Physics-Constrained Adaptive Flow Matching for Climate Downscaling
Authors: Kevin Debeire, Aytaç Paçal, Pierre Gentine, Luis Medrano-Navarro, Nils Thuerey, Veronika Eyring
Organizations: Deutsches Zentrum für Luft- und Raumfahrt (DLR), Institut für Physik der Atmosphäre, Oberpfaffenhofen, Germany · School of Computation, Information and Technology, Technical University of Munich, Germany · LEAP Center, Columbia University, New York, NY, 10027, USA · Institute of Environmental Physics (IUP), University of Bremen, Bremen, Germany
Regional climate information at kilometer scales is essential for assessing the impacts of climate change, but generating it with global climate models is too expensive due to their high computational costs. Machine learning models offer a fast alternative, yet they often violate basic physical laws and degrade when applied to climates outside of their training distribution. We present Physics-Constrained Adaptive Flow Matching (PC-AFM), a generative downscaling model that addresses both problems. Building on the Adaptive Flow Matching (AFM) model of Fotiadis et al. (2025) as our baseline, we add soft conservation constraints that keep the downscaled output consistent with the large-scale input for precipitation and humidity, and use gradient surgery via the ConFIG algorithm to prevent these constraints from interfering with the generative objective. We train the model on Central Europe climate data, evaluate it on a 10-time downscaling task (63km to 6.3km) over six variables (near-surface temperature, precipitation, specific humidity, surface pressure, and horizontal wind components) across a comprehensive set of metrics including bias, ensemble skill scores, power spectra, and conservation error, and test the generalization on two held-out climate regions. Within the training distribution, PC-AFM reduces conservation errors and improves ensemble calibration while matching the baseline on standard skill metrics. Outside the training distribution, where unconstrained models develop large systematic errors by extrapolating learned statistics, PC-AFM halves precipitation wet bias, reduces conservation error and improves extreme-quantile accuracy, all without any information about the target climate at inference time. These results indicate that physical consistency is a practical requirement for deploying generative downscaling models in real-world applications.
Figures & tables
Category
Variables
Levels
Surface input fields (10 variables)
tas , uas , vas , pr
Surface
huss , ps
Surface
rlds , rlus , rsds , rsus
Surface
Pressure-level input fields (24 variables)
ta , ua , va , zg
850, 700, 500, 300, 200 hPa
Table 1: Input and output variables. The input comprises 37 channels: 10 surface fields, 24 pressure-level fields, and 3 static invariants. The output comprises 6 surface variables.
Figure 1: Geographic domains used for training and evaluation. Evaluation diagnostics for all three domains are computed using ESMValTool. The Central Europe domain (blue) is used for training. The Iberian Peninsula (green) and Northern Europe region (orange) are withheld from training and used exclusively for out-of-distribution evaluation.
Figure 2: Overview of the PC-AFM architecture and training procedure. Top: At inference, the low-resolution input (32 × 32, 63 km) is bilinearly upsampled and passed through the learned encoder Eψ to produce an initial high-resolution estimate x^0 . A stochastic interpolant xt=(1−t)x^0+tx1+σtε is constructed and refined by the denoiser Dθ , conditioned on the low-resolution input and noise level σt . Fifty denoising steps yield 12 ensemble members at 320 × 320 resolution (6.3 km). Both networks use no positional embeddings to support geographic generalization. Bottom: During training, gradients from the generative objective ( LAFM+λencLenc ) and the physics-constrained conservation objective ( Lphys ) are combined via the ConFIG operator, which ensures non-negative progress on both objectives simultaneously. The conservation penalty is down-weighted at high noise levels and disabled during a 200k-sample warmup period.
Figure 3: Relative performance of PC-AFM versus AFM-baseline for the Central Europe training region. Each cell shows the ratio of PC-AFM to AFM-baseline; values below 1 (green) indicate improvement. Conservation error is not applicable (“–”) for variables without an explicit conservation constraint. Bold entries summarize row and column averages.
Figure 4: Precipitation ( pr ) evaluation for the Central Europe training region. (A) Spatial maps of time-mean bias, relative bias, CRPS, and conservation error for AFM-baseline (top) and PC-AFM (bottom). (B) Radially averaged power spectral density. (C) Log-transformed marginal PDF. (D) Rank histograms with MCB. PC-AFM halves the conservation error and improves ensemble calibration (MCB: 0.767 to 0.523) while maintaining comparable bias and CRPS.
Figure 5: Quantile MAE relative performance (PC-AFM / AFM-baseline) for impact-relevant diagnostics in the Central Europe training region. Average ratio: 0.71.
Figure 6: Relative performance of PC-AFM versus AFM-baseline for the Northern Europe region (unseen during training). Layout as in Figure 3 .
Figure 7: Precipitation evaluation for the Northern Europe region (unseen during training). Layout as in Figure 4 . PC-AFM halves the wet bias (49.9% to 24.6%), reduces CRPS by 23%, and cuts conservation error from 0.037 to 0.018 mm h -1 . Ensemble calibration improves substantially (MCB: 1.006 to 0.772).
Figure 8: Quantile MAE relative performance for the Northern Europe region (unseen during training). Average ratio: 0.77.
Figure 9: Relative performance of PC-AFM versus AFM-baseline for the Iberian Peninsula (unseen during training). Layout as in Figure 3 .
Figure 10: Precipitation evaluation for the Iberian Peninsula (unseen during training). Layout as in Figure 4 . PC-AFM reduces conservation error (0.012 to 0.009 mm h -1 ) and improves ensemble calibration (MCB: 1.104 to 0.879).
Figure 11: Quantile MAE relative performance for the Iberian Peninsula (unseen during training). Average ratio: 0.80.
Generative machine learning is an increasingly important complement to dynamical downscaling for producing high-resolution precipitation projections, with diffusion models currently the leading approach. Flow matching is a related generative framework that has recently achieved strong results across image, video and other domains, and shown early promise for downscaling. We train a flow matching model to map daily precipitation from 8 km to 2 km over a convective-scale domain centred on Singapore, and benchmark it against CPMGEM, a score-based diffusion model. Flow matching achieves consistently better spatial skill: higher fractions skill score at every precipitation threshold and neighbourhood scale tested, and tighter structure and amplitude components of the SAL score with comparable location skill. However, flow matching underestimates the upper tail of the precipitation distribution, resulting in a dry bias in the climatological mean. These results suggest that flow matching is a competitive generative framework for convective-scale precipitation downscaling, particularly well suited to capturing spatial structure.
Decisions about heat, frost and flooding are made locally, yet reanalyses and climate models resolve tens of kilometres. Generative downscalers add kilometre-scale detail with uncertainty, but are tied to a grid, costly to sample and lack a tractable likelihood. Here we introduce CD-FFM, a functional flow-matching model that treats the high-resolution field as a continuous random field that one model can query at any location and resolution. Its velocity acts pointwise, so spatial variability comes from a noise prior fitted to the observed spectrum and every observation has an exact likelihood. Downscaling ERA5 to 1 km daily temperature over Fennoscandia, CD-FFM lowers the continuous ranked probability score by 27-31% against regression and 12-18% against a forty-times larger neural operator in each of three test years, and outperforms a CorrDiff-style diffusion model while sampling about fifty times faster. The same model transfers to precipitation and makes large kilometre-scale ensembles affordable.
Zhi-Song Liu, Helmi Toropainen, Ammar Kheder +1
Department of Computational Engineering, LUT University, Lappeenranta, Finland. · Institute for Atmospheric and Earth System Research (INAR), University of Helsinki, Helsinki, Finland. · Atmospheric Modelling Center, Lahti (AMC-Lahti), Lahti, Finland.
Effective adaptation and mitigation strategies for climate change require high-resolution projections to inform strategic decision-making. Conventional global climate models, which typically operate at resolutions of 150 to 200 kilometers, lack the capacity to represent essential regional processes. IPSL-AID is a global to regional downscaling tool based on a denoising diffusion probabilistic model designed to address this limitation. Trained on ERA5 reanalysis data, it generates 0.25 degree resolution fields for temperature, wind, and precipitation using coarse inputs and their spatiotemporal context. It also models probability distributions of fine-scale features to produce plausible scenarios for uncertainty quantification. The model accurately reconstructs statistical distributions, including extreme events, power spectra, and spatial structures. This work highlights the potential of generative diffusion models for efficient climate downscaling with uncertainty