Soft Physics, Hard Evidence: VeinCast Routes Weather Knowledge Without Writing the Equations

TL;DR for operators

Scientific ML teams often face an awkward knowledge problem: experts may know which variables should interact more reliably than they know the exact equation that should govern every interaction under every operating regime. Hard-coding approximate equations can import modeling assumptions; ignoring expert structure leaves the model to reconstruct useful relationships from data alone.

VeinCast offers a third option. Instead of enforcing governing equations, it gives the model a fixed registry of plausible atmospheric relationships, allows a small number of additional state-dependent connections, and then uses the resulting graph to decide how information from different meteorological fields should be combined.¹

The ablations are more informative than the headline benchmark ranking. Full VeinCast records average normalized RMSE/ACC of 0.59/0.807. Removing the dynamic graph worsens this to 0.69/0.778; retaining the graph but removing graph-conditioned fusion gives 0.69/0.779. A static physical graph reaches 0.64/0.787, while a capacity-matched wider baseline remains at 0.69/0.754.

For teams designing scientific or operational ML systems, the transferable idea is not “add physics.” It is more specific: encode relationships you trust, permit bounded learned exceptions, and let those relationships influence information routing rather than leaving them as descriptive metadata. The evidence supports that design inside VeinCast’s deterministic 1.5° ERA5 setting. It does not establish that learned edges are physical causes or that the architecture will retain its advantage in production, at higher resolution, or in probabilistic forecasting.

When experts know the relationships but not the equations

A domain expert may be confident that two quantities should influence one another while being much less confident that a simplified equation captures that influence accurately across all relevant regimes.

Weather forecasting makes this tension unusually visible. Atmospheric variables are physically coupled, but equation-level constraints introduced into a learned model can inherit approximation, closure, parameterization, and discretization choices. At the other extreme, treating 69 meteorological fields as variables whose useful interactions must be inferred entirely from data discards knowledge scientists already possess.

VeinCast places expert knowledge between these extremes. Its model predicts a shared six-hour transition across five upper-air variables at 13 pressure levels plus four surface variables, then recursively applies that transition to produce forecasts out to 14 days.

The physical contribution is not an equation-based loss. Training uses a masked, latitude-weighted Huber forecasting objective with no equation-based physical regularizer. Instead, physics enters through assumptions about relationships among fields.

That is what the paper calls relational-level physical guidance.

The graph is a permission structure, not a physical simulator

Inside each local spatial window, VeinCast represents meteorological fields as graph nodes. A predefined registry supplies relationships including vertical coupling, wind–geopotential links, temperature–moisture links, moisture transport, and surface–atmosphere connections.

Those registered connections are always retained. But they do not form a closed map of allowable interactions.

For every target field, sample, and spatial window, the model can also select the four highest-scoring nonregistered residual edges. Their scores depend on the current field representations as well as learned relation biases and an additive preference for registered physical relations.

This creates two kinds of authority:

• the expert registry says which interactions deserve persistent support;
• the learned residual channel can add interactions whose relevance depends on the current atmospheric state.

The static-graph ablation shows why both matter. Replacing the dynamic graph with physical relations alone improves over having no graph, reaching 0.64 normalized RMSE and 0.787 ACC versus 0.69/0.778 without the dynamic graph. But the full model reaches 0.59/0.807.

At lead times of at least 168 hours, the separation becomes larger: full VeinCast records 0.81/0.704, compared with 0.88/0.681 for the static physical graph and 0.96/0.668 with no dynamic graph.

This is evidence for combining persistent prior structure with state-dependent correction. It is not evidence that the model has discovered a better set of physical laws.

Representing relationships is not enough; they must control fusion

A graph can easily become architectural decoration: the model computes relational features, but the main information pathway continues largely unchanged.

VeinCast goes further. Its graph changes spatial attention and also influences how fields enter a shared latent representation.

The model calculates a form of source-node centrality from the dynamic adjacency matrix. Fields that behave as stronger information sources receive a learned bias when four fusion latents aggregate cross-field information. Graph-enriched field representations are therefore not merely carried alongside the forecast computation; they help determine which information enters the shared representation and with what priority.

The corresponding ablation is unusually clean. Removing graph-conditioned fusion while retaining graph information elsewhere degrades average performance from 0.59/0.807 to 0.69/0.779. At 168 hours and beyond, performance changes from 0.81/0.704 to 0.95/0.672.

That makes the architectural contribution more precise. The gain is associated not only with identifying relationships, but with giving those relationships decision rights over cross-field information routing.

VeinCast also limits how much shared information can overwrite field-specific representations. Its learned feedback gate is bounded below 0.25, while the decoder’s latent-information gate is bounded below 0.5. Replacing adaptive gates with fixed coefficients changes results only modestly—0.60/0.802 versus 0.59/0.807—so adaptive gating appears to refine the system rather than explain most of its advantage.

The ablations rule out two simpler explanations

The component tests are valuable because they test competing explanations for the improvement.

Progressive rollout training is another contributor, but it addresses a different problem. Supervising one, then two, then four autoregressive steps improves average normalized RMSE/ACC from 0.64/0.785 at Stage 1 to 0.59/0.807 at Stage 3. At 168 hours and beyond, the corresponding metrics improve from 0.86/0.678 to 0.81/0.704.

This supports multi-step training as a way of reducing the mismatch between short-horizon training and recursive long-horizon inference. It should not be confused with evidence for the relational graph itself.

What scientific-ML teams can carry into other systems

The business relevance is architectural rather than meteorological.

For teams building models around engineering, industrial, environmental, biological, or financial systems, expert knowledge often arrives in an inconvenient form. People know reliable dependencies, exclusions, hierarchies, or pathways, but the exact transformation governing those relationships may be disputed, approximate, or regime-dependent.

VeinCast suggests a design pattern for that situation:

Stable prior relationships → bounded learned exceptions → relationship-conditioned information routing.

The affected decision is how much authority to assign domain knowledge inside the model. A fixed relational scaffold can narrow the interaction space without dictating exact numerical dynamics. A learned residual channel can handle conditions the registry did not anticipate. Routing mechanisms can then ensure that the encoded knowledge actually changes computation.

The potential operational benefit is easier architectural inspection: teams can examine the registry, residual-selection mechanism, fusion pathway, and gate bounds as distinct design decisions. But the paper does not measure debugging cost, governance outcomes, serving latency, infrastructure expense, or downstream economic value. Those remain hypotheses for deployment teams to test.

Diagnostic graphs are not discovered atmospheric causality

VeinCast’s internal diagnostics are interesting but easy to overread.

Across 160 test samples spanning four seasons and lead times of 24, 168, and 336 hours, approximately 0.46 of learned adjacency mass goes to vertical relations, 0.16 to learned residual edges, and 0.11 to self edges. Fusion gates also differ by variable type.

These observations show how the trained network allocates internal relational weight. They do not establish that an edge represents a causal atmospheric mechanism. The authors explicitly treat this analysis as diagnostic rather than causal.

That boundary is important if the architecture is reused in regulated or scientifically sensitive systems. Inspectable routing is not the same thing as causal interpretability.

The evidence is strong inside a narrow deployment envelope

VeinCast is evaluated on deterministic global forecasting at 1.5° resolution, using WeatherBench2 ERA5 data, a fixed registry of 69 fields, training from 1979–2017, validation in 2018, and testing in 2020.

Within that setting, the benchmark and ablation evidence is substantial. VeinCast is competitive with FengWu, GraphCast, Pangu-Weather, ARROW, and FuXi, and its component tests consistently favor the full relational architecture. It is not uniformly best: Pangu-Weather retains lower Z700 RMSE at 10- and 14-day horizons.

Outside that setting, several questions remain open. The model is closed-set rather than able to accept arbitrary variables or pressure levels. The paper does not test probabilistic forecasting, higher-resolution operation, multiple independent reanalysis products, operational forecast systems, production reliability, or deployment economics.

So the broader lesson should travel more cautiously than the benchmark score.

VeinCast provides evidence that domain knowledge does not need to become an equation before it can receive architectural authority. When experts know the structure of plausible interaction better than they know its exact dynamics, a relational prior can serve as a stable scaffold while learned residual connections handle state-dependent exceptions. The paper’s more consequential contribution is making that scaffold control information flow—and then showing through ablation that this control contributes beyond either static structure or added capacity alone.

Cognaptus: Automate the Present, Incubate the Future.