Spatiotemporal Pyramid Flows (SPF) for Climate Emulation

• Spatiotemporal Pyramid Flows (SPF) are generative models that hierarchically partition climate trajectories into cascading stages, capturing coarse decadal signals to fine-scale variability.
• The method employs learned downsampling and upsampling operators to efficiently refine spatial and temporal resolutions, significantly reducing computation compared to autoregressive approaches.
• By conditioning on physical forcings, the unified flow-matching framework achieves robust and accurate climate emulation, ensuring stable long-horizon rollouts across diverse Earth System Models.

Spatiotemporal Pyramid Flows (SPF) are a class of flow-matching generative models introduced for climate emulation, with the specific aim of modeling Earth System Model (ESM) outputs across climate-relevant timescales while retaining probabilistic uncertainty and avoiding the computational cost of weather-scale autoregressive rollout (Irvin et al., 1 Dec 2025). In this formulation, the generative trajectory is partitioned into a spatiotemporal pyramid that jointly cascades in space and time, so that coarse, low-frequency structure is modeled first and progressively refined into higher-resolution and higher-frequency outputs. The method is explicitly designed to support direct sampling at multiple temporal levels, conditioning on prescribed physical forcings such as greenhouse gases, aerosols, and intervention-related signals, and was presented together with ClimateSuite, a large multi-model climate-emulation dataset (Irvin et al., 1 Dec 2025).

1. Motivation and problem setting

SPF was proposed against a background in which ESMs “faithfully resolve processes from hours-to-centuries but at prohibitive computational cost” (Irvin et al., 1 Dec 2025). The paper identifies two limitations of recent machine-learning surrogates based on weather-scale autoregressive emulation. First, such systems suffer from “error accumulation and climatological drift under nonstationary forcings,” including increasing greenhouse gases and geoengineering interventions. Second, they are “extremely slow” for long-horizon sampling; the paper gives the example that “a 10-year trajectory can take hours of GPU time” (Irvin et al., 1 Dec 2025).

The work also distinguishes SPF from regression-style emulators that map forcings directly to long-term means. Those approaches “lack the ability to generate high-frequency uncertainty ensembles” (Irvin et al., 1 Dec 2025). SPF is presented as a bridge between these regimes by modeling climate sequences hierarchically across space and time in a single flow-matching framework, conditioning each stage on prescribed physical forcings, and enabling direct, parallel sampling at arbitrary timescales without stepwise autoregression (Irvin et al., 1 Dec 2025).

A central conceptual point is that SPF treats climate emulation as a generative transport problem rather than as a sequence of weather-scale prediction steps. This design targets long-horizon stability under nonstationary forcing trajectories while preserving access to uncertainty-aware ensemble generation. A plausible implication is that SPF is intended not merely as an acceleration technique, but as a reformulation of the temporal abstraction used in climate surrogate modeling.

2. Spatiotemporal pyramid architecture

The architecture generalizes “pyramidal flows” from the image and video literature by breaking the generative trajectory into $K$ stages that jointly cascade in spatial resolution and temporal scale (Irvin et al., 1 Dec 2025). In the climate-emulation instantiation described in the paper, $K=3$:

• Stage 0 (coarsest): Decadal resolution, $T_0=10$ years on a downsampled spatial grid.
• Stage 1 (mid): Yearly resolution, $T_1=12$ months aggregated to annual on a medium grid.
• Stage 2 (finest): Monthly resolution, $T_2=12$ monthly frames on the full spatial grid.

At each stage $k$, SPF defines learned downsampling and upsampling operators, $\mathrm{Down}_k(\cdot)$ and $\mathrm{Up}_k(\cdot)$, with per-stage spatial factors $r_k^h=r_k^w=2$ and temporal factors $r_k^t\in\{10,12\}$ (Irvin et al., 1 Dec 2025). During training and inference, stage $K=3$0 operates on a noised latent of the form

$K=3$1

where $K=3$2 linearly interpolates over the segment $K=3$3 (Irvin et al., 1 Dec 2025).

The resulting hierarchy is described as a “denoise and refine” cascade. Stage 0 captures the forced, low-frequency decadal signal on a small grid; Stage 1 upsamples in space and time to recover interannual variability on a medium grid; and Stage 2 upsamples further to reconstruct monthly weather fluctuations on the full grid (Irvin et al., 1 Dec 2025). This progression matters because it makes the coarse-scale climatic signal an explicit part of the generative path rather than a byproduct of long autoregressive rollout.

A common misconception would be to regard SPF as only a spatial super-resolution scheme. The formulation in fact couples spatial refinement with temporal refinement, and the paper’s framing is explicitly spatiotemporal rather than only multiresolution in space.

3. Flow-matching formulation

SPF is trained as a conditional flow-matching model with a single neural network $K=3$4, implemented in the reported instantiation with “a DiT transformer backbone,” to approximate the conditional velocity field that transports a standard Gaussian prior $K=3$5 to the data distribution $K=3$6 (Irvin et al., 1 Dec 2025). The paper defines correlated endpoints for stage $K=3$7 as

$K=3$8

and minimizes the squared error between the network’s instantaneous velocity and the ground-truth interpolation vector through the pyramid flow-matching loss

$K=3$9

This objective encodes the stagewise transport from coarse-to-fine representations while preserving a single-network formulation (Irvin et al., 1 Dec 2025).

To support direct sampling at intermediate timescales, the method introduces binary indicators $T_0=10$0 that “freeze or refine the temporal dimension per stage” (Irvin et al., 1 Dec 2025). The resulting multi-timescale loss is

$T_0=10$1

where the endpoints $T_0=10$2 follow the chosen spatial-only or spatiotemporal path (Irvin et al., 1 Dec 2025).

This multi-timescale objective is central to SPF’s claimed functionality. It is what allows annual averages, for example, to be sampled directly “without first generating monthly data” (Irvin et al., 1 Dec 2025). This suggests that temporal abstraction is built into the transport objective itself, rather than being delegated to post hoc aggregation.

4. Conditioning on physical forcings and generation procedure

At each stage, SPF is conditioned “via cross-attention on the forcings aggregated or averaged to that stage’s timescale” (Irvin et al., 1 Dec 2025). The forcings listed in the paper are cumulative CO$T_0=10$3 and CH$T_0=10$4 emissions, SO$T_0=10$5 and black carbon emission maps, and stratospheric aerosol optical depth (AOD) for geoengineering experiments (Irvin et al., 1 Dec 2025). These forcings enter as learnable patch embeddings with positional and timescale embeddings.

The stated purpose of this conditioning design is to enable the model to capture “slow trends (forced climate change) without autoregressive rollouts through historical sequences” (Irvin et al., 1 Dec 2025). In the climate-emulation setting, this is significant because forcing trajectories are exogenous and often nonstationary; conditioning them stagewise aligns the temporal abstraction of the model with the temporal aggregation of the drivers.

The sampling procedure follows from the ODE-flow formulation. Generation is “parallel in both space and time”: the learned velocity field $T_0=10$6 is integrated from $T_0=10$7 to $T_0=10$8 within each stage in parallel for all $T_0=10$9 frames (Irvin et al., 1 Dec 2025). After finishing stage $T_1=12$0, the method applies a closed-form “rescaling–renoising” correction at the jump to stage $T_1=12$1 to ensure distributional continuity under arbitrary spatial and temporal upsampling factors (Irvin et al., 1 Dec 2025). The paper also introduces a temporal funneling mechanism so that only a subset $T_1=12$2 of frames—often $T_1=12$3—need be generated at coarser stages, yielding a $T_1=12$4 reduction in memory and FLOPs when annual or decadal outputs are sufficient (Irvin et al., 1 Dec 2025).

Direct coarse-timescale sampling is therefore explicit: to sample at a coarser timescale $T_1=12$5, the method stops ODE integration at the end of stage $T_1=12$6, applies the rescaling–renoising for the remaining spatial upsampling, and denoises once more, “completely avoiding fine-scale generation” (Irvin et al., 1 Dec 2025). This is an important distinction from weather-scale emulators, because the avoided computation is tied not only to fewer frames but also to the elimination of stepwise temporal dependence.

5. ClimateSuite dataset and empirical results

The paper introduces ClimateSuite as “the largest ML-ready climate emulation dataset to date” (Irvin et al., 1 Dec 2025). It comprises 10 CMIP6 ESMs, including the NorESM2-LM used by ClimateBench and nine additional models, together with 276 non-SAI simulations and 39 stratospheric aerosol injection experiments (Irvin et al., 1 Dec 2025). The dataset includes “over 33,700 simulation-years of monthly, $T_1=12$7 latitude–longitude fields, plus temporally aligned forcings” (Irvin et al., 1 Dec 2025). The inclusion of geoengineering runs under nonstationary forcings is highlighted as making ClimateSuite “uniquely suitable for robustly training and stress-testing probabilistic emulators” (Irvin et al., 1 Dec 2025).

On the held-out ClimateBench scenario, SSP2-4.5 for 2015–2100, SPF is reported to outperform deterministic baselines, pre-trained foundation models, and pixel-space flow baselines (Irvin et al., 1 Dec 2025). The paper gives the following results for a 100M-parameter SPF:

Setting	SPF	Comparator
Yearly CRPS	0.238	PixelFlow 0.247
Yearly RMSE	0.565	0.549
Yearly sampling	3 s	19 s
Monthly CRPS	0.462	PixelFlow 0.474
Monthly RMSE	1.100	1.101
Monthly sampling	6 s	84 s

A 200M-parameter SPF is reported to further lower RMSE and CRPS while remaining “$T_1=12$8 faster than autoregressive flows” (Irvin et al., 1 Dec 2025). With scale and pre-training on ClimateSuite, a 600M-parameter SPF fine-tuned from ClimateSuite pre-training achieves yearly CRPS 0.216 versus 0.222 without pre-training, and monthly CRPS 0.432 versus 0.453 (Irvin et al., 1 Dec 2025).

The multi-model generalization results are especially emphasized. On held-out SSP2-4.5 across 10 ESMs and a held-out UKESM1-0-LL SAI experiment, SPF “beats a 600 M-parameter UNet in both RMSE and CRPS for every model,” with average CRPS 0.256 versus 0.393 on standard scenarios and 0.315 versus 0.466 on the SAI run (Irvin et al., 1 Dec 2025). The paper interprets these results as demonstrating stable, accurate rollouts under nonstationary forcings and strong transfer across structural model variability.

In computational terms, SPF is reported to achieve “$T_1=12$9 speedups over autoregressive pyramidal flow baselines when sampling ten-year sequences at yearly resolution,” and “$T_2=12$0 speedups at monthly resolution” (Irvin et al., 1 Dec 2025). The coarse stages are described as especially inexpensive; the paper gives the example that “a decadal sample on a $T_2=12$1 grid costs only a few seconds, versus minutes for full monthly sampling on $T_2=12$2” (Irvin et al., 1 Dec 2025).

6. Contributions, limitations, and research directions

The paper identifies three principal contributions: a unified spatiotemporal pyramid flow-matching framework supporting arbitrary resolution resampling and direct multi-timescale sampling; efficient, parallel climate emulation in pixel space without VAEs or stepwise autoregression; and ClimateSuite as a multi-model, multi-scenario dataset for uncertainty-aware emulation, including stratospheric aerosol injection interventions (Irvin et al., 1 Dec 2025).

The main limitations are also stated explicitly. The authors note the “lack of explicit physical constraint enforcement,” giving energy or mass conservation as examples, and the model’s “reliance on existing ESM parameterizations,” which may leave gaps for extreme or out-of-distribution forcings (Irvin et al., 1 Dec 2025). These caveats are significant because SPF emulates simulated climate trajectories rather than solving governing equations directly; fidelity is therefore bounded by both the training distribution and the physical consistency built into the model class.

Future directions proposed in the paper include integrating “physics-informed ODE priors or conservation layers,” extending SPF to daily or hourly downscaling, and exploring “joint spatiotemporal latent representations” for greater efficiency and accuracy in climate risk assessment (Irvin et al., 1 Dec 2025). This suggests that SPF may be understood as a foundation for a broader class of multi-timescale probabilistic emulators, rather than as a final architecture.

Taken together, SPF defines a climate-emulation paradigm in which coarse climatic structure, forcing-conditioned evolution, and fine-scale variability are learned within a single staged transport process. Its distinctive claim is not only higher sampling efficiency, but the unification of probabilistic generation, hierarchical temporal abstraction, and nonstationary forcing control in a flow-matching framework (Irvin et al., 1 Dec 2025).