Adaptive Flow Matching for Resolving Small-Scale Physics
Authors: Stathi Fotiadis
Noah D Brenowitz
Tomas Geffner
Yair Cohen
Michael Pritchard
Arash Vahdat
Morteza Mardani
Venue: ICML 2025 poster (2025)
Keywords: flow matching
diffusion models
weather downscaling
distribution misalignment
Primary area: applications

Abstract: Conditional diffusion and flow models are effective for super-resolving small-scale details in natural images. However, in physical sciences such as weather, three major challenges arise: (i) spatially misaligned input-output distributions (PDEs at different resolutions lead to divergent trajectories), (ii) misaligned and distinct input-output channels (channel synthesis), (iii) several channels with diverse stochasticity scales (multiscale). To address these, we propose to first encode inputs into a latent base distribution that is closer to the target, then apply Flow Matching to generate small-scale physics. The encoder captures deterministic components, while Flow Matching adds stochastic details. To handle uncertainty in the deterministic part, we inject noise via an adaptive noise scaling mechanism, dynamically adjusted by maximum-likelihood estimates of the encoder’s predictions. Experiments on real-world weather data (including super-resolution from 25 km to 2 km scales in Taiwan) and in synthetic Kolmogorov flow datasets show that our proposed Adaptive Flow Matching (AFM) framework outperforms existing methods and produces better-calibrated ensembles.

Adaptive Flow Matching for Resolving Small-Scale Physics
Stathi Fotiadis*
Noah D Brenowitz?
Tomas Geffner
Yair Cohen? 2 Michael Pritchard Arash Vahdat" Morteza Mardanit
*Work done during an internship at NVIDIA. Equall advising.
Abstract
has recently been extended to super-resolving small-scale physics (Aich et al. [024] Ling et al. (024). However, this task faces significant challenges: (C1) Input and target data are often spatially misaligned due to distinct PDE solu- tions operating various resolutions, leading divergent trajectories. For instance, the eye-of-tyhhoon is spatially misaligned between low and high resolution in CWA data due to different dynamical models used to simulate each scale (see Figure D. Additionally, the input and target vari- ables (channels) often represent different physical quanti- ties, causing further misalignment. (C2) the data exhibit multiscale dynamies, where certain large-scale processes are more deterministic (e.g. the propagation of midlatitude storms), while small-scale physics, such as thunderstorms, are highly stochastic; and (C3) the record of Earth observa- tions is limited compared to natural image datasets.
Conditional diffusion and flow-based models can super-resolve small-scale details in natu- ral images, but weather and other physical do- mains face three unique challenges: (1) spa- tially misaligned inputs and outputs (different PDE resolutions yield divergent trajectories), (ii) mismatched and distinct input-output channels (channel synthesis), and (tit) channel-dependent stochasticity. To address these challenges, we first encode the inputs into latent base dis- tribution that reconstructs the large-scale and more deterministic parts of the target. Next, we handle the remaining uncertainty by inject- ing noise via an adaptive noise scaling mech- anism, informed by maximun-likelihood esti- mates of the encoder's RMSE. Finally, we ap- ply Flow Matching to refine the latent state and add fine-seale physics. Experiments on real- world weather data (c.g., 25 km to 2 km super- resolution in Taiwan) and synthetic Kolmogorov flow indicate that our Adaptive Flow Matching (AFM) framework provides improvements over prior baselines- tic channels and consistently achieves better- calibrated ensembles.
Few efforts have been made to directly address these chal- lenges generative learning. Prior work typically relies on residual learning approaches (Mardani instance, the method proposed in (Mardani et al. 2023) in- troduces two-stage process where the deterministic com- ponent is first learned through regression, followed by ap- plying diffusion on the residuals to capture the small-scale physics. While this approach offers a way to separate deter- ministie and stochastic components, poses a significant risk of overfitting. The initial regression stage may over- fit the training data, and thus fails adequately represent the variability of the small-scale dynamics when training the diffusion model in the second stage (see Appendix AA. Additionally, this two-stage method lacks a principled way to handle the uncertainty inherent in both the deterministic and stochastic components.
2023). For
for more stochas-
1. Introduction
Resolving small-scale physics is crucial in many scientific applications (Wilby et al. 1998 Rampal et al. 2022 2024). For instance, in atmospheric sciences, accurately capturing small-scale dynamics is essential for local planning and disaster mitigation. The success of conditional diffusion models in super-resolving natural images and videos (Song Batzolis et al.
To address these limitations, we propose an end-to-end ap- proach based Flow Matching. The key elements of our method are as follows: First, an encoder maps the coarse- resolution input data to a latent space that is closer to the target fine-resolution distribution. Flow matching is then applied starting from this encoded distribution to generate the target distribution. The encoder captures the determin- istic component, which then augmented with noise to in-
al. 2021
2021 Hoogeboom et al. 2023
'Imperial College London FNVIDIA. Correspondence Morteza Mardani <mmardani @ com
Proceedings of the 1nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Adaptive Flow Matching for Resolving Small-Scale Physics
Noise w
y
Encoder E
2 = E(y) + OzE
Flow Matching
x
X Z
FM ODE dx,= = vX t)dt
XT
Figure I: Overview of Adaptive Flow Matching (AFM). The encoder transforms (coarse-res inputs into a latent distribution more aligned with the (fine-res.) target. It generates channels absent in the input and corrects both spatial and channel misalignments, such as repositioning the typhoon's eye its accurate location, and generating radar data. From the latent space, FM generates small-scale physics by transporting samples from p(z) to p(x) via the velocity field v(x, t).
troduce uncertainty. The deterministic prediction is based on the idea that physical processes occur different time scales, with larger-scale physics having longer-term, more deterministic effects. We refer to this method as Adap- rive Flow Matching (AFM), where the stochasticity is con- trolled by the noise injected at the encoder output. Proper tuning of the noise scale is critical to balance determinis- tic and stochastic dynamics. To achieve this, we propose maximum likelihood procedure that adjusts the noise scale based on the encoder's error, dynamically tuning it on the fly according to the encoder mismatch.
the degree of misalignment grows, conditional diffu- sion/flow models become progressively worse, while AFM with adaptive noise scaling still performs well.
2. Related Work
Conditional diffusion and flow models. Conditioning is powerful technique for paired image-to-imann translation in diffusion/flow models (Batzolis et al. 2022 Kawar et al.) 2022 Xingjian Xie 2023). It is commonly used in im- age restoration tasks such as super-resolution and deblur- ring, where the goal is map a low-quality input high-quality target, with corresponding pixel associations between input and target. However, our experiments demonstrate (see Section 5). plain conditional models of- ten fail when the data severely misaligned.
AFM can be viewed through the lens of diffusion models, efficiently implemented using stable denoising objective. We conduct extensive experiments on both synthetic and the same data Mardani al. (2023), the best estimates of the 25km and 2km observed atmospheric state available from meteoro- logical agencies, centered around region containing Tai- wan. Additionally, we synthesize dynamics from a multi- resolution variant of 2D fluid-flow, where we can control the degree of misalignment. Our results show that AFM consistently outperforms existing methods across various skill metrics. Overall, main contributions are summa- rized as:
real datasets. For the real data,
Diffusion Bridges and Stochastic Interpolants. Diffu- sion bridges (De Bortoli et al.) 2024; Shi et al., 2023
Liu et al., 2023 Pooladian et al. 2023 facilitate transi-
tions between distributions but rely strong assumptions, such local alignment in I (Liu et al. 2023).
suitable for misaligned data. Stochastic interpolants (Al-
bergo et al. 2023: Albergo & Vanden-Eijnden 2023
man et al. 2022 Liu et al. 2022) assume smooth trans- port and are often trained with independent coupling be- tween noise and data, even for conditional problems. In
Adaptive Flow Matching (AFM): A method for matching spatially misaligned data (plus misaligned channels) with multiscale physics.
partieular, (Albergo et al. 2023) couples base and target distributions. Chen et al. (2024 applies stochastic inter- polants for fluid-flow forecast. But, our AFM suits sce- narios with misaligned input-output variables, where the encoder learns large-scale deterministic dynamics, while adaptively balancing the contribution of deterministic and stochastic components.
Adaptive Noise Scaling: Derived based MI prin- ciple, which optimally balances the learning of deter- ministic and stochastic components between the en- coder and Flow Matching.
Experiments on Real and Synthetic Data: Our re- sults show AFM outperforms existing alternatives. As
Co-training generative models with encoders. Co-
2

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Adaptive Flow Matching for Resolving Small-Scale Physies
Table I: Comparison between AFM and alternatives for learning the generative map between misaligned data (y. x). Define e (0,, 1). := (E(y) -x//s,, and r := x E|x The noise scale ds is the ML noise estimate ensuring Bi|le| 1. CDM and
CPM represent conditional diffusion and flow models, respectively.
Scheme
Perturbation Kernel
Score
Train Loss (Denoising)
CFM
(1 1)e + ix
Vx xx log pa (x.lx)
El | Da(xi))) () x]]²]
CDM
X, = x + ofe
Vx log pa (x.lx)
E [Da 0r) - x
CorrDiff'
= r oze
Vr, log pa (ralrr)
B[lell] - B [[IiDa - -
AFM (Ours)
Xx.Pe(xe)
[(( ( x11 + xl|ell
training encoders with diffusion models has been explored in various domains. DifíCast (Yu et al., 2024) uses an en- coder to predict the mean precipitation for nowcasting with diffusion model that handles the residuals. Similarly, Grad-TTS (Popov et al.] 2021) and Bridge-TTS (Chen et al. 2022) integrate encoders with diffusion-based pro- cesses for text-to-speech synthesis. These methods focus on temporal generation and operate on single channel. In contrast, our AFM framework tackles superresolution and channel synthesis with spatial misalignment for mulitvari- ate data with different stochastic charactristics.
challenges learning the joint distribution.
Before delving into details, is useful to review Flow Matching as a critical component of our approach.
Flow Matching. It transports samples from source dis- tribution po (x) target distribution pi (x) by learning velocity field. Flows are often trained using linear inter- polant between noise and data. The forward process simply follows the ODE:
dx = v(x),))
dt
Atmospheric super-resolution (downscaling). In atmo- spheric sciences, recovery of the fine-resolution from the coarse resolution is known as downscaling. Several works have explored statistical downscaling; see, e.g., (Rampal 1998; Rampal et al. 2022). For instance, instance,CornDif (Mardani et al. 2023) learns a diffusion model on residuals left from a deterministic prediction. It a two-stage approach where the overfitting of the regression error propagates to the diffusion in second stage, leading to reduced generation quality with severely under-calibrated ensembles. In AFM this is mitigated using the balance be- tween the deterministic and stochastic errors and the adap- tive noise scaling per channel.
where v(x;; () represents the velocity field over time [0, 1]. For the linear interpolant, the true velocity that gen- erates single data sample X is given by V'me (X,, t) Xo. The goal is to minimize the discrepancy between the learned velocity field vo(x,, t) and the true velocity as:
let al. 2024 Wilby et al.
[1 t) - (x1 - -xx) x))]
min Ex
(2)
where X2 := (1 - t)xo + -x1- Upon convergence, the learned velocity is used generate samples xo po(x)
by solving the ODE in Eq.
It is also useful to recognize the connection between Flow Matching and denoising diffusion models. This is useful as one can utilize the effective and scalable (denoising) training of diffusion models such as the celebrated eluci- dated diffusion models (EDM) (Karras et al., 2022) to sta- bly learn velocity fields. we deffer the discussion and con- nections to the Appendix (see section C).
3. Background and Problem Statement
Consider the task of learning the conditional distribution p(xly) from finite set of paired data {(y,, x)) > where y E gexhxx and x E Xx represent the input and target, respectively. For example, in atmospheric sciences, is a coarse-resolution forecast from the Global Forecast System (GFS) at 25 km resolution, while x is the fine- resolution target from a high resolution regional weather simulation system at km.
4. Adaptive Flow Matching
To learn the conditional distribution p(xly), a natural approach is to use conditional diffusion/flow models. They have succeeded in image-to-image tasks e.g.. image restoration or super-resolution, where conditioning pro- vides rich information about the target (Saharia et al., 2022). However, traditional methods struggle when the in- put y and target are significantly misaligned (see Sec- tion 5 for evidence). To address this, we propose a multi- scale approach:
This task is particularly challenging because the pairs (y.x) are often misaligned. First, these pairs might rep- resent solutions to partial differential equations (PDEs) at significantly different spatial and temporal discretizations. This can lead to completely different temporal or spatial trajectories, including due to effects of internal chaotic dy- namics. Second, the input and target may involve sev- eral different channels, e.g., corresponding distinct at- mospheric variables with intrinsic correlations, that further
The input y is encoded into a latent variable z = E This encoding serves two pur- poses: (i) it first matches the large-scale, mainly determin-
Deterministic Dynamics:
3

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Adaptive Flow Matching for Resolving Small-Scale Physies
Algorithm 1 AFM training
Algorithm 2 AFM sampling
1: Input: X, ((y;,)))) M
1: Input: y, At, σ2 Dp. E
2: Initialize dz, 0,
Sample noise E ~ (.
3: repeat
Form latent z = E O&E
4: Sample o, ~ =4[0, oz] and ~ I0.
4: Initialize X, 2
5: Compute error: e == (E x)/o
5: for / = 0: Al do
6: Perturb input: x; = x + oe(c + c)
6: 0r = (1 t)Ø
7: Take a gradient step on:
7: va(x,, = (Do(x,, 02) 100xxx,,11 xx)\(11 -t)
8: X(+A: = X, + vo( t). At
8: Vo.
(Do(x,, σz) - x|| + x ||el|
end for
10: return X1
9: Every M steps: u v R2 Xval E(y 12
10:
0= + B)0= + _0
(1 (1- 1) the residual ervor by e: (E(y) -x// and the noise e ~ N(, 1), the Flow Matching for joint training of the encoder and flow reduces to the denoising objective:
11: until convergence
istic dynamics of the input and output, aligning the spa- tially misaligned large-structures due to diverging trajecto- ries, and (ii) it aligns the channels by projecting the input into the same space as the output (since y and x represent different weather variables).
min -x,,0,- 8.0
(x((x,,0) [I
-x1²] (6)
Intuitively, this denoising objective addresses not only the Gaussian noise typical of diffusion models but also residual errors introduced during the deterministic encoding pro- cess. Note that the residual error e conveys the essential in- formation about the input conditioning y required for gen- erating the target x. Therefore, it is crucial carefully balance the influence of this error by appropriately select- ing σ₂ and applying regularization to the encoding process. These considerations will be discussed next.
Flow matching is then used to transform the base distribution p(z) into the target distri- bution p(x). To account for uncertainty in the encoding phase, we perturb the latent variable with Gaussian noise:
Generative Dynamics:
z = Ely) + o.c, 6 6~ I0,
(3)
In the following sections, we detail the learning process for both the encoder and Flow Matching.
4.2. Adaptive Noise Scaling
4.1. Training
It is essential to tune the noise parameter dg based on the data before applying diffusion denoising. Specifically, we consider the latent variable:
The objective is to jointly learn the encoder and the Flow Matching model. To achieve this, we delve into Flow Matching in the latent space. Specifically, it establishes a linear interpolant defined as x, = (1 - t)z + tx, where z is the encoded state and x is the target, for t E [0. Consequently, based on Equation (2). the Flow Matching objective is formulated as:
E(ly) -0,6
where we observe x and y, and aim to adjust o= in maxi- mum likelihood (ML) sense so that z aligns closely with x. In this context, 0₂ controls the scale of the noise added to the encoder's 's output & E(y). By leveraging the ML estima- tor for a., we can derive it as the root-mean- -a---m---uuuuu-rrrrn (RMSE) of the unnormalized residual error, namely
min E, _.x.x [1)o(xx,)) (x -=)1 8,0
(4)
Due to the stochasticity in z and the stability of the EDM framework for training diffusion models-along with its funin--free characteristics- is advantageous to incorpo- rate denoising through EDM when training Flow Matching. To this end, the first step is to recast the linear interpolant as Gaussian diffusion process (cf. Equation 33)
a2 = -E[|x=E(y>]13)
(8)
In practice, to prevent overfitting, a= progressively updated using the encoder's RMSE on a validation set. Intuitivil the deterministic regression model over- fits a the training dataset, the validation RMSE will grow, and thus our AFM scales up the noise in the base distribution for the Flow Matching, During training we up- date o₂ using an exponential moving average (EMA) ev- ery M = 10k training steps to avoid sudden changes in stochasticity:
x = (1 - ()) + 1x + (1 - 1)a.e,
LE 10. (5)
The following proposition simplifies the task of learning the velocity field as a denoising process.
Proposition 1. For the perturbation model x; = = x OCE, where the noise standard deviation given by at ==
02 f- (1 - 1-3); + cur
4

---

Adaptive Flow Matching for Resolving Small-Scale Physies
An illustration of the evolving values of og during training is provided in Figure(5) 5 Once training completes we fix the final value of og for inference.
Table 2: AFM vs. baselines for CWA downscaling: Values in bold show the best performance, AFM outperforms baselines except for the deterministic temperature variable, where CorrDiff' excels. Temperature, being the most deterministic, benefits from CorrDiff's fully deterministic predictions. While AFM could match ConDiiff using a UNet encoder and higher A, this would
Alternative Stochastic Encoders. One can alternatively model uncertainty using VAE. It essentially imposes KL. regularization on the encoder output to predict μ and o, where o can drive the noise for FM. While valid, we opted for a simpler ML method that offers a closed-form and in- tuitive solution for a (see (Rybkin et al. 2021)). Although the VAE allows learning of a from the data, it requires tun- ing the KL regularization and suffers from the prior hole problem.
compromise stochastic predictions for other variables.
Variable
Metric
CorrDiff
CFM
CDM
UNet
AFM
RMSE
5.08
5.06
5.70
5.09
4.90
CRPS 1
1.89
1.88
2.39
1.78
Radar
MAE 4
2.50
2.46
2.78
2.50
2.42
SSR 1
0.38
0.33
0.46
0.44
RMSE
0.83
0.93
0.95
0.86
1.00
CRPS 1
0.50
0.58
0.54
0.52
Temp.
Deterministic to Stochastic Regime. If the data is purely deterministic, encoder perfectly matches the data and thus O2 - or 0. This means there is no work left for the generative FM. Alternatively, if data is purely stochastic, encoder returns large error (02, oe )) which leads to base noise distribution with large variance, that triggers FM.
MAE
0.60
0.72
0.70
0.62
0.67
SSR
0.36
0.41
0.52
0.47
RMSE
1.47
1.45
1.62
1.49
1.44
CRPS 1
0.85
0.82
0.93
0,80
E. Wind
MAE
1.07
1.06
1.24
1.09
1.07
SSR 1
0.43
0.50
0.61
0.61
RMSE
1.66
1.61
1.84
1.66
1.61
4.3. Encoder Regularization
N. Wind
CRPS 1
0.95
0.90
1.06
0.88
MAE +
1.20
1.16
1.41
1.21
1.17
Another effective approach to control the residual error is through regularization. Specifically, one can impose re- construction regularization on the encoder, encouraging the encoder output to approximate the target, i.e., x ~ E(y), thus minimizing the residual error e. In an ideal scenario, this would lead to e se 0. However, enforcing perfect matching can adversely affect the generalization capability of the model, leading to overfitting. To mitigate this, a soft regularization is applied, controlled by a penalty weight X. This weight balances the trade-off' between reducing the residual error and maintaining the generalization ability of the model. The resulting objective function becomes:
SSR
0.41
0.49
0.58
0.58
dataset, designed to capture variable degrees of misalign- ment. Both datasets present significant downscaling chal- lenges with local and large-scale misalignment, mixed- scale dynamics, and channel-specific variability.
Baselines. AFM is compared against several baseline de- terministic and stochastic methods:
Regression: A standard UNet is trained to predict high- res. output from low-res. input using MSE training. This serves as a purely deterministic approach, representing a baseline for direct super-resolution.
(e=)
min Ex (x 8,0
|Do(xe, σt) - x1|² + X ||el
Conditional Diffusion Model (CDM): CDM maps Gaussian noise to the high-res. data while conditioned on the low-res. input. CDM does not explicitly model the deterministic component in the data.
4.4. Sampling
Once the velocity field ( t) = (x), (1) -x))-1)
is learned using Algorithm [ sampling requires integrat- ing the flow forward in time based on the ODE in Equa- tion (T). The forward integration process from = to
Conditional Flow Matching (CFM): A variant of Flow Matching that interpolates between a Gaussian sample and a data sample, building deterministic mappings.
t is expressed as:
Corrective Diffusion Models (CorrDiff) (Mardani et al., 2023): A supervised regression is first trained to learn the mean E[xly]. The residual e == x -[xly] is then used to train the diffusion model. Early stopping used to mitigate the overfitting for the regression stage.
X1
( t) dt
(10)
In practice, this integration can be approximated using Eu- ler steps, which are detailed in Algorithm
Evaluation Metrics. We report performance using stan- dard metrics such as RMSE, Continuous Ranked Proba- bility Score (CRPS), Mean Absolute Error (MAE), and Spread Skill Ratio (SSR). These metrics provide compre- hensive assessment of both the accuracy and uncertainty quantification of the model's predictions. RMSE, CRPS, and MAE measure the estimation error while SSR evalu- ates the model calibration. To calculate CRPS and SSR we
5. Experiments
We evaluate performance of the proposed Adaptive Flow Matching (AFM) model on two datasets: i) A regional weather downscaling dataset with real-world meteorolog- ical observations from Taiwan's 's Central Weather Adminis- tration (CWA) ii) a synthetic multiscale Kolmogorov flow
5

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Adaptive Flow Matching for Resolving Small-Scale Physies
AFM
Irput
UNN
CHH
CDM
CorDiif
A
Ground Trush
Max Kudar helictee
Ricker
Ladbeadd Wer 30m
Nartheard Wiree 30m
32w
Figure 2: AFM vs. baselines for different weather variables. AFM generates more physically consistent outputs, while UNet (regression) output appears blurre,, and CDM struggles to accurately reconstruct radar reflectivity, Note that radar reflectivity is not present in the input data and is entirely generated as a new channel.
10
Power
Gerund
Trsh Uite
CBM
16"
Condel
CFM
10"
10
10²
10²
Frequency
Frequency
Frequency
Frequency
(a) Radar Reflectivity
(b) Temperature
(c) Eastward Wind
(d) Northward Wind
Figure 3: AFM power spectra vs. baselines for CWA downscaling, AFM exhibits superior spectral fidelity, closely aligning with the ground truth across all variables, with a particularly strong fidelity for the purely generated radar reflectivity. It consistently outperforms CorrDiff. especially in capturing high- tigh-frequency details across all variables.
produce 64 ensemble members using different seeds. These metrics are discussed in Appendix F
Weather Administration (CWA) 2021b. For evaluation, we use a common set of 205 randomly selected out-of-sample date and time combinations from 2021. Metrics and spec- are computed compare AFM with baseline models. We utilize 32-membee ensemble; larger ensembles do not significantly alter the key findings. A detailed data descrip- tion is provided first, followed by our observations.
Network Architecture, Training, and Sampling. For dif- fusion model training and sampling, we adopted EDM (Karras et al.) 2022). continuous-time diffusion model nspired by SDEs with effective auto-tuning (see Table in (Karras et al. 2022)). We adopt most of the hyperpa- rameters from EDM and make modifications as detailed in Appendix D Note that EDM is used for CDM/CFM as well in consistent manner.
Dataset. The dataset is derived from ERA5 reanalysis data (Hersbach et al.] 2020), focusing on 12 variables includ- ing temperature, wind components, and geopotential height at two pressure levels, well surface-level variables e.g., 2-meter temperature and total column water vapor. The target output data (Central Weather Administration (CWA) 202I) covers a 900x900km region around Taiwan on a 448 448 grid. Hourly observations span four years 2018-20211) split into training (2018-2020) and evalua- tion (2021) sets. Input (ERA5) data is upsampled using bi-linear interpolation to match the target domain dimen-
5.1. Regional Downscaling for Taiwan
We focus on the task of super-resolving (downscaling) mul- tiple weather variables for the Taiwan region, a challenging meteorological regime. The input cossstireoolution data at 25 km scale comes from ERA5 (Hersbach al.) 2020). while the target fine-resolution 2 km scale data is sourced from the Central Weather Administration (CWA) (Central
6

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Adaptive Flow Matching for Resolving Small-Scale Physies
Table 3: AFM vs. baselines for Kolmogorov Flow for var- ious misalignment degrees AFM consistently demonstrates superior performance across varying levels of data misalignment, showcasing its robustness. While CDM exhibits greater variabil- ity, this comes at the expense of significantly reduced fidelity. Note that for deterministic models, CRPS is equivalent to MAE.
Table 4: Effect of adaptive og. Experiments were conducted on the CWA 112 x 112 datasct with adaptive o; and À 0.0. For both encoders adaptive os significantly enhances performance across all variables and metrics.
Encoder
1x 1 Conv. /
UNet
Adapt. 02
X
X
Metrie
CFM
CDM
UNet
AFM
RMSE 4
5.01
4.82
5.06
4.95
RMSE +
0.98
1.13
1.15
0.73
Radar
CRPS
1.88
1.66
1.83
1.78
CRPS |
0.52
0.58
0.37
SSR + 1
0.45
0.72
0.38
0.44
MAB
0.69
0.80
0.82
0.51
SSR 1
0.54
0.69
0.62
RMSE $
1.13
0.99
1.01
0.94
Temp.
CRPS
0.66
0.59
0.57
0.54
RMSE 4
0.96
0.94
1.14
0.76
SSR + 1
0.39
0.47
0.36
0.40
CRPS 4
0.52
0.48
0.40
5
MAE
0.69
0.67
0.82
0.54
RMSE +
1.50
1.42
1.48
1.49
SSR + 1
0.58
0.70
0.58
E. Wind
CRPS
0.86
0.78
0.85
0.85
SSR
0.48
0.68
0.46
0.53
RMSE 4
1.22
1.24
1.36
1.09
CRPS 1
0.67
0.65
0.65
RMSE +
1.64
1.58
1.64
1.58
MAEL 4
0.89
0.89
1.00
0.77
N. Wind
CRPS
0,94
0.89
0.93
0.95
SSR
1
0.56
0.76
0.23
SSR ->
0.47
0.66
0.46
0.50
5.2. Multiscale Kolmogorov-Flow
Our Multiscale Kolmogorov Flow dataset offers sim- plified simulation of atmospheric dynamics, focusing on downscaling from a coarse to a fine grid while preserv- ing physical structures. Kolmogorov flow (KF) is a well- known scenario where 2D fluid flow in doubly periodic domain is forced by spatially varying source of momen- tum. To mimic the structure of the down-scaling problem we couple the KF flow ground truth to an otherwise un- forced fluid system representing a oparse-resolutinn atmo- spheric simulation. The strength of this coupling T controls how well the coarse simulation tracks the ground truth.
sions (CWA) (Hu et al. 2019; Zhang et al. 2018 See Appendix G.I for the detailed discussion of the datasets.
Observations. The performance of AFM is compared with various alternatives, and both deterministic and probabilis- tic skills are reported in Table We examine three variants of AFM: with (i) small versus large encoders; (ii) with additional use of adaptive noise scaling; and (ii) condi- tioning in the large encoder limit. Notably, temperature is the most deterministic variable among the four listed, while radar is the most stochastic one.
Dataset. The dataset is constructed by simulating dynam- ies governed by system of partial differential equations involving vorticity fields Çi and Çr. coupled through pa- rameters like 7 and influenced by steady-state forcing. The simulation uses a pseudo-spectral method on a 512 x 512 grid with a 3rd-order ferent 7 values (3, 5, 10) are used to generate training and test sets where higher values of T looser coupling which translates to higher misalignment between low and high res simulations. Detailed descriptions of the equations and pa- rameters are provided in Appendix G.2
Our main finding is that AFM consistently outperforms existing alternatives across different metrics for non- deterministic channels (radar and winds). For temperature, although AFM is not the top performer, one can tune X (larger value) in AFM to be as good as CorrDiff. This however compromises the stochastic prediction for non- deterministic channels; see the ablation in the Appendix section Table [2] Ablations are deferred to Appendix [ due to space limitations.
Bashff-th time stepper. Dif-
Spectral analysis is crucial for assessing fidelity at differ- ent scales in weather prediction. Fig. 3 shows that AFM's spectra closely match the ground truth across variables. In contrast, the UNet-based regression scheme fails to gener- ate high-frequency components. Interestingly, the condi- tional diffusion model (CDM), commonly used for image super-resolute resolutio,,,,,, also lacks spectral fidelity. Regarding the calibration of the generated ensemble i.e., SSR, AFM pro- vides the best balance, being closest to 1.0. While AFM does not acheive perfect calibration, it does improve the balance berween the spread and RMSE skill of the gener- ated ensemble, especially for surface wind channels.
Observations. AFM consistently outperforms other meth- ods across various skill metrics for different degrees of mis- alignment, denoted by т (Table 3. Interestingly, while CDM appears to be the most calibrated, AFM demonstrates superior performance overall. In this case study, we use a x 1 convolutional architecture for the AFM encoder. The ablations are deffered to Appendix due to limited space. This ad advattgee is further supported by the spectral analysis in Fig. [3 where AFM's spectra most closely align with the ground truth, highlighting its robustness in preserving physical structures even under significant misalignment.
7

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Adaptive Flow Matching for Resolving Small-Scale Physies
Input
Ground Truth
UNet
CFM
CDM
AFM
10
10
Figure 4: AFM vs. baselines for Kolmogorov Flow and т 10:
when the figures are zoomed in, it is apparent that AFM aligns closer to the ground truth, and the presence of high figuefryy artifacts in the baseline models becomes more noticeable.
Role of regularization X. We also investigated how the AFM model's performance depends on the À parameter, which balances encoder error and variability, For the sim- pler 1 x 1 Conv. encoder (with only 60 parameters), the model achieves its best accuracy when À 0, indicating this encoder is lightweight enough and regularization is need for fitting the data. In contrast, the more com- plex UNet encoder benefits from moderate regularization (A = 0.25), suggesting that overfiuting is more likely with larger-capacity models and can be alleviated by incorporat- ing some regularization. Overall,
2.0
Radar Reflectivity
Temperature
1.5
E Wind
N Wind
6 1.0
0.5
400k
Training Steps
Figure 5: Adaptive os values over training steps differ- ent channels. The plot corresponds to AFM with 1 x Conv. encoder and À = 0.25. a increases early training due to high en- coder and subsequently converges the encoder improves. Stochastic channels e.g., radar reflectivity exhibits the highest a. indicating its stochastic nature, while temperature shows the low- est, reflecting its deterministic nature.
choice X should be informed by the complexity of the encoder: simpler net- works require weaker to regularization, while more pressive architectures can benefit from modest regulariza- tion term. Further ablations and experiments can be found in the Appendix; see Appendices I and K
6. Conclusions
5.3. Ablations
Role of noise scaling (ax). To investigate the effect of adaptive noise scaling, we tracked the per-channel noise scale during training (Figure 5) for 1 x 1 Conv. encoder with X = 0.25 on the CWA dataset. Initially set to 1, the noise scales grow in the early stages due to high encoder er- ror, then stabilize and converge as training progresses and the encoder improves. Notably, radar reflectivity exhibits the highest σ= values throughout the training process, flecting its stochastic nature. In contrast, the temperature channel consistently shows the lowest noise values, align- ing with its more deterministic nature. These variations in noise across channels highlight the necessity of an adap- tive noise scaling approach, particularly when compared to datasets (e.g., natural images) that exhibit more uniform noise statistics. To further validate this, we ablated the use of adaptive ... As shown in Table incorporating adaptive noise scaling significantly improves performance for both the 1 x Conv and UNet encoders. This adaptability is cru- cial for handling misaligned data with varying degrees of stochasticity, ultimately enhancing the AFM model's 's over- all performance and reliability multiscale physics.
We introduced Adaptive Flow Matching (AFM) to address misaligned data distributions in scientific super-resolution tasks (e.g., downscaling). AFM combines deterministic coding of large-scale dynamics with Flow Matching, ef- fectively capturing both deterministic and stochastic com- ponents the data. Experiments on synthetic and real- world datasets demonstrated that AFM outperforms exist- ing methods, especially when input and target distributions are significantly misaligned. Future avenues to explore in- clude: (i) extending AFM to unpaired misaligned datasets and to other domains (e.g. MRI subsampling in medical imaging, natural images), (ii) theoretical analysis of its convergence, especially due to adaptive noise scaling, and (iii) leverage alternative stochastic encoders such as VAEs.
Impact Statement
This work advances machine learning for scientific data, introducing generative framework capable of resolving fine-scale physical phenomena from coarse, misaligned in- puts. A primary application is in atmospheric downscal-
8

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Adaptive Flow Matching for Resolving Small-Scale Physies
Table 5: Effect of encoder regularization A. Experiments were conducted the CWA 112 x 112 dataset with adaptive Reg- ularization with 0.25 works best for UNet.
Chen, Y., Goldstein, M., Hua, M., Albergo, M. S., Boffi,
N. M., and Vanden-Eijnden, E. Probabilistic forecasting with stochastic interpolants and follmer processes. arXiv [cs. [c), March 2024.
Variable
Metric
x Cony.
UNet 0.25
0.00
0.25
1.00
0.00
1.00
Chen, Z., He, G., Zheng, K., et al. Bridge-tts: Text-to- speech synthesis with schrödinger bridge. arXiv preprint arXiv:2203.cm 0.661,, 2022.
RMSE
4.82
5.10
5.04
5.06
4.95
5.11
Radar
CRPS |
1.66
1,83
1.83
188
1.78
1.89
SSR
0.72
0.42
0.40
0.38
0.44
0.41
RMSE
0.99
0.85
1.05
1.01
0.94
1.07
Temp-
CRPS J.
0.59
0.50
0.60
0.57
0.54
0.62
De Bortoli, V., Malkin, N., Tong, A., and Harchaoui, Z. Schrödinger bridge flow for unpaired data translation. arXiv preprint (exxxx:249.903493 2024.
SSR +
0.47
0.43
0.34
0.36
0.40
0.36
RMSE
1.42
1.46
1.50
1.48
1.49
1.53
E. Wind
CRPS |
0.78
0.84
0.88
0.85
0.85
0,91
SSR +
0.68
0.46
0.44
0.46
0.53
0.41
Dosovitskiy, A. An image is worth 16x16 words: Trans-
RMSE
1.58
1.63
1.67
1.64
1.58
1.74
formers for image recognition at scale. arXiv preprint arXiv:2010. 1929, 2020.
N. Wind
CRPS 1.
0.89
0.92
0.95
0.93
0.95
1.03
SSR 0.66
0.46
0.44
0.46
0.58
0.40
Gneiting, T. and Raftery, A. E. Calibrated probabilistic
ing, where improved resolution and uncertainty quantifica- tion can enhance early detection of extreme weather events, contributing to disaster preparedness and public safety. We are not aware of direct negative societal impacts of this re- search.
forecasting using ensemble model output statistics and minimum erps estimation. Monthly Weather Review, 132 (10):2921-2930, 2004.
Hersbach, H., Bell, B., Berrisford, P., Hirahara, S., Horányi, A., Muñoz-Sabater, J., Nicolas, J., Peubey, C., Radu, R., Schepers, D., et al. The era5 global reanalysis. Quarterly Journal of the Royal Meteorological Society, 146(730):1999-2449 2020.
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Adaptive Flow Matching for Resolving Small-Scale Physies
A. AFM versus CorrDiff
Here we illustrate experimentally how the two-stage workflow of CofrDiff relates to decreased diversity and under- calibration. We focus on the training dynamics of the UNet regression model, by training one and keeping various check- points (0.5M, 2M and 50M steps) along training. Then for each checkpoint we learn diffusson model predict the residual. However, this approach is prone overfitting in the first stage, limiting the overall performance. As shown in Table when the regression model is trained for too many steps (e.g., 50M), the variability is significantly reduced. This reduction in SSR suggests that the regression component becomes overly confident, producing residuals that are narrowly concen- trated around zero. Consequently the diffusion model only need to model limited variability, which diminishes its ability to generate diverse corrections at test-time. By contrast, an undertrained (0.5M steps) UNet leaves more stochasticity to be modeled, which leads better-calibrated ensembles.
To further elucidate the issue we plot the training and validation losses of the encoder during training in Figure 6 As we can observe, the UNet regressor show a generalization gap that widens after approximately 500k training steps. This is evident from the divergence between the training and validation MSE losses across all channels. We also note that the radar and temperature channels have a wider generalization gap.
These observations highlight a critical limitation of CorrDiff's two-stage methodology: the initial regression stage is prone to overfitting. which in turn constrains the diffusion model's capacity to generalize effectively. To address this issue, the authors of CofrDiff use early-stopping to chose a less overfit UNet.
Our proposed AFM method addresses this limitation by adopting end-to-end training setup. AFM simultancously bal- ances deterministic and stochastic objectives, preserving sufficient variability in the signal, reducing the risk of overfitting while at the same time produces more accurate results. An important part of our method is that it takes into account the different stochasticity levels between channels (see Figure 9 using the adaptive noise scaling, and overall mitigates the overfitting issue in a more principled and effective way.
To gather the above results for CofDDiff we use the same UNet from the CorrDiff paper which corresponds to the 89M- parameter model (XI. in T). This makes the RMSE and CRPS slightly different the rest of our experiments where used 22M-araaeter UNet (L).
Table 6: Effect of two-stage training on ensemble diversity. This table shows how the number of training steps for the UNet regressor in the first stage influences the final performance of the entire CorrDiff pipeline. In the second stage, different diffusion models are trained using UNet checkpoints obtained after 0.5M, 2M, and 50M steps. Results indicate that using less-trained UNet models yields better calibration and higher stochastic variability (SSR), while heavily-trained UNet models reduce ensemble diversity. This suggests that once UNet becomes specialized biased, the second- stage diffusion model struggles to correct it and maintain variability in its outputs.
U-Net Training steps
Variable
Metric
0.5M
2M
50M
RMSE J
5.08
5,28
5.13
Radar
CRPS +
1.81
1.89
2.10
SSR
0.52
0.35
0.14
RMSE +
0.96
0.93
0.91
Temp.
CRPS L
0.57
0.58
0.53
SSR
0.41
0.28
0.31
RMSE 4
1.51
1.53
1.48
E. Wind
CRPS |
0.83
0.89
0.87
SSR +
0.60
0.42
0.39
RMSE 4
1.72
1.70
1.65
N. Wind
CRPS 1
0.97
1.01
0.99
SSR
0.56
0.39
0.37

---

Adaptive Flow Matching for Resolving Small-Scale Physics
10a
MSE
10
valid-radar
valid-t2m
valid-u10
valid-v10
train-radar
train-t2m
train-u10
train-v10
10
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Training Steps (M)
Figure 6: UNet train and validation loss during training CWA 448 x 448 data. Evolution of training and validation MSE for the UNet regressor across training steps. We observe that the UNet starts overfitting early as 500k steps. Furthermore, certain variables like radar reflectivity(blue) and lemperature(yellow) show higher amounts of overfitting, This behavior is the reason that two-stage approaches like CorrDiff utilize early stopping to avoid overfitting the training data, because this very difficult to correct on the second stage. AFM resolves this issue by leveraging end-to-end training determinsticn loss weghting as well adaptive noise scaling.
B. Proof of Proposition
Proposition 1. For the perturbation model x o,e + or6, where the noise standard deviation is given by o. := (1 - t) the residual error by e = (Ely) - x and the noise e ~ NO, 1), the Flow Matching for joint training of the encoder and flow reduces to the denoising objective:
min E [x.yorr-((0.oo (0z/ (D (0s5/a))]\/xxxxxx x|[]]
(11)
Proof. Consider the linear interpolant connecting z ~ N(E(y)) III) to the target distribution p(x). This process can be expressed as:
X4 (1 ()(E()) o.c) ttx
(12)
(1 t)E(y) + + (1 t)o.c (1 ()(E(y) - x) +x (1 t)a
(13)
(14)
x + (1 (1-t),, te,
(15)
where e (E(y) x)/
Define 0, (1-t)o; and reinterpret time t in terms of the noise level or, as is typical in continuous noise sampling (c.g.. EDM). The perturbation kernel then becomes = x + oe + c).
Now, consider the training objective in 4
min ((x., 8,0
[xe, (x 2)]1
where the velocity field expressed as a denoiser (or equivalently score form (Vincent 2011))
v t) DD X 1-t
(16)
which is the reverse-time version of Equation 20p. since in Flow Matching, time progresses from = 0 to = 1.
Substituting v t) into the Flow Matching objective ( and simplifying using x; = (1 - t)z + tx and 0./0, = 1 -t
we obtain the denoising objective in [T). completing the proof.
12

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Adaptive Flow Matching for Resolving Small-Scale Physies
C. Flow Matching and Diffusion Models
C.1. Diffusion Models (DM)
Diffusion models generate data by transforming a base distribution, often Gaussian noise, into the target data distribution Po(x). the forward diffusion process, Gaussian noise with standard deviation is added the data, producing a sequence of distributions Po (x; r). As increases, the data distribution approaches pure noise. The backward process then denoises samples, starting from noise drawn from (O, o distribution.
max I) and progressively reducing the noise recover the data
Considering the variance-exploding elucidated diffusion model (EDM), both the forward and backward processes are deseribed by SDEs. The forward SDE is: dx:- = 22 dwr.
(17)
while the backward SDE is:
dx; -2 Xx, log p(x o4) dt + v 20ro dur..
(18)
In EDM, denoising score matching is used to learn the score function Vx p(x; o), essential for the reverse diffusion process. A denoising neural network D,(x; a) is trained as:
min p...p....r [I [I + x11]]
(19)
and the score function is constructed via Vx, logp(x;; = (Do(xi: - x)/o²²
For completeness, we note that the ODE version of this backward process (used in the main text as Eq. 20) directly links EDM to Flow Matching-basse based approaches.
C.2. Connections Between Flow Matching and EDM
While both EDM and Flow Matching methods aim transform distributions, Flow Matching does so using ODEs for deterministic evolution, whereas elucidated diffusion models (EDM) leverage SDEs for stochastic denoising. An ODE formulation for diffusion models bridges the two methods, allowing Flow Matching to benefit from formulations and network parameterizations introduced for diffusion models (Karras et al. 2022 Song et a 2021). For the simple noise schedule d= = vf, the ODE for continuous-time EDM is:
dx,
X - Do((,, ot)
(20)
dt
where the right-hand side acts as the velocity field, linking diffusion dynamics to Flow Matching. Note that the diffusion process runs backward in time from ( = to / 0, whereas the Flow Matching process proceeds in the opposite direction.
D. Connection of AFM to Residual Learning
Consider the case where we have pre-trained encoder E, which has been trained using supervised regression loss, for instance. In this scenario, starting from the forward diffusion process in Equation (5) if we subtract both sides by the encoder output Ey) and define the residual error as et == X2 - -(y), we arrive at a form that closely resembles a standard Flow Matching forward process with Gaussian noise as the base distribution. This is expressed as:
e, te (1 t)e,
t E [0, 1].
(21)
where e represents the residual error between the target x and the encoder output Ely), and e ~ NN(0, 1) is the noise.
This simple process facilitates the construction of backward process, where one can learn the velocity field by minimizing the Flow Matching loss in Equation ( This approach closely mirrors the CorrDiff method proposed in (Mardani et al., 2023). which leverages residual learning train diffusion models. CorrDiff has demonstrated considerable success in capturing small-scale details in generative tasks, particularly where precise reconstruction of fine structures is required.
However, as discussed in Section the initial supervised training of the encoder often leads to near-perfect matching between Ey) and x. While this may seem desirable, it can result in overfitting and reduced model calibaration, as also evidenced by the results in Table discussed in Appendix A Therefore, balancing this residual learning approach with appropriate regularization is critical to maintaining the model's 's ability to generalize effectively to unseen data.
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Adaptive Flow Matching for Resolving Small-Scale Physies
Table 7: Details of different encoder sizes used in the experiments. For the UNet the channel multipliers are applied to the base channel size of 32 at each layer.
Encoder
Channel Multipliers
Number of Parameters
XL (only used in Appendix A)
[1, 4, 8] [1,2, 2, 4,
89M
L
12M
M
[1,2,
5M
S
[,,
IM
XS
1,,1,
0.2M
1 x 1 Conv.
60
E. Network Architecture and Training
For diffusion model training and sampling, we use EDM (Karras et al. 2022). a continuous- diffusion model available with a public codebase. EDM provides a insspiee design based on ODEs, auto-tuned for our scenario (see Table in (Karras al. 2022)). We adopt most of the hyperparameters from EDM and make modifications as listed below.
Architecture of Diffusion Model: To cover the large field-of-view 448 x 448, we adapt the UNet from (Song & Ermon
2019) by expanding it to 5 encoder and 5 decoder layers. The base channel size is 32, multiplied by [1, 2, 2. 4, 4] across layers. Please note this is a scaled down version of the model used in (Mardani et al. 2023) due to computational constrains. Attention resolution set to 28. Time representation is handled via positional embedding, though this is disabled in the regression network, as no probability flow ODE is involved. No data augmentation is applied. The UNet has 12 million parameters, and we add 4 channels for sinusoidal positional embedding to improve spatial consistency, following practices in (Dosovitskiy 2020
Carion et al. 2020).
Architecture of Encoder Model For the encoder E, we use two architectures: (1) simple I x convolution layer, and (2) UNet similar to the diffusion UNet but without time embedding. For the UNet encoder we ablate various parameter counts from 0.2M to 12M, architectural details these can be found in Table and the results the albations in Appendix []] For the experiment where only the UNet was used for regression, the model has 12M parameters (L). For CorDDiff then same 12M model was used for the regression network and the diffusion model. For the results in CorDDiff under-calibration (Appendix A) we used the same model as in (Mardani al. 2023) with 89M parameters (XL).
Optimizer: We use the Adam optimizer with learning rate of 10 4, 0.9, B2 0.99. Dropout is applied with rate of 0.13. Hyperparameters follow the guidelines in EDM Karras et al. 2022).
Noise Schedule: For AFM and CFM, we use a continuous noise schedule sampled uniformly a E E az). For CDM and ConDifff we use EDM's optimized log-normal noise schedule, a ~ lognormal( 112, 1.2).
Training: The regression network receives 12 input channels from the ERA5 data, while diffusion training concatenates these 12 input channels with 4 noise channels. EDM randomly selects noise variance aiming denoise samples per mini-batch. CFM, CDM, CFM and CorrDiff are trained for 50 million steps, whereas the regression UNet is trained for 20 million steps. For AFM we evaluate the encoder's s RMSE every 10k steps and update a= using EMA with a 0.9. 0.9.Training is distributed across 8 DGX nodes, each with 8 A100 GPUs, using data parallelism and total batch size of 512.
Conditioning: AFM starts from the encoder output and learns the stochastic dynamics directly in the latent space. CDM and CFM map Gaussian noise to the high-resolution output space while being conditioned on the low-resolution input. Conditioning works by concatenating the low-resolution input with the noise, as described in Batzolis et al. (202I) and (Saharia al. (2022). AFM also be conditioned on the low-resolution input, the effect of which we explore in the ablations in Appendix
Sampling: Our sampling process employs Euler integration with 50 steps across all methods. We begin with a maximum noise variance Fmx and decrease to a minimum of mmin = 0.002. The value of Fmax varies depending on the method: for CDM and CorrDiff, we use Fmax = 800, as per the original implementation in (Mardani et al. 2023); for CFM, we set max 1, as specified in (Lipman et al. 2022); and for AFM, we use the a= value learned during training.
14

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Adaptive Flow Matching for Resolving Small-Scale Physies
F. Evaluation Metrics
F.I RMSE
The Root Mean Square Error (RMSE) is a standard evaluation metric used to measure the difference between the predicted values and the true values (Chai & Draxler 2014). In the context of our problem, let x be the true target and be the predicted value. The RMSE is defined as:
RMSE E [|| x|]
(22)
This metric captures the average magnitude the residuals, i.c. the difference between the predicted and true values. A lower RMSE indicates better model performance, suggests the predicted values are closer to the true values on average. RMSE is sensitive to large errors, making it ideal choice for evaluating models where minimizing large deviations is critical.
F.2. CRPS
The Continuous Ranked Probability Score (CRPS) is a measure used to evaluate probabilistic predictions (Wilks 2011 I.. compares the entire predicted distribution F F(x) with the observed data point For a probabilistic forecast with cumulative distribution function (CDF) F, and the true value x, the CRPS is given by:
CRPS(F, = -- (F(y) - TTyy 2 x))² dy,
(23)
where T(.) is the indicator function. Unlike RMSE, CRPS provides and spread of the predicted distribution. A lower CRPS indicates better match between the forecast distribution and the observed data. It especially useful for probabilistic models that output a distribution rather than single point prediction.
comprehensive evaluation of both the location
When applying CRPS to a finite ensemble of size m approximating with the empirical CDF incurs an O(1 (1/m)) bias favoring models with less spread. For small m unbiased versions of the formulas should be used instead (Zamo & Naveau 2018), but for the ensemble sizes here this small effect, so we used the more common biased formulas.
F.3. Spread Skill Ratio
The Spread-Skill Ratio (SSR) evaluates the reliability the predicted uncertainty by comparing the spread (variance) of the predicted distribution with the accuracy of the predictions (Gneiting & Raftery 2004). Let 0g be the standard deviation of the predicted distribution and RMSE defined above. The SSR defined as:
0% RMSE
SSR
(24)
An SSR value close to indicates that the predicted uncertainty (spread) is well-calibrated with the model's predictive skill. If the SSR is less than 1, the model underestimates uncertainty, while an SSR greater than indicates that the model overestimates uncertainty. This metric is particularly useful in evaluating the quality of probabilistic forecasts in terms of their sharpness (spread) and accuracy (skill).
G. Details of the Datasets
Further details and visualizations of the ERA5-CWA and KF datasets, used throughout the paper, are presented here.
G.I. ERAS-CWA Dataset
Table Table g summarizes the input-output channels and the corresponding resolutions. It is evident that the input and output channels generally differ, and even those that do overlap, such as (Temperature, East Wind, North Wind), are not perfectly aligned. For instance, comparing the Eastward Wind (10m) in the contour plots reveals the eye of the typhoon located northeast of the Taiwan region; see Fig. 7 Notably, the typhoon's 's eye shifts in the output due to the datasets
15

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Adaptive Flow Matching for Resolving Small-Scale Physies
Input and target variables for the ERA5 to CWA downscaling task include both single- level and pressure-level variables, the latter at 850 and 500 hPa.
Table 8: ERA5-CWA Variables:
Description
Input
Output
Pixel Size
36 x 36
448 x 448
Total Column Water Vapor
Maximum Radar Reflectivity
Temperature at 2 Meters
Temperature at 2 Meters East Wind at 10 Meters
Single-Levvel Channels
East Wind at 10 Meters
North Wind at 10 Meters
North Wind at 10 Meters
Temperature
Geopotential
Pressure-Level Channels
East Wind
North Wind
originating from two different simulations, which solve distinct sets of partial differential equations (PDEs) at significantly different resolutions, resulting in divergent trajectories.
Input Data (ERAS). This data for this study are derived from the ERA5 reanalysis, which provides a comprehensive set of atmospheric variables at various vertical levels (Hersbach et al. 2020). For our analysis, we selected a subset of 12 variables. These include four variables (temperature, East and North components of the horizontal wind vector, and geopotential height) two pressure levels (500 hPa and 850 hPa). Additionally, we incorporated single-level variables: 2-meter temperature, 10-meter wind vector components, and total column water vapor.
Target Data (CWA). The horizontal range of these data encompasses 900 x 900-km region containing Taiwan, with a horizontal resolution of approximately many output variables. We focus on four variables, three common to the input data surface temperature and ruface horizontal wind components - and one of which is distinctly related to precipitating hydrometeors, the composite synthetic radar reflectivity at time of data assimilation. We represent the high-dimensional target data as x E R #xxxC where H W 448 and C' 4. See table
The dataset encompasses approximately four years (2018-2021) of observations, sampled hourly intervals. For model development and validation, the data were partitioned chronologically. The training set comprises observations from 2018 to 2020, totaling 24,601 data points. The remaining data from 2021, consisting of 6,501 data points, were reserved for evaluation purposes. This temporal split allows for an assessment of the model's performance on future, unseen data, simulating real-world application scenarios.
Lastly, we upsample the input data to a 448 448 grid using bilinear interpolation to match the output resolution, a common practice with residual networks for consistency (Hu al., 2019 Zhang al. 2018) (see Figure 7b for an example of input target misalignment).
G.2. Kolmogorov-Flov Dataset
representative input-output sample of the KF dataset is shown in Fig. for different misalignment degrees т. Dataset description. We construct toy dataset by simulating the dynamics given by:
Sn + Jwhn (n) F vn V GTT²"
S1 + J(vr, (i) - (G (R) + -v GIT
(25)
V2th = Çi
42Lh = Çh-
Here, g(f.g) 1-gy-fae ffg9 is the Jacobian operator. The stream function is related to the velocity field by Vy (-u, implying that Veb. (u, v) 0, so that velocity points along contours of the stream-function. Çr. represents the vorticity.
The & field represents a coarse-resolution simulation nudged towards a high-resolution ÇA. The parameter controls the coupling strength between the G and Ç. fields. A steady-state forcing F = 10 cos(10z) injects energy into the small-scale field Sr but not the low-resolution Şt, mimicking the injection of energy by sub-grid processes like convection or flow over topography. Stronger dissipation ii > Ub used limit the effective resolution of the large-scale field. A small amount
16

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Adaptive Flow Matching for Resolving Small-Scale Physies
of Rayleigh damping Ty 100 is added to limit the pile-up of energy large-scales.
These equations are solved using standard pscudo-spectral method on the GPU. The 3rd-order stepper is used for all but the hyper-viscosity terms; for these stiff terms, we use an backward Euler time stepper. The resolution is 512 x 512 and the timestep dt = 0.001. A 2/3 de-aliasing filter is applied in spectral space every timestep (Orszag 1971 Outputs are saved every 5 = 0.2 time units.
time
We create datasets for different r values: 3, 5, 10, 20. Higher T corresponds to greater misalignment between the coarse and high-resolution simulations. This variation allows us assess the robustness of our method across different levels of coupling and identify potential thresholds in r beyond which certain downscaling approaches may become unreliable. For each T value, we generate a dataset comprising 100, 000 training points and 500 test points.
G.3. Misalignment
A key challenge in both datasets is the misalignment between input and output fields, arising from differences in the PDEs used at each scale. These discrepancies lead to both large-scale and fine-scale shifts.
In the ERAS-CWA dataset (Figure for example, storm centers in the eastward and northward wind fields are misaligned by several grid points- etween the input and target. An additional difficulty in this dataset is that the radar reflectivity channel is only present in the target and must be entirely reconstructed from the other input variables.
The Kolmogorov-Flow dataset was specifically designed to study varying degrees of misalignment. Here, the degree of mismatch is controlled by the coupling parameter т. For smaller values (e.g.. T 3), the input and target fields remain largely aligned, with only minor local discrepancies and some loss of fine-scale detail. As increases (e.g., T 10), the misalignment becomes more pronounced, introducing clear positional shifts between corresponding features, shown in This setup provides a controlled way assess how different levels of misalignment affect model performance.
Figure 8
H. Analysis of Adaptive 0g during Training
To evaluate the behavior of the adaptive noise scaling mechanism in our Adaptive Flow Matching (AFM) model, we monitored the sigma values across different channels during the training process. Results depicted in Figure Figure correspond to the model with Ixl conv. encoder and X 0.25 trained the CWA dataset. The sigma values initially set to I for all channels. During the early stages of training. sigma increases across the channels, due to the high encoder error. As training progresses and the encoder's 's performance improves, the sigma values begin to stabilize and converge towards their final values.
Notably, the radar reflectivity channel exhibits the highest sigma values throughout the training process, reflecting its in- herently stochastic nature. This consistent with our understanding that radar data contains significant variability and uncertainty. In contrast, the temperature channel consistently shows the lowest sigma values, aligning with its more de- terministic characteristics. These variations in sigma across channels underscore the effectiveness of our adaptive noise scaling approach, as it allows the model to appropriately adjust noise levels based on the inherent uncertainty of each chan- nel. This adaptability is crucial for managing misaligned data with differing degrees of stochasticity, thereby enhancing the overall performance and reliability of the AFM model in multiscale physics applications.
I. CWA Ablation Studies
To evaluate the effectiveness of the AFM model and understand the impact of its components, we conducted ablation studies on the CWA weather downscaling task at 112 x 112 resolution. We focused varying the À parameter, different encoder types, the use of adaptive dx and y conditioning. The results are summarized in Tables 9 TO and 11
L.1. Effect of X Parameter and Encoder Type
Table 9 presents the performance of the AFM model with different À values and encoder types. The À parameter controls the trade-off between data fitting and uncertainty regularization in the AFM model.
1 x 1 Conv. Encoder: For the x 1 Conv. encoder, setting À = 0 achieves the best performance across most variables, particularly for radar reflectivity, eastward wind, and northward wind. This indicates that the model benefits from minimal regularization when using a simpler encoder, allowing to focus on fitting the data closely. The low parameter count (only 60 parameters) of the 1 Cony, encoder might limit its ability capture all the information in the low-resolution input,
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Adaptive Flow Matching for Resolving Small-Scale Physies
making less regularization advantageous.
UNet Encoder: In contrast, the UNet encoder shows improved performance with small regularization parameter of = 0.25. This suggests that the more complex architecture of the UNet benefits from some regularization that prevents overfitting and leads to better accuracy and uncertainty estimates across multiple variables.
1.2. Impact of adaptive oz
Table 10 examines the effect of enabling or disabling adaptive oz for both encoder types.
Findings: Enabling adaptive a consistently enhances the model's performance across all variables and both encoder types. The improvements in RMSE, CRPS, and SSR metrics suggest that adaptive og allows the model to better capture the underlying uncertainty in the data. This adaptive approach provides the flexibility to model variable levels of uncertainty across different regions and variables, leading more accurate and reliable predictions.
I.3. Effect of Conditioning
Table 11 assesses the impact of enabling or disabling y conditioning the AFM model for both encoders.
1 Conv. Encoder: Disabling y conditioning provides better results across most metrics for this encoder. Given the simplicity of the 1 x 1 Conv. encoder and its limited parameter count, it may not effectively utilize the additional information provided by y conditioning. The model performs better when focusing on directly mapping the input to the output without the added complexity.
Enabling conditioning yields the best results for the UNet encoder. The more complex architecture of the UNet can leverage the additional context from conditioning to improve its predictions. This demonstrates the capacity of the UNet encoder to capture and utilize supplementary information, enhancing both accuracy and uncertainty estimation.
UNet Encoder:
1.4. Summary
These ablation studies highlight the strengths of the AFM model and its components in weather downscaling tasks:
Effectiveness of Adaptive oz: Enabling adaptive og consistently improves model performance across both encoder types and all variables. This underscores the importance modeling spatially varying uncertainty in complex weather data.
Encoder Choice and Regularization: The 1 x 1 Conv. encoder performs best without regularization (A 0), indicating that minimal regularization benefits simpler models. For the UNet encoder, a small regularization parameter (A 0.25) yields better results, suggesting that regularization helps prevent overfitting in more complex models.
Impact of y Conditioning: y conditioning enhances performance for the UNet encoder but not for the 1 1 Conv. encoder. This suggests that the effectiveness of incorporating additional context depends on the model's capacity to utilize that information.
Overall, the AFM model demonstrates strong performance and flexibility in weather downscaling tasks. By carefully selecting model components such encoder type, adaptive ox and regularization parameter X, the AFM can be tailored to balance predictive accuracy and uncertainty estimation effectively. These findings highlight the potential of the AFM model as a powerful tool for probabilistic weather downscaling.
J. CWA Ensemble Analysis
For few representative samples, several ensemble members and the ensemble mean shown in ?? 11a-J. The generated samples using different random seeds exhibit notable diversity, particularly for channels like Radar Reflectivity. This diversity confirms the model's ability to produce well-dispersed ensemble, which is crucial for achieving a calibrated and reliable probabilistic forecast. Additionally, the ensemble mean closely aligns with the true target, indicating that the model successfully captures the underlying physical processes while preserving uncertainty across channels.
Animated PNGs of ensemble members for different models and 7 are provided at https: bttps://t..o 1y/20q97
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Adaptive Flow Matching for Resolving Small-Scale Physies
Table Encoder and À Ablations on CWA 112 X 112 Dataset. For this table we keep the best configurations for each value based on radar reflectivity. We separate per Conv, and UNet encoder to elucidate differences. Best results for each encoder are highlighted in bold. No regularization (A 0) works best for the x 1 Conv. encoder, while for UNet, small À 0.25 value yields the best results across multiple variables. Overall the 1 x Conv. without regularization is the best configuration.
Radar
Temperature
Eastward Wind
Northward Wind
Encoder
RMSE
CRPS |
SSR
RMSE
CRPS J
1.66
0.99
1.42
0.78
0.68
1.58
0.66
1 Cony.
0.25
5.10
1.83
0.42
0.85
0.50
0.43
1.46
0.84
0.46
1.63
0.92
0.46
1.00
5.04
1.83
0.40
1,05
0.60
0.34
1,50
0.88
0.44
1.67
0.95
0.44
0.00
5.06
1.83
0.38
1.01
0.57
0.36
1.48
0.85
0.46
1.64
0.93
0.46
UNet
0.25
4.95
1.78
0.44
0.94
0.54
0.40
1.49
0.85
0.53
1.58
0.95
0.50
1.00
1.89
0.41
1.07
0.62
0.36
1.53
0.91
0.41
1.74
1.03
0.40
Table 10: Adaptive og Ablation on CWA 112 x 112 Dataset for 1 x 1 Conv. and UNet Encoders. This table examines the effect of enabling (V) or disabling (X) Adaptive o₂ across both x 1 Conv. and UNet encoders. Best results for each encoder and metric are highlighted in bold. Results indicate that the proposed adaptive 0₂ consistently improve results.
Encoder
Adapt
Radar
Temperature
Eastward Wind
Northward Wind
RMSE
CRPS
SSR
RMSE
CRPS
SSR +
RMSE
CRPS
SSR
RMSE
CRPS
SSR
1x 1 Cony.
5.01
1.88
0.45
1.13
0.66
0.39
1.50
0.86
0.48
1.64
0.94
0.47
1.66
0.59
0.47
1.42
1.58
0.66
5.06
1.83
0.38
1.01
0.57
0.36
1.48
0.85
0.46
1.64
0.93
0.46
UNet
4.95
1.78
0.44
0.94
0.54
0.40
1.49
0.85
0.53
1.58
0.95
0.50
Table 11: y conditioning ablation CWA 112 112 Dataset for x Conv. and UNet Encoders. This table assesses the impact of enabling (V) disabling (X) conditioning across both 1 x 1 Conv. and UNet encoders. Best results for each encoder and metric are highlighted in bold. Results indicate that for the 1 encoder the conditioning is beneficial while for UNet the opposite stands. This makes sense since the 1 x 1 Conv. encoder has only 60 parameters and might not capture all the information in the low-resolution input.
Encoder
cond.
Radar
Temperature
Eastward Wind
Northward Wind
RMSE
CRPS
SSR
RMSE J
CRPS J
SSR
RMSE 1
CRPS
SSR
RMSE :
CRPS
SSR ->
1x I Conv. x
4.82
1.66
0.72
0.99
0.59
0.47
1.22
0.78
0.68
1.63
0.89
0.66
5.06
1.82
0.34
0.89
0.54
0.40
1.46
0.82
0.45
1.59
0.92
0.46
5.06
1.83
0.57
0.36
0.85
0.46
0.46
UNet
/
4.95
1.78
0.44
0.94
0.54
0.40
1.49
0.85
0.53
1.67
0.95
0.50
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Adaptive Flow Matching for Resolving Small-Scale Physics
Eastward Wind S00hPa
Eastward Wind B50hPs
Eastward Wind 10m
Geopotential 500nPa
Ar My Saali Luite
2016
Mu Ad
Preie Mi Loak
STOOE
300EC
-10
20
Northwerd 500
Northwward Wind 050hPe
Northword dl Wind
Geaputentiel 850hPP
J--ha Pa PlauAo
Artua My Kuring
SAGSE
-30
-ja
22%
-20
-2o
Temperature
Tempereture 2m
Vapor
Eoa Musting
220
271
224
2119
(a) Input variables
Max Radar Reflectivity
Temperature
Eastward Wind
Northward Wind 10m
2P'
50
mis
Troa 30th Aui
20
24'W
21'
121'5
121'
112'5
121'
(b) Target variables
Figure 7: Visualization of ERAS-CWA Dataset Variables. The top row shows input variables such as temperature and wind at coarse-resolution. while the bottom row presents the corresponding fine-resolution target variables. The maximum radar reflectivity is absent from the input variables and must be constructed by the model. This key misalignment between the low- and high-resolution data increases the complexity of the problem beyond standard super-resolution tasks.
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Adaptive Flow Matching for Resolving Small-Scale Physies
T= 3
T= 5
T= ==10
-10
Input
5
Ground Truth
Figure 8: Kolmogorov Flow Dataset. Visualization of input and target Kolmogorov Flow dataset for varying levels of misalignment (r = 3, 5, 10). As increases, the discrepancy between the coarse and fine-resolution fields grows, offering a controlled environment to test downscaling performance.
Radar Reflectivity
Radar Reflectivity
Temperature
1.00
Temperature
E Wind
Wind
N Wind
Wind
0.75
0.50
0.25
200k
400k
600k
200k
400k
600k
Training Steps
Training Steps
(a) 1 Conv.
(b) UNet
Figure 9: Adaptive Oz values over training steps for different channels. The plot corresponds to the AFM model using 1 x 1 Conv. (a) and UNet (b) encoders with À 0.25. With 1 convolution, d. increases during early training due to high encoder error and subsequently converges as the encoder improves. With UNet, a₂ starts decreasing with training. In both cases, radar reflectivity exhibits the highest ar, indicating its stochastic nature, while temperature shows the lowest, reflecting its deterministic characteristics. The varying sigma values across channels demonstrate why adaptive noise scaling is effective managing misaligned data with differing levels of stochasticity.
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Adaptive Flow Matching for Resolving Small-Scale Physies
Table 12: Complete ablation results for the CWA 112 x 112 dataset. Best two models in each variable/metric in bold.
Model
Adapt.
Use Xao
Radar
Temperature
Eastward Wind
Northward Wind
RMSE 4
CRPS
SSR
RMSE
CRPS
SSR
RMSE
CRPS
SSR
RMSE.
CRPS
SSR
X
X
5.16
1.88
0.44
1.12
0.65
0.36
1.53
0.88
0.47
1.72
0.98
0.46
0.00
/
x
4.82
1.66
0.72
0,99
0.59
0.47
1.42
0.78
0,68
1.63
0.89
0,66
/
5.06
1.82
0.34
0.87
0.52
0.44
1.44
0.81
0.49
1.60
0.89
0.49
X
x
5.10
1.83
0.42
0.85
0.50
0.43
1.46
0.84
0.46
1.63
0.92
0.46
AFM
0.25
Conv.
x
5.11
1.85
0.37
0.89
0.56
0.28
1.53
0.92
0.36
1.71
1.02
0.34
/
5.12
1.87
0.29
0.89
0.54
0.40
1.46
0.84
0.45
1.59
0.90
0.46
X
5.04
1.83
0.40
1.05
0.60
0.34
1.50
0.88
0.44
1.67
0.95
0.44
1.00
1.85
0.36
0.92
0.58
0.30
1.52
0,92
0.35
1.71
1.01
0.34
5.12
1.88
0.30
0.86
0.50
0.41
1.43
0.82
0.46
1.58
0.90
0.45
x
1.01
0.57
0.36
1.48
0.85
x
1.84
0.43
0.85
0.52
0.37
1.43
0.81
0.50
1.60
0.89
0.48
/
0.36
0.85
0.43
x
1.88
0.45
1.13
0.39
1.50
0.86
1.64
0.94
0.47
AFM UNet
x
5.01
1.82
0.32
0.85
0.52
0.31
1.43
0.85
0.40
1.58
0.93
0.39
4.95
1.78
0.44
0.94
0.54
0.40
1.49
0.85
1.67
0.95
0.50
x
x
5.11
1.89
0.41
1.07
0.62
0.36
1.53
0.91
0.41
1.74
1.03
0.40
1.00
x
5.04
1.85
0.29
0.84
0.53
0.26
1.45
0.87
0.35
1.62
0.97
0.34
/
/
5.07
1.85
0.31
0.88
0.54
0.29
1.46
0.88
0.38
1.65
0.98
0.37
Table 13: Performance Comparison of Models CWA 112 112 Dataset. The AFM model has 1 x 1 Conv. encoder, = 0, adaptive and no conditioning. Overall, the AFM model exhibits strong performance across different metrics and variables, particularly excelling in its calibration (variability). Best results for each metric are highlighted in bold. Note that for deterministic models, CRPS equals MAE.
Model
CFM
CDM
UNet
AFM
RMSEŞ
5.06
4.95
4.94
4.82
CRPS
1.84
1.74
1.66
Radar
MAEJ
2.41
2.49
2.45
2.63
SSR
0.36
0.52
0.72
RMSE
0.86
0.87
0.87
0.99
CRPS
0.50
0.52
0.59
Temperature
MAE
0.64
0.64
0.64
0.74
SSR
0.45
0.38
0.47
RMSE
1.42
1.44
1.42
1.42
CRPS
0.81
0.81
0.78
East. Wind
MAE
1.04
1.05
1.05
1.05
SSR
0.48
0.49
0.68
RMSE
1.59
1.59
1.60
1.63
CRPS
0.89
0.89
0.89
North. Wind
MAE
1.14
1.14
1.16
1.19
SSR
0.47
0.48
0.66
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Adaptive Flow Matching for Resolving Small-Scale Physies
Table 14: AFM ablations for À on the CWA weather downscaling task at full resolution 448 x 448. For this ablation, a X 1 Conv. encoder was used, oN was set to 1, and no y conditioning was employed. Overall, À = seems to produce better estimates except for temperature, whose deterministic nature benefits from the added regularization.
X
Variable
Metric
0.0
0.01
0.1
0.25
0.5
1.0
2.5
RMSE +
4.90
5.11
5.03
4.97
5.00
5.25
5.27
CRPS
1.78
1.92
1.88
1.85
1.87
1.97
1.96
Radar
MAE
2.42
2.66
2.47
2.47
2.47
2.59
2.75
SSR
0.44
0.49
0.39
0.42
0.41
0.39
0.52
RMSE
1.00
1.00
0.87
0.86
0.82
1.00
0.85
CRPS
0.52
0.62
0.54
0.52
0.48
0.56
0.48
Temperature
MAE
0.67
0.78
0.66
0.64
0.60
0.68
0.62
SSR
0.47
0.48
0.38
0.40
0.41
0.32
0.45
RMSE +
1,44
1.52
1,49
1.48
1.49
1.55
1,48
CRPS
0.80
0.86
0.87
0.87
0.88
0.92
0.86
East. Wind
MAE
1.07
1.13
1.09
1.09
1.09
1.14
1.08
SSR
0.61
0.55
0.42
0.40
0.41
0.38
0.43
RMSE
1.61
1.61
1.66
1.67
1.67
1.72
1.65
CRPS
0.88
0.88
0.95
0.96
0.96
1.00
0.95
North. Wind
MAE
1.17
1.17
1.19
1.20
1.19
1,24
1.18
SSR
0.58
0.61
0.41
0.41
0.40
0.38
0.43
Table 15: AFM ablations for encoder size on the CWA weather downscaling task at full resolution 448 x 448. For this ablation, we use À = 0, a= = 1, and no y conditioning. Larger encoders improve performance for complex spatial data like radar, while for temperature and wind data, smaller encoders are adequate and sometimes even slightly better. The optimal encoder size depends on the specific variable being predicted but overall the difference are not significant.
Encoder
Variable
Metric
L
M
S
XS
RMSE
4.93
4.96
4.98
4.98
CRPS
1.82
1.84
1.85
1.85
Radar
MAE
2.44
2.52
2.52
2.46
SSR
0.40
0.43
0.42
0.39
RMSE
1.01
1.00
0.99
1.02
CRPS
0.55
0.54
0.54
0.56
Temperature
MAE
0.68
0.68
0.68
0.70
SSR
0.34
0.38
0.40
0.38
RMSE
1.48
1.48
1.47
1.50
CRPS +
0.85
0.83
0.83
0.86
East. Wind
MAE 1
1.10
1.08
1.08
1.11
SSR
0.49
0.53
0.53
0.51
RMSE
1.64
1.64
1.63
1.66
CRPS +
0.92
0.92
0.91
0.93
North. Wind
MAE J.
1.19
1.20
1.19
1.21
SSR
0.48
0.51
0.52
0.50
23

---

Adaptive Flow Matching for Resolving Small-Scale Physics
Table 16: AFM ablations for conditioning for the CWA weather downscaling task at full resolution 448 x 448. For this ablation, we use adaptive dz, x = 0.25 and a UNet encoder guided by the ablations in the 112 x 112 resolution. Including conditioning consistently improves performance across all metrics. RMSE, CRPS, and MAE are lower when y conditioning used, and SSR values are closer
y conditioning
Variable
Metric
X
RMSE
5.09
4.90
CRPS
1.88
1.80
Radar
MAE 4
2.33
2.24
SSR
0.24
0.25
RMSE
0.92
0.89
CRPS
0.55
0,50
Temperature
MAE 4
0.67
0.64
SSR
0.33
0.43
RMSE
1.49
1.45
CRPS
0.91
0.86
East. Wind
MAE 4
1.10
1.07
SSR
0.34
0.41
RMSE
1.66
1.61
CRPS
1.00
0.94
North. Wind
MAE
1.20
1.18
SSR
0.33
0.41
24

---

Adaptive Flow Matching for Resolving Small-Scale Physies
AFM UNet
Impur
UNel
CPM
CDM
Doff
A 0.0
Ground Truth
Refeeciitty
No Reder
Temporature
Wind
20
Narthward
1084
110%
110%
(a)
AFM UNet
Imput
UNet
CPM
CDM
DiftDiT
A 0.0
Ground Truth
No Redwr
Temporature
val
vah
Sch
Eastward
Northward
110%
115%
(b)
Figure 10: (Cont.)
25

---

Adaptive Flow Matching for Resolving Small-Scale Physics
AFM Uleet
Impur
UNet
CPM
CDM
Don
A- 0.0
Ground Truth
Temporak Refleciitty
No Reder
Eastward Win
Northward
110%
(c)
AFM UNet
Imputt
UNett
GFM
CDM
Corbiff
A.0.0
Ground Truth
Reflectiity
No Reew
Tomporature
Eastward Win
23%
Worthward
(d)
Figure 10: (Cont.)
26

---

Adaptive Flow Matching for Resolving Small-Scale Physies
Inpur
UNet
CFM
CDM
CanDif
- 0.0
Ground Truth
Terperauue Refectiity
Teder
271
Eastward Wind
Northward
(c)
SFM A 0.0
Impuc
UNst
CFM
CDM
CorrDif
Ground Truth
Terpe 2on
20
Easeward Wind 10m
2799
(f) (f)
Figure 10: (a-e) Visual Comparison of All Models for CWA Weather Data. The AFM model demonstrates superior reconstruction quality, particularly in capturing fine-scale details, while other baselines show blurring or misalignment in key areas. (a-e) show the results for different models.
27

---

Adaptive Flow Matching for Resolving Small-Scale Physies
Sample
Sample
Sample
379
27M
Mox Roster
110%
2015
104
209
27M
1
Eastward
(
Ensemble Meen
Sample
Sample
Sample
Ground Truth
379
ping
Pooter
Iemperrture 2m
224
22
Eastwarrd
10m
Miu
115
(b)
Figure 11: (Cont.)
28

---

Adaptive Flow Matching for Resolving Small-Scale Physies
Sample
Sample
Sample
Mox Reflectivvtt
77
279
27M
11.%
2014
1241
271
100
Enstward
(c)
Ensemble Meen
Sample
Sample
Sample
Ground Truth
279
prg
Pooter
22%
lemreerature 2m
224
Eastwarr
Northwarrd 10m
112
(d)
Figure 11: (Cont.)
29

---

Adaptive Flow Matching for Resolving Small-Scale Physies
Sample
Sample
Sample
Reflectiiity
1347
271
27%
lemperratrr
24%
11410
2011
1201
10m
271
Wind
22n
Ersembla
Sample
Sample
Ground Truth
Max Radar nettectvvtty
2416
1200
1216
Temperature 2m
209
3019
3016
309
2210
2019
3316
Eastward Wind 10m
Northward Wind 10m
1324
127
50
Figure 11: Ensemble predictions for CWA Weather Data. Results demonstrate AFM's ability to capture variable dy- namics. (a-e) show the results for different points in time.
30

---

Adaptive Flow Matching for Resolving Small-Scale Physies
K. Kolmogorov Flow Ablations
Different hyperparameters of both the dataset and the model are ablated, with the resulting metrics reported in Table 17 The results show that, in this scenario, where the data is relatively misaligned, a smaller encoder tends achieve better generalization performance, possibly due to its capacity to focus on the most relevant features while avoiding overfitting, Additionally, the experiments demonstrate that conditioning the AFM with the coarse-resolutin input enhances the predic- tive skill, highlighting the importance of incorporating multiscale information for improved downscaling accuracy. These findings provide valuable insights into the optimal model configurations for handling misaligned data.
Table 17: Kolmogorov Flow Ablation Study for AFM: This table examines the effect of different hyperparameters performance across misalignment levels (7). A smaller encoder with conditioning consistently performs better for highly misaligned data. Additionally, adaptive noise scaling (0z) enhances performance when conditioning AFM on coarse- resolution input data (y).
Encoder
Iconv
UNet
Adapt. dz
X
X
/
cond.
X
/
X
/
X
X
RMSE
1.22
0.91
1.22
0.73
1.15
1.11
1.17
1.17
CRPS
0.63
0.48
0.62
0.37
0.65
0.63
0.69
0.70
3
MAE
0.83
0.65
0.83
0.51
0.81
0.78
0.83
0.84
SSR 1
0.56
0.58
0.58
0.62
0.37
0.34
0.31
0.30
RMSE
1.17
0.78
1.17
0.76
1.16
1.01
1.09
1.07
CRPS
0.62
0.42
0.61
0.40
0.67
0.58
0.66
0.64
5
MAE 4
0.82
0.56
0.82
0.54
0.83
0.72
0.80
0.78
SSR
0.58
0.58
0.60
0.58
0.35
0.36
0.33
0.33
RMSE
1.36
1.06
1.39
1.09
1.35
1.36
1.36
1.28
CRPS
0.71
0.57
0.78
0.65
0.79
0.79
0.80
0.74
10
MAE
0.96
0.78
0.98
0.77
0.98
0.99
1.00
0.95
SSR
0.69
0.63
0.43
0.23
0.36
0.37
0.38
0.45
L. Kolmogorov Flow Ensemble Analysis
Representative KF samples along with the generated ensemble members are depicted in Figs, 14a 14b and 14c These figures illustrate the variability captured by the ensemble across different forecast lead times. The diversity in the ensemble members indicates the model's ability to represent the inherent uncertainty in the system. Furthermore, the alignment of ensemble mean with the observed samples suggests that the model not only captures central tendency but also effectively characterizes the stochastic nature of the dynamics.
31

---

Adaptive Flow Matching for Resolving Small-Scale Physics
Input
Ground Truth
UNet
CFM
CDM
AFM
10
7= 5
-10
7= 3
Figure 12: AFM vs. Baselines for Different Misalignment Levels in Kolmogorov Flow Downscaling: Each row corresponds to a different misalignment level (left side). From top to bottom, the rows represent = 10, = 5, and 3. As misalignment increases, the AFM significantly outperforms baseline models by generating samples that better align with the target distribution. Additionally, note the presence of high- fighfrreuency artifacts the baseline models, which noticeable when the figures zoomed in.
10
1
10
Power
10
Irput
COM
10
AFM
10
10º2
10
10
10
10
Frequency
Frequency
Frequency
(c) T 10. Figure 13: AFM spectra vs. baselines for the Kolmogorov Flow. AFM maintains superior fidelity to the ground truth across different values, highlighting its robustness in preserving both small and large-scale structures under various misalignment conditions. The small bump around the middle is caused by the energy that is added in the system (see
(a) T
(b) 5.
Appendix G.2
32

---

Adaptive Flow Matching for Resolving Small-Scale Physies
Input
Ground Truth
Mean
Sample 1
Sample 2
Datapoint
Datapoint 2
(a)
Input
Ground Truth
Mean
Sample 1
Sample 2
Datapoint 1
Datapoint
( =5
Input
Ground Truth
Mean
Sample 1
Sample 2
Datapoint 1
Datapoint 2
(c) 10
Figure 14: Ensemble Predictions of AFM for Kolmogorov Flow at Different - Values. AFM ensemble predictions
shown for different values (r 3. 5, 10), illustrating the model's ability to capture the variability and dynamics of the Kolmogorov flow across increasing levels of misalignment.
33


Reviews:
[{"review_id": "M20M0UKNTe", "reviewer": "Reviewer_eJPb", "summary": "The paper addresses image 'super-resolution' in the context of atmospheric physics. The super-resolution aims at generating small stochastic scales into the input data while preserving and aligning large scale physics. The authors propose to adapt and apply flow matching for this purpose. To this end, the authors show that it is possible to jointly train a time dependent decoder and the flow. An adaptive noise scaling factor allows to adjust the noise level of the diffusion process to the input data Experimental results are illustrated on two meteorological datasets. Quantitative results and comparisons with related work are reported.\n\n\"## update after rebuttal\nThe rebuttal provided by the authors has addressed my few concerns. The paper has probably a limited audience amongst the ML/CV community, but nevertheless illustrates a very relevant practical use case (and associated challenges) beyond the known benchmarks.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their positive and constructive feedback, and for recognizing our clear motivation and rigorous methodology.\n\n*1. In figure 2, 4: there is a confusion between the acronyms: SFM -> AFM* \n\n**Response.**\n Thank you for noticing the typo. In the revised manuscript, we have used AFM consistently. \n\n*2. The authors could be more specific when describing the 'misalignment' between the input / output. How this mis-alignment could be characterized: is it global or local, subpixel or few pixels scale?*\n\n**Response.**\nThank you for the suggestion. We have clarified the notion of misalignment in the revised manuscript. It arises in two ways: (1) channel-level mismatch, where input and output contain different modalities (e.g., forecasts vs. radar reflectivity); and (2) spatial misalignment due to differences in the underlying PDEs between coarse and fine models. Spatially, the misalignment can be both global and local. For instance, in Fig. 7, storm centers in wind fields are displaced by several grid points (large-scale shift), while in Fig. 8, smaller PDE discrepancies ($\\tau{=}3$) lead to subpixel-to-few-pixel level local misalignment.  In the revised manuscript we have added a section G.3 discussing this in further detail and also adapted Section 5 to more clearly define misalignment.\n\n*3. I understand that the input low resolution data is first rescaled by bilinear interpolation to the size of the output. Is it correct?*\n\n**Response.**\nThank you for the question. Indeed, we apply bilinear upsampling to align dimensions, then let the encoder refine and align further. This is explained in Section 5.1 of the manuscript (and in more detail in G.1 of the Appendix). In the revised version we added further clarifications regarding this in the main text (Sec 5.1).\n\n*3. Could you envisage different fields of application? Could this approach be useful for other use cases?*\n\n**Response.**\nCertainly. Our AFM framework applies wherever partial or misaligned data requires super-resolution or channel synthesis, e.g., MRI subsampling in medical imaging. We highlight broader applicability in the Conclusion of the revised manuscript.\n\nWe hope our responses and revisions have adequately addressed your questions.", "claims_and_evidence": "The authors claim that small physics scales can be accounted for with a diffusion denoising model while the large scale discrepancy between output/input can be tackled with an determinist encoder. Experimental setting and comparison with standard approaches demonstrate promising results.", "methods_and_evaluation_criteria": "The motivation of the approach is clearly exposed and the mathematical derivation of the problem clearly presented. The steps of the  approach and reasoning are well justified and discussed. \nPerformance is evaluated in terms of RMSE, Continuous Ranked Probability Score (CRPS), Mean Absolute Error (MAE), and Spread Skill Ratio (SSR).Ablation studies are performed. Thorough comparisons are reported.", "theoretical_claims": "The mathematical derivations (main paper and supplementary material) are correct tmk", "experimental_designs_or_analyses": "Experimental results are limited but convincing. One real data and one synthetic data are used for training and evaluation. Details of the datasets are reported in appendix. Thorough and detailed visual and quantitative results are reported. It is interesting in particular to observe the discrepancy of quantitative results for stochastic channels (ie radar and wind) vs deterministic (ie temperature) channels.", "relation_to_broader_scientific_literature": "This paper is at the frontier between applied atmospheric physics and machine learning. It introduced most relevant material, without being exhaustive, as required.", "essential_references_not_discussed": "no missing reference, to my knowledge.", "other_strengths_and_weaknesses": "A well presented paper, with a clear 'flow' , a well justified approach, and limited but convincing results.", "other_comments_or_suggestions": "- In figure 2, 4: there is a confusion between the acronyms: SFM -> AFM \n- The authors could be more specific when describing the \"mis-alignment\" between the input / output. How this mis-alignment could be characterized: is it global or local, subpixel or few pixels scale?\n- I understand that the input low resolution data is first rescaled by bilinear interpolation to the size of the output. Is it correct?", "questions_for_authors": "Could you envisage different fields of application? Could this approach be useful for other use cases?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethical_review_concerns": "no ethical concern"}}, {"review_id": "36jxNYBvwF", "reviewer": "Reviewer_xRHs", "summary": "The paper focuses on tackling small-scale physical science problems (e.g., weather super-resolution). It proposes the joint encoder and flow-matching training objective over the prior two-stage methodologies to improve the overfitting. Specifically, this work introduces Adaptive Flow Matching (AFM) with the help of adaptive noise scaling to introduce stochasticity in the process. At last, authors compare w.r.t. Various baselines on regional downscaling and multiscale Kolmogorov-Flow tasks.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their comments and for finding our paper interesting.\n\n*1. In the abstract itself, it is claimed that AFM achieves SOTA on regional downscaling. However, this is not the case. According to Table 2 and Figure 2, improvements over CFM are arguable. However, on Kolmogorov-Flow, task performance seems to be much better. But overall, it feels marginally better.*\n\n**Response.**\nWe thank the reviewer for their remark. Our claim of “SOTA” is based on the consistent gains observed across datasets, channels and metrics, particularly in those with greater stochastic variability. In Table2, for instance, *radar reflectivity* sees a clear improvement when using AFM over CorrDiff, and AFM’s ensemble calibration (e.g., SSR) is superior across all channels. Similarly in Table 13 AFM outperforms all other models in most metrics and channels and has better calibration by a large narging.  \nAlso, in the Kolmogorov Flow experiments (Table 3), AFM clearly outperforms CFM and CorrDiff and the gap become wider as the misalignment increases (tau 3 -> 10). While the margins may appear moderate on more deterministic channels (e.g., temperature), these results overall indicate that AFM consistently delivers better calibration and uncertainty modeling, which is crucial in scientific tasks. We clarify in the revised abstract that our method excels especially under higher stochasticity and improves overall ensemble calibration.\n\n---\n\n*2. As flow matching maps the any to any distribution, there should be a baseline that just does that along with traditional CFM. Basically, direct $\\mathcal{X} \\to \\mathcal{Y}$ mapping. Similarly, other methods, such as Diffusion Schrödinger bridge matching,' could also be added. Because of these prior works (and missing baselines), I am unsure if the claim for Section 4 Line 149-152 is valid.*\n\n**Response.** Thank you for the comment. In our setting, direct $\\mathcal{X} \\to \\mathcal{Y}$ flow matching is not applicable due to both spatial and channel misalignments—$\\mathcal{X}$ and $\\mathcal{Y}$ often have different modalities (e.g., $\\mathcal{X}$ includes forecast features, while $\\mathcal{Y}$ includes precipitation and radar reflectivity). As such, intermediate alignment via an encoder is necessary. The closest viable baseline is Conditional Flow Matching (CFM), which we include. Methods like diffusion Schrödinger bridges (e.g., I2SB) also assume aligned domains and are similarly inapplicable here.\n\n---\n\n*3. Paper often misrepresents the Flow matching as a stochastic process (Abstract Line 25), which is misleading. Meanwhile, stochasticity comes from noise term in eq. (5).*\n\n**Response.** Thank you for pointing this out. You are correct—flow matching is inherently deterministic (ODE-based), and the stochasticity in our method arises from the noise injected into the encoder output. We have clarified this distinction in the abstract of the revised manuscript.\n\n---\n\n*4.Typos in Figures 2,3 & 4 where SFM is written instead of AFM. “scientific” type in conclusion*\n\n**Response.**\nThank you for noticing the typos, that are now fixed in the revised manuscript. \n\n---\n\n*5. What is the final $\\sigma_z$ value during the inference? Is it the one result at the end of training (Figure 5)?*\n\n**Response.**  Yes, that is correct. During inference, we use the final converged value of \\(\\sigma_z\\) obtained at the end of training, as shown in Figure~5.\n\nWe hope our responses and revisions have adequately addressed the remaining of your concerns.", "claims_and_evidence": "In the abstract itself, it is claimed that AFM achieves SOTA on regional downscaling. However, this is not the case. According to Table 2 and Figure 2, improvements over CFM are arguable. However, on Kolmogorov-Flow, task performance seems to be much better. But overall, it feels marginally better.", "methods_and_evaluation_criteria": "Methods: Yes, the proposed method makes sense, as spatially misaligned input/output could be challenging for the existing generative model formulations.\n\nEvaluations: Due to a lack of reviewer expertise, it is hard to verify the evaluation procedure. However, it seems exhausting.", "theoretical_claims": "The review has verified Proposition 1  and the resulting sampling procedure. Both seem to be accurate.", "experimental_designs_or_analyses": "The reviewer verified all experimental designs and analyses. There are no visible issues.", "relation_to_broader_scientific_literature": "The proposed approach to AFM might be of interest to the broader community as it attempts to perform joint training of the encoder and the flow model.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Weaknesses:\n- The paper contains many typos and lacks clarity (see Suggestions).\n- As flow matching maps the any to any distribution, there should be a baseline that just does that along with traditional CFM. Basically, direct $y$ to $x$ mapping. Similarly, other methods, such as “Diffusion Schrödinger bridge matching,” could also be added. \n    - Because of these prior works (and missing baselines), I am unsure if the claim for Section 4 Line 149-152 is valid.", "other_comments_or_suggestions": "- Paper often misrepresents the Flow matching as a stochastic process (Abstract Line 25), which is misleading. Meanwhile, stochasticity comes from noise term in eq. (5). \n- Typos in Figures 2,3 & 4 where SFM is written instead of AFM.\n- “scientific” type in conclusion", "questions_for_authors": "- What is the final $\\sigma_z$ value during the inference? Is it the one result at the end of training (Figure 5)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "y1zjcDIPj9", "reviewer": "Reviewer_zyDu", "summary": "The paper introduces stochastic flow matching (SFM) for super-resolving small-scale physics in weather data, tackling challenges such as data misalignment, multiscale dynamics, and limited data availability. The approach employs an encoder to project coarse-resolution inputs into a latent space, followed by flow matching to generate fine-resolution outputs. Additionally, adaptive noise scaling is used to balance deterministic and stochastic components. Experimental results show that SFM surpasses existing methods, including conditional diffusion and flow models.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their constructive comments. Below are our responses to your remarks.\n\n*1. In my opinion, VAE is a more natural choice ...* (we shorten the questions due to the character limit)\n\n**Response.** Thank you for the insightful question. As clarified in our Remark in Sec 4.2, while VAEs are a natural choice for stochastic encoding, we intentionally opt for a maximum-likelihood-based approach with a deterministic encoder and post-hoc noise injection. This choice is motivated by both **simplicity** and **interpretability**.\n\nFrom a modeling standpoint, our approach yields a closed-form expression for the noise variance—simply the RMSE of the regression error—which enables straightforward tuning using validation data to control generalization. In contrast, VAEs require careful balancing of the KL term and reconstruction loss, which is known to be challenging and prone to issues such as posterior collapse and the prior hole.\n\nFrom a physics perspective, particularly in applications like weather downscaling, large-scale structures are largely deterministic. A deterministic encoder is therefore well-suited to capture these coherent features, while the stochasticity in small scales is effectively modeled via calibrated noise.\n\nWhile we agree that VAE-style encoders are a valid alternative, our design provides a principled and practical path that aligns well with physical intuition and avoids some known drawbacks of VAEs.\n\n---\n\n*2. The novelty of the paper is questionable...*\n\n**Response.** We respectfully disagree. While our method shares surface similarities with CorrDiff, AFM introduces key innovations that address fundamental limitations of CorrDiff and generalize it as a corner case.\n\n*Joint Training vs. Two-Stage*: CorrDiff relies on a two-stage pipeline, first training a deterministic encoder, then fitting a diffusion model to the residuals. This separation causes overfitting in the encoder. In consequence, the residuals shrink and become uninformative, leaving the generative model. In contrast, AFM performs joint training of the encoder and the generative model, maintaining informative residuals throughout optimization.\n\n** Uncertainty Modeling**: CorrDiff lacks an uncertainty-aware encoder. AFM introduces an adaptive noise injection mechanism that dynamically increases the encoder noise when the validation error rises, signaling overfitting. This prevents collapse and encourages the encoder to retain uncertainty.\n\n** Channel-wise Adaptivity**: AFM performs per-variable noise scaling—crucial for scientific data where variables (e.g., temperature vs. radar reflectivity) exhibit different stochastic behaviors. CorrDiff applies uniform noise, which fails to capture such heterogeneity.\n\nThese design choices lead to significantly improved calibration and spread-skill performance (see Tables 2–3, 6, 13; Figures 5–6, 8).\n\n---\n\n\n*3. Lack of detailed explanation of noise scaling...* \n\n**Response.** Thank you for the remark. We have expanded our explanation of the noise scaling mechanism in Sec. 4.2, Appendix E, and Algorithm 1 of the revised manuscript.\n\nBriefly, we estimate $\\sigma_z$ using a maximum-likelihood criterion linked to the encoder's RMSE on validation data (see Eq. 8). As training progresses, if the encoder begins to overfit—reflected by rising validation error—our method increases $\\sigma_z$, injecting more uncertainty into the latent space. This prevents the encoder from collapsing into a deterministic solution and ensures that the flow-matching model remains calibrated and robust to out-of-distribution inputs.\n\nUnlike fixed or manually tuned noise levels, our approach adaptively adjusts $\\sigma_z$ per variable (channel-wise), which is crucial for scientific data where different physical variables exhibit distinct noise scales and predictability (e.g., temperature vs. radar reflectivity, as shown in Fig. 5 and Fig. 8).\n\nIn practice, to avoid abrupt changes, we update $\\sigma_z$ using an exponential moving average (EMA) every $M=10{,}000$ training steps $\\sigma_z \\leftarrow (1{-}\\beta)\\, \\sigma_z + \\beta\\, \\sigma_z^{\\mathrm{cur}}$\nThe final $\\sigma_z$ is fixed for inference. An ablation in Sec. 5.3 (Table 4) demonstrates the effectiveness of this adaptive noise scaling in improving ensemble calibration and spread-skill alignment.\n\n---\n\n*4. Too many typos...*\n\n**Response.** Thank you for noticing the typos. We have corrected them in the revised manuscript.\n\n---\n\n\n*5.Where is definition of  $\\lambda $?...\"*\n\n**Response.** Thank you for your comment. Currently  $\\lambda $ is introduced in Sec. 4.3. It balances the deterministic regression and stochastic diffusion terms. In the limit  $\\lambda \\rightarrow \\infty $, AFM reduces to a purely deterministic model. We have added this clarification to the main text in the revised manuscript.\n\n--- \n\nWe hope our responses and revisions have adequately addressed your concerns.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes, I checked all the mathematical derivations and proofs, including Appendix B and C for the derivation of denoising objective.", "experimental_designs_or_analyses": "I reviewed the experimental design and analyses presented in the paper, including the methodology for super-resolving small-scale physics in weather data using stochastic flow matching (SFM). Specifically, I examined how the authors addressed challenges such as data misalignment, multiscale dynamics, and limited data availability. I also assessed the effectiveness of the encoder-latent space mapping, flow matching process, and adaptive noise scaling in balancing deterministic and stochastic components.", "relation_to_broader_scientific_literature": "The paper's contribution lies in applying the well-established techniques of flow matching and denoising diffusion modeling to enhance the resolution of small-scale details in natural images, particularly in weather science. While similar to CorrDiff, the proposed method introduces joint training of a regression encoder and residual diffusion, effectively mitigating potential overfitting issues that may arise in the two-stage training approach adopted in CorrDiff.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Weakness.\n1. In my opinion, VAE is a more natural choice for stochastic encoding; however, the authors employ a deterministic encoder and subsequently inject noise into the latent variable. I find the intuition behind this approach questionable, as the explanation provided in the paper is not entirely convincing to me.\n2. The novelty of the paper is questionable given the well-established connection between flow matching and diffusion modeling. The proposed approach bears a strong resemblance to CorrDiff, although the authors argue that CorrDiff employs a two-stage training process, where the regression encoder is trained first, followed by the diffusion of residual components. In contrast, this paper appears to adopt a joint training strategy, with the final loss formulated as the sum of both components.\n3. Lack of detailed explanation of noise scaling, since it is one of the main contributions of the paper, as stated by the authors.", "other_comments_or_suggestions": "1. Too many typos occurs in the submission, e.g. Conclusions Para.2. \n2. Where is definition of $\\mathcal{D}_{\\theta}$? I highly recommend to clarify it in the main context.", "questions_for_authors": "See \"Weaknesses\".", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3xGl2AaLJb", "reviewer": "Reviewer_X7XZ", "summary": "Applying conditional diffusion (CDM) and flow matching (FM) to natural images is very effective for super-resolving small-scale details, like the image semantic- or geometric information. However, CDM and FM will have difficulties with physical sciences, particularly for weather, mainly due to 1) spatially misaligned input-output (super-res weather input); 2) misaligned and distinct input-output channels (various channels sensor data); 3) stochasticity in channels resulting in multi-scales issue. To alleviate these challenges for weather application scenarios, this paper proposes first to encode various inputs to a latent distribution that is closer to the target and reduce input differences. Then the authors map the small-scale physics with Flow Matching. So Flow Matching adds stochastic details while the encoder adds deterministic ones, an adaptive noise scaling mechanism is utilized to dynamically adjust the noise scale at specific iterations to reduce the residual error and maintain the generalization. Extensive experiments on benchmarks, like real-world weather data  (25 km to 2 km scales in Taiwan) and in synthetic Kolmogorov flow datasets, validate the effectiveness of the proposed method, Adaptive Flow Matching.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their remarks and for recognizing the extensive experimental validation of our work.\n\n*1. When we compare the visually generated images between SFM and CorrDiff, it seems CorrDiff is better. Can the authors provide more convincing explanations about the advantages of CorrDiff?*\n\n\n**Response.** While a single visual sample (e.g., Fig. 2) may suggest CorrDiff looks better, such examples can be misleading and not representative. A more robust comparison is provided in Fig. 3, where the spectral analysis shows that CorrDiff underperforms AFM at both low and high frequencies. Moreover, quantitative results in Table 2 demonstrate that AFM achieves better reconstruction accuracy and significantly improved uncertainty calibration.  In the revised manuscript appendix, we have included another case in each of Figures 9 and 10 to shed more light in this comparison.\n\n---\n\n*2a. Though they obtain competitive performances and support theoretical proofs to demonstrate their method, it is not clear whether this method can be generalized or employed to other tasks, both physical science and nature image tasks. So this leaves a doubt about the wide application of this paper and I suggest the author can make some more demonstrations.*\n\n*2b.I am also curious whether the proposed adaptive flow matching works for nature images, and I do expect to see such validated experiments to make this paper stronger.*\n\n*2c.The applications to weather scenarios are limited, or we need to see large-scale validations to make this paper more convincing.*\n\n*2d. How do the main contributions of adaptive flow matching from interpolating the latent from common flow-matching based nature image methods?*\n\n**Response.** Thank you for the suggestion. While our method was motivated by challenges in scientific data, it makes no domain-specific assumptions and is broadly applicable. To demonstrate generality, we include two diverse testbeds: (1) real-world weather downscaling using Taiwan’s CWA dataset, and (2) a synthetic PDE-based Kolmogorov flow dataset. Extending AFM to natural image tasks is a promising direction, but beyond the scope of this already extensive study. We have added this discussion to the conclusion of the revised manuscript.\n\n---\n\n*4a. Regarding Fig 2, I guess the SFM shall be the method proposed in this paper.*\n\n*4b. SFM in Figures shall be well denoted, since we may think it shall be AFM.*\n\n**Response.** Thank you for noting this. We have fixed the AFM label in the figures of the revised manuscript. \n\nWe hope our responses and revisions have adequately addressed your remarks.", "claims_and_evidence": "Yes, the authors have supported necessary claims and evidence with step-by-step proofs.", "methods_and_evaluation_criteria": "Although the main contribution of this paper is to apply conditional diffusion and flow matching into a very downstream task, whether physical science, they propose some technical methods to resolve the problems raised by the application tasks and obtain competitive performances.", "theoretical_claims": "I have not checked each proof detail, presented in the Appendix, but the proofs in the main paper sound reasonable. For Eq (9), imposing interpolation like this formulation has been widely used in flow-matching based diffusion methods for nature image tasks.", "experimental_designs_or_analyses": "Regarding Fig 2, I guess the SFM shall be the method proposed in this paper.\n\nWhen we compare the visually generated images between SFM and CorrDiff, it seems CorrDiff is better. Can the authors provide more convincing explanations about the advantages of CorrDiff?\n\nI am also curious whether the proposed adaptive flow matching works for nature images, and I do expect to see such validated experiments to make this paper stronger.", "relation_to_broader_scientific_literature": "This paper aims to apply CDM and FM to physical science weather and conduct experiments on regional weather data, from 20km to 2km, and synthesis data. Though they obtain competitive performances and support theoretical proofs to demonstrate their method, it is not clear whether this method can be generalized or employed to other tasks, both physical science and nature image tasks. So this leaves a doubt about the wide application of this paper and I suggest the author can make some more demonstrations.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "Strengths:\n\n1. The presentation of this paper is well and extensive experiments on weather scenario validation have been conducted.\n\n2. The writing of this paper is also well-following.\n\nWeaknesses:\n\n1. The applications to weather scenarios are limited, or we need to see large-scale validations to make this paper more convincing.\n\n2. Whether the proposed adaptive flow matching works for the nature image task and needs to conduct further experiments.", "other_comments_or_suggestions": "1. SFM in Figures shall be well denoted, since we may think it shall be AFM.\n\n2. How do the main contributions of adaptive flow matching from interpolating the latent from common flow-matching based nature image methods?", "questions_for_authors": "Please see my overall comments and questions above, and I suggest the authors address them well.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper presents an adaptive flow matching approach for defining a super-resolution method adapted to small-scale physics. An application is proposed on a real world weather dataset in addition to synthetic Kolmogorov flow datasets.

During the evaluation strong points have been raised: the method is well presented and discussed, improvement convincing while limited, the paper illustrates a relevant practical use case. 
The following negative points have also been noticed: limited experimental evaluation, other baselines could have been considered, differences unclear with other methods notable CorrDiff.  

During rebuttal, authors have answered to reviewers' remarks, precising better the scope of the contribution and differences with other methods (notably CorrDiff).   
After the rebuttal and discussion, reviewer zyDu improves his score to weak accept, reviewers xRHs and X7XZ maintained their scores to weak accept and reviewer eJPb kept his score as accept.    

While a part of the methodology and some experiments were evaluated as a bit limited, the contribution, there was a consensus for saying that the results are convincing, the method is sound and well justified and the contribution can be interesting for a broad audience. The positive points outweigh the negative ones.  
I propose then acceptance.
Decision: Accept (poster)