FlowCast-ODE: Continuous Weather Forecasting

• The paper introduces FlowCast-ODE as a continuous flow-based framework that models atmospheric evolution with ODE integration to produce coherent hourly forecasts.
• It employs dynamic flow matching and a two-stage 6h-to-1h training strategy to address error accumulation and temporal discontinuities from assimilation cycles.
• Using a low-rank AdaLN-Zero mechanism in a 3D Swin transformer, FlowCast-ODE reduces model size by 15% without compromising forecast accuracy and spatial detail.

FlowCast-ODE is a framework for continuous hourly weather forecasting that models atmospheric state evolution as a continuous flow and combines dynamic flow matching with Ordinary Differential Equation (ODE) integration to generate temporally coherent forecasts from previous atmospheric states rather than from noise (He et al., 18 Sep 2025). It was proposed to address two difficulties identified for hourly prediction: the rapid accumulation of errors in autoregressive rollouts and temporal discontinuities within the ERA5 data’s 12-hour assimilation cycle. The framework trains on 6-hour data and is then refined on hourly data, while a lightweight low-rank AdaLN-Zero modulation mechanism reduces model size by 15% without compromising accuracy. Empirically, it is reported to yield lower root mean square error (RMSE), better energy conservation, reduced blurring, more fine-scale spatial detail, and comparable performance to the state-of-the-art model in forecasting extreme events like typhoons, while alleviating temporal discontinuities associated with assimilation cycle transitions (He et al., 18 Sep 2025).

1. Problem setting and conceptual basis

FlowCast-ODE targets accurate hourly weather forecasting, a regime in which recent deep learning models had demonstrated strong capability on 6-hour intervals but had not resolved the instability of fine-resolution autoregressive prediction (He et al., 18 Sep 2025). The stated motivation is that hourly forecasting is especially vulnerable to rapid error accumulation in rollouts and to temporal discontinuities in ERA5 induced by the 12-hour assimilation cycle.

The central conceptual move is to represent atmospheric evolution as a continuous conditional flow. Each atmospheric field $X_k \in \mathbb{R}^{C \times H \times W}$ evolves according to a learned velocity field, and the model learns a trajectory from the current state $X_k$ to the future state $X_{k+6}$ by defining intermediate states $x_t = (1-t) X_k + t X_{k+6}$, with $t \in (0,1)$ (He et al., 18 Sep 2025). The governing equation is

$\frac{dx(t)}{dt} = v_{\theta}(x_t, t, c)$

where $v_{\theta}$ is the instantaneous velocity model parameterized by a neural network and $c$ denotes conditioning features such as land/sea mask, geopotential, soil type, lat/lon, local time, and time of year (He et al., 18 Sep 2025).

A recurrent point in the formulation is that the transport begins from the known previous atmospheric state rather than from a random latent variable. The paper explicitly contrasts this with diffusion models that evolve from noise and argues that the conditional-flow formulation aligns more naturally with physical dynamic systems and the Markovian property of the underlying physics (He et al., 18 Sep 2025). This suggests that the framework is intended not merely as a generative sampler, but as a physically motivated time-evolution model for geophysical state variables.

2. Dynamic flow matching formulation

The training objective is based on Dynamic Flow Matching, described as a variant in which the model directly transports from the known prior to the target future state instead of sampling from noise (He et al., 18 Sep 2025). For two atmospheric states $x_0 = X_k$ and $x_1 = X_{k+6}$, the conditional probability path is specified by

$X_k$0

so that intermediate samples lie on a linear interpolation with no added noise (He et al., 18 Sep 2025). The corresponding target velocity is $X_k$1, and the loss is given as

$X_k$2

with $X_k$3 (He et al., 18 Sep 2025).

Within the paper’s framing, this construction has several practical implications. It is presented as physically aligned because the path starts from the true prior atmospheric state. It also removes the need for multiple sub-models, in contrast to Pangu-Weather, which is described as using multiple models for multiple horizons (He et al., 18 Sep 2025). The same section further states that the continuous formulation with explicit ODE integration is inherently more stable and avoids spurious high-frequency or nonphysical patterns, and that using 6-hour targets in this flow-matching regime skips assimilation-induced jumps during training (He et al., 18 Sep 2025).

A common misconception would be to read FlowCast-ODE as a conventional diffusion-style weather generator with a renamed objective. The formulation does not support that reading. The model is explicitly defined as learning a conditional flow from the previous atmospheric state, and the paper presents this departure from noise-based generation as one of its main methodological differences (He et al., 18 Sep 2025).

3. ODE integration and the coarse-to-fine hourly forecasting strategy

The transition from 6-hour supervision to hourly prediction is handled through a two-stage training procedure. In the coarse stage, the velocity model is trained via dynamic flow matching on 6-hourly intervals at 00, 06, 12, and 18 UTC. In the fine stage, the pre-trained model is fine-tuned on hourly data, using the model as an ODE integrator to produce and train on sequences of hourly steps through autoregressive rollouts for 6, 12, and 18 hours; gradient checkpointing is used to enable long-sequence training with manageable memory overhead (He et al., 18 Sep 2025).

The ODE solver is an explicit Euler integrator. Given the atmospheric state at time $X_k$4,

$X_k$5

with $X_k$6 (He et al., 18 Sep 2025). Repeated application yields hourly states at arbitrary resolution. The paper attributes temporal coherence to the fact that adjacent steps are linked by a continuous learned velocity field rather than by independently trained one-hour transition operators.

This coarse-to-fine structure is central to the handling of ERA5 discontinuities. The coarse stage avoids direct exposure to the ERA5 12-hour assimilation cycle discontinuities, while the fine stage imposes hourly consistency through ODE rollout (He et al., 18 Sep 2025). The paper’s rationale is that this avoids overfitting to assimilation artifacts and helps the model generalize physical dynamics across temporal scales.

A plausible implication is that FlowCast-ODE uses temporal scale separation as both a statistical and physical prior: 6-hour learning is used to suppress discontinuity-driven supervision artifacts, whereas hourly integration is used to reconstruct smooth intermediate evolution. That interpretation is consistent with the stated claim that the framework alleviates temporal discontinuities associated with assimilation cycle transitions (He et al., 18 Sep 2025).

4. Architecture and low-rank time modulation

FlowCast-ODE is built on a 3D Swin transformer with time-modulated blocks for temporal conditioning (He et al., 18 Sep 2025). Time conditioning is implemented through adaptive layer normalization, written as

$X_k$7

where $X_k$8 is the normalized feature and $X_k$9 and $X_{k+6}$0 are functions of the time embedding (He et al., 18 Sep 2025).

The paper’s architectural modification is a low-rank decomposition of the adaLN-Zero projection. Instead of a full-rank $X_{k+6}$1 projection, the mapping is factorized as

$X_{k+6}$2

where $X_{k+6}$3, with $X_{k+6}$4 given as an example (He et al., 18 Sep 2025). The reported consequence is a parameter reduction of approximately 15%, specifically from 54.2M to 45.7M, without loss in accuracy (He et al., 18 Sep 2025).

In the paper’s presentation, this low-rank modulation serves two roles. First, it reduces model size and computational cost. Second, because the time embedding controls the continuous flow field, it provides a mechanism for conditioning every transformer block on ODE time. The low-rank factorization therefore operates not as a generic compression device but as a targeted simplification of temporal modulation.

The paper explicitly states that the reduction in parameters does not compromise accuracy (He et al., 18 Sep 2025). That claim is restricted to the reported ablation context; it should not be generalized beyond the experiments described there.

5. Empirical behavior: RMSE, energy conservation, spatial fidelity, and extreme events

The empirical evaluation reports that FlowCast-ODE achieves lower RMSE than both Pangu-Weather at $X_{k+6}$5 and $X_{k+6}$6 and ClimODE on most variables, including MSLP, Z500, T2M, U10M, Q850, and T850, especially in the first 72 hours and often over 120 hours, despite significantly more iterative steps from hourly rolling rather than Pangu’s 24-hour rolling (He et al., 18 Sep 2025). RMSE is computed as a latitude-weighted global mean:

$X_{k+6}$7

(He et al., 18 Sep 2025).

A second line of evidence concerns energy conservation and temporal continuity. The paper states that kinetic and internal energy trajectories from FlowCast-ODE more closely track the temporal evolution of ERA5 total energy and avoid the nonphysical jumps caused by the 12-hour assimilation cycle (He et al., 18 Sep 2025). The total energy formula is given as

$X_{k+6}$8

where $X_{k+6}$9 is specific heat at constant volume, $x_t = (1-t) X_k + t X_{k+6}$0 is temperature, $x_t = (1-t) X_k + t X_{k+6}$1 is latent heat of vaporization, $x_t = (1-t) X_k + t X_{k+6}$2 is specific humidity, $x_t = (1-t) X_k + t X_{k+6}$3 is geopotential, and $x_t = (1-t) X_k + t X_{k+6}$4 is kinetic energy (He et al., 18 Sep 2025).

The spatial-fidelity analysis relies on power spectra. The paper reports that FlowCast-ODE preserves higher spectral energy at short-to-mid wavelengths than Pangu-Weather, reducing blurring and maintaining finer spatial features, though some overestimation is observed at scales corresponding to the training patch size (He et al., 18 Sep 2025). Visual comparisons are described as showing fields that are less smoothed and closer to ground truth in both large-scale structure and small-scale detail.

For extreme events, the paper examines six 2021 west Pacific typhoons, including SURIGAE and IN-FA. The reported result is that tropical cyclone track Mean Absolute Error is slightly higher than Pangu-Weather at $x_t = (1-t) X_k + t X_{k+6}$5 in the first 48 hours, but thereafter the performances are comparable (He et al., 18 Sep 2025). The significance attributed in the paper is that a $x_t = (1-t) X_k + t X_{k+6}$6 single-model system can reproduce cyclone trajectories with accuracy similar to high-resolution, multi-model systems.

6. Temporal discontinuities, interpretation, and relation to adjacent “FlowCast” research

A central contribution of FlowCast-ODE is its explicit treatment of temporal discontinuities in ERA5. The paper identifies hourly jumps at 09/21 UTC due to 12-hour data assimilation windows and argues that conventional hourly autoregressive models are trained to replicate these jumps, leading to nonphysical artifacts, error accumulation at window boundaries, and degraded long-range accuracy (He et al., 18 Sep 2025). The proposed mitigation is threefold: avoiding direct training on hourly transitions crossing assimilation windows, using a two-stage 6h-to-1h training strategy, and generating hourly forecasts by continuous flow and ODE integration (He et al., 18 Sep 2025).

The broader literature contains other systems with the name “FlowCast,” but they address different tasks. “FlowCast: Trajectory Forecasting for Scalable Zero-Cost Speculative Flow Matching” is a training-free speculative generation framework for accelerating inference in ODE-based flow matching generative models by extrapolating velocity and verifying it with an MSE threshold (Bajpai et al., 1 Feb 2026). “FlowCast: Advancing Precipitation Nowcasting with Conditional Flow Matching” applies Conditional Flow Matching to radar-based precipitation nowcasting, using a direct noise-to-data mapping in latent space and emphasizing rapid, high-fidelity sample generation with drastically fewer function evaluations than diffusion (Ribeiro et al., 12 Nov 2025). FlowCast-ODE shares the vocabulary of continuous flow and ODE integration with these works, but its problem setting is continuous hourly weather forecasting from previous atmospheric states rather than speculative generation acceleration or radar nowcasting (He et al., 18 Sep 2025).

This naming overlap can produce a second misconception: that FlowCast-ODE is merely an ODE-integrated version of one of the earlier FlowCast frameworks. The available descriptions do not support that simplification. FlowCast-ODE is defined by dynamic flow matching from a known prior atmospheric state, a coarse-to-fine 6-hour to hourly training scheme, explicit hourly Euler integration, and low-rank AdaLN-Zero modulation in a 3D Swin transformer backbone (He et al., 18 Sep 2025). The connection to the other FlowCast papers is therefore methodological at the level of flow-based generative modeling, not identity of application or training recipe.

In the paper’s own terms, the framework’s overall significance lies in combining single-model efficiency, smooth temporal coherence, and mitigation of assimilation-induced artifacts while remaining competitive with strong baselines on RMSE, energy conservation, spatial fidelity, and extreme-event forecasting (He et al., 18 Sep 2025). A plausible implication is that the method is positioned as a continuous-time alternative to horizon-specific or discontinuity-sensitive weather models, particularly when hourly coherence is as important as endpoint accuracy.