACC across all forecast lead times and variables. VeinCast consistently achieves the best or second-best rank for most variables, with especially strong dominance at 120 h and 168 h, indicating superior and stable performance across the full variable set.

Ablation Study

We conduct ablation studies to examine the contributions of the dynamic field graph, graph-conditioned latent fusion, adaptive fusion gates, and model capacity. Table 3 summarizes the performance averaged over all evaluated lead times, together with the long-range results at lead times of at least 168 hours.

The complete VeinCast consistently provides strong overall and long-range forecasting performance.

Dynamic field graph: Removing the dynamic field graph leads to a pronounced degradation in both normalized RMSE and ACC, with the performance gap becoming more evident at longer lead times. Retaining only the predefined physical relations recovers a substantial portion of this loss, demonstrating that the physical relational scaffold provides an effective structural prior. Nevertheless, the remaining gap to the complete model indicates that state-dependent residual edges complement the fixed physical graph by capturing cross-field interactions that vary with the local atmospheric state.

Graph-conditioned fusion: Disabling graph conditioning in the latent fusion module also causes a clear performance drop. In this variant, the dynamic graph remains available to the encoder, but its contextual representation and centrality information are not used to guide cross-field aggregation. The resulting degradation shows that relational information is most effective when it directly conditions the fusion and feedback processes, rather than remaining isolated within the graph-enhanced encoder.

Adaptive fusion gates: Replacing the adaptive alpha and beta gates with fixed coefficients produces performance close to that of the complete model. This relatively small difference suggests that adaptive gating is not the primary source of the overall improvement. Instead, it serves as a fine-grained mechanism for regulating field-specific feedback and the use of fusion memory under different atmospheric states and forecast lead times.

Model capacity: The capacity-controlled Wide Baseline increases the encoder width while removing both the dynamic graph and the latent fusion pathway, resulting in a parameter count comparable to that of VeinCast. Despite this increased capacity, it remains substantially inferior, particularly in terms of long-range ACC. This result confirms that the improvements of VeinCast arise from its relational modeling and graph-conditioned fusion mechanisms rather than from a simple increase in model size.

Progressive rollout training: Table 4 shows that the rollout fine-tuning in Stages 2 and 3 consistently improves forecasting performance over the one-step model. By progressively training the model on multi-step predictions, these stages better align the training and inference processes and mitigate error accumulation during recursive forecasting. The results therefore confirm the effectiveness of progressive rollout fine-tuning, particularly for improving the stability of long-range forecasts.

Analysis of Learned Graph Structure and Fusion Gates

We analyze the learned adjacency matrix, edge-type weights, and fusion gates to characterize how the proposed modules organize cross-variable information. The results are shown in Fig. 4.

Learned field graph: The adjacency matrix exhibits clear block-wise structures across variable families. Strong weights concentrate within the same family and between vertically related atmospheric variables, whereas unrelated variable pairs receive weaker connections. Non-zero interactions between temperature and moisture, as well as between wind and pressure-related variables, further show that the graph captures structured cross-variable dependencies rather than uniformly mixing all fields.

Edge-type weight distribution: Vertical edges account for the largest proportion of the adjacency mass, confirming that cross-level atmospheric coupling is the dominant relation in the learned graph. The learned residual receives the second-largest contribution, showing that the model supplements the predefined relations with data-driven connections. Self edges and the remaining physically defined relations retain non-zero weights, indicating that both prior structure and learned interactions contribute to graph construction.

Fusion feedback gate: The feedback gate alpha assigns larger values to the horizontal wind and surface variables than to geopotential height, temperature, and moisture. This variable-dependent pattern shows that the feedback pathway modulates different atmospheric fields unequally, with stronger modulation for wind and surface-related predictions.

Fusion query gate: The query gate beta exhibits stronger variable selectivity than alpha. The largest gate values occur for the horizontal wind variables, whereas geopotential height receives the smallest value. The remaining variables lie between these two extremes. This ordering indicates that the query pathway adjusts the contribution of fused information according to the target variable family.

Overall, the learned graph combines structured atmospheric relations with complementary data-driven connections, while the two fusion gates apply distinct variable-dependent modulation. These results show that the proposed modules perform selective information integration across atmospheric variables.

Conclusion

This paper introduced VeinCast to model structured and state-dependent interactions among meteorological fields in global medium-range weather forecasting. The Physics-Guided Dynamic Field Graph combines predefined relations with state-dependent residual edges, while its context adapts Earth-window attention and guides field-to-latent aggregation through Graph-Conditioned Latent Fusion. Experiments show competitive accuracy across surface and upper-air variables, particularly at extended lead times. Ablations and graph analyses demonstrate that the physical scaffold, residual connections, and graph-conditioned aggregation are complementary, and that the gains do not arise from model capacity alone. These findings support relational-level physical priors as a middle ground between implicit data-driven mixing and rigid equation-level constraints. The current evaluation is limited to deterministic forecasts at 1.5◦ resolution over a fixed 69-field set. Future work will consider higher-resolution and probabilistic forecasting.

Improvements to AI Systems Based on VeinCast:

1. Relational-Level Physics Integration: Instead of embedding full governing equations (which risk model-form bias), AI systems can incorporate physics as a relational scaffold—predefined edges between variables (e.g., wind–geopotential, temperature–moisture) that provide stable inductive bias while allowing state-dependent residual edges to capture regime-specific couplings. This balances physical grounding with data-driven flexibility.

2. Dynamic Graph Construction with Top-K Residual Edges: AI systems can learn a graph per local spatial window where nodes are variables (not grid points), combining a fixed physics-defined backbone with the top-K highest-scoring residual edges based on local atmospheric state. This captures both persistent and transient cross-field interactions without requiring all possible connections.

3. Graph-Conditioned Attention Modulation: Rather than using graph context as a separate branch, AI systems can use it to adapt existing attention mechanisms—e.g., generating gated keys that blend base spatial attention with graph-informed keys, and applying per-head gates. This preserves native spatial representations while injecting relational context.

4. Source-Node Centrality for Aggregation: When fusing multiple fields into latent representations, AI systems can compute source-node centrality from the learned adjacency matrix (how much each source field contributes to others) and use it as a bias in cross-attention. This prioritizes locally influential fields during aggregation, improving information routing.

5. Bounded Fusion Feedback with Variable-Specific Gates: AI systems can write fused latent context back to field representations using bounded gates (e.g., 0–0.25 range) that are conditioned on field metadata and current state. This prevents over-mixing and preserves field-specific information, with gates learned to modulate differently per variable (e.g., stronger for winds, weaker for geopotential).

6. Relation-Aware Decoder Queries: For forecasting, AI systems can construct decoder queries from variable identity, pressure level, and surface indicators, then use a fixed relation bias (favoring same-field, vertically related, and physically coupled variables) during cross-attention over field memory. This ensures the decoder retrieves physically relevant context.

7. Progressive Rollout Training with Curriculum: AI systems can be trained in stages—first on single-step predictions, then progressively fine-tuned on multi-step rollouts (e.g., 2-step, 4-step) to align training with inference and reduce error accumulation during autoregressive generation.

8. Graph-Guided Multi-Scale Latent Backbone: AI systems can use a U-shaped latent backbone where fusion latents (initialized with learnable embeddings + geography + calendar) alternate between spatial attention within each latent and cross-attention across latents, with downsampling/upsampling to capture multi-scale dependencies—all conditioned on graph context from the encoder.

What the Improved AI System Can Do:

• Forecast 69 heterogeneous atmospheric fields (surface + upper-air at 13 pressure levels) with competitive or superior accuracy up to 14 days ahead, outperforming models like GraphCast, Pangu-Weather, and FuXi on most variables.

• Explicitly model state-dependent cross-field interactions (e.g., temperature–moisture coupling that strengthens in convective regimes) without rigid equation constraints, adapting to local atmospheric conditions.

• Maintain field-specific fidelity during information fusion via bounded, variable-dependent gates, preventing over-smoothing while still sharing useful context across variables.

• Achieve stable long-range performance through progressive rollout training, reducing error accumulation in autoregressive forecasting.

• Provide interpretable relational structure—the learned graph reveals dominant vertical couplings, complementary residual edges, and variable-specific fusion behavior, aiding scientific analysis.

• Scale efficiently by using local window graphs and latent bottlenecks, avoiding full pairwise interactions across all grid points and variables.