Stochastic Flow Matching for Resolving Small-Scale Physics

## Abstract

Abstract Conditioning diffusion and flow models have proven effective for super-resolving small-scale details in natural images. However, in physical sciences such as weather, super-resolving small-scale details poses significant challenges due to: ( i ) 𝑖 (i) ( italic_i ) misalignment between input and output distributions (i.e., solutions to distinct partial differential equations (PDEs) follow different trajectories), ( i ⁢ i ) 𝑖 𝑖 (ii) ( italic_i italic_i ) multi-scale dynamics, deterministic dynamics at large scales vs. stochastic at small scales, and ( i ⁢ i ⁢ i ) 𝑖 𝑖 𝑖 (iii) ( italic_i italic_i italic_i ) limited data, increasing the risk of overfitting. To address these challenges, we propose encoding the inputs to a latent base distribution that is closer to the target distribution, followed by flow matching to generate small-scale physics. The encoder captures the deterministic components, while flow matching adds stochastic small-scale details. To account for uncertainty in the deterministic part, we inject noise into the encoder’s output using an adaptive noise scaling mechanism, which is dynamically adjusted based on maximum-likelihood estimates of the encoder’s predictions. We conduct extensive experiments on both the real-world CWA weather dataset and the PDE-based Kolmogorov dataset, with the CWA task involving super-resolving the weather variables for the region of Taiwan from 25 km to 2 km scales. Our results show that the proposed stochastic flow matching (SFM) framework significantly outperforms existing methods such as conditional diffusion and flows.

## 1 Introduction

Resolving small-scale physics is crucial in many scientific applications (Wilby et al., ; Rampal et al., ; ). For instance, in the 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 et al., ; Batzolis et al., ; Hoogeboom et al., ) has recently been extended to super-resolving small-scale physics (Aich et al., ; Ling et al., ). However, this task faces significant challenges:
(*C*1) Input and target data are often spatially misaligned due to differing PDE solutions operating at various resolutions, leading to divergent trajectories. Additionally, the input and target variables (channels) often represent different physical quantities, causing further misalignment. (*C*2) the data exhibits multiscale dynamics, 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 (*C*3) the record of Earth observations is somewhat limited compared to the natural image datasets.

Few efforts have been made to directly address these challenges in generative learning. Prior work typically relies on *residual* learning approaches (Mardani et al., ; Zhao et al., ). For instance, the method proposed in Mardani et al. () introduces a two-stage process where the deterministic component is first learned through regression, followed by applying diffusion on the residuals to capture the small-scale physics. While this approach offers a way to separate deterministic and stochastic components, it poses a significant risk of overfitting. The initial regression stage may overfit the training data, leading to poor generalization, especially as the data is limited, and thus fails to adequately represent the variability of the small-scale dynamics when training the diffusion model in the second stage. Additionally, this two-stage method lacks a principled way to handle the uncertainty inherent in both the deterministic and stochastic components.

To address these limitations, we propose an end-to-end approach based on 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 deterministic component, which is then augmented with noise to introduce uncertainty. The deterministic prediction is based on the idea that physical processes occur on different time scales, with larger-scale physics having longer-term, more deterministic effects. We refer to this method as stochastic flow matching (SFM), where the stochasticity is controlled by the noise injected at the encoder output. Proper tuning of the noise scale is critical to balance deterministic and stochastic dynamics. To achieve this, we employ a 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.

Figure: Figure 1: Overview of the Stochastic Flow Matching (SFM) Method. 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 to its more accurate location, and generating radar data. From the latent space, FM generates small-scale physics by transporting samples from $p({\mathbf{z}})$ to $p({\mathbf{x}})$ via the velocity field $\boldsymbol{\nu}(\mathbf{x},t)$.
Refer to caption: x1.png

SFM can be viewed through the lens of diffusion models, efficiently implemented using a stable denoising objective. We conduct extensive experiments on both idealized and realistic datasets. For the realistic data, we use the same data as Mardani et al. (), the best estimates of the 25km and 2km observed atmospheric state available from meteorological agencies, centered around a region containing Taiwan.
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 SFM consistently outperforms existing methods across various skill metrics.

All in all, the main contributions of this paper are summarized as follows:

- •
Stochastic Flow Matching (SFM): A method for matching *spatially* misaligned data (plus misaligned *channels*) with multiscale physics, specifically tailored for data-limited regimes in physical sciences.
- •
Adaptive Noise Scaling: We design an adaptive noise scaling based on the maximum likelihood criterion, which optimally balances the learning of deterministic and stochastic components between the encoder and flow matching.
- •
Extensive Experiments: We conduct extensive experiments on multi-scale weather data products as well as synthetic PDE datasets. Our results show that as the degree of misalignment increases, conditional diffusion and flow models perform progressively worse, while our end-to-end SFM framework with adaptive noise avoids overfitting.

**Table 1: Comparison between SFM and alternatives for learning the generative map between misaligned data (${\mathbf{y}},{\mathbf{x}}$). Define $\boldsymbol{\epsilon}\sim\mathcal{N}(0,1)$, $\boldsymbol{e}:=(\mathcal{E}({\mathbf{y}})-{\mathbf{x}})/\sigma_{z}$, and $\mathbf{r}:={\mathbf{x}}-\mathbb{E}[{\mathbf{x}}|{\mathbf{y}}]$. The noise scale $\sigma_{z}$ is the ML noise estimate ensuring $\mathbb{E}[|\boldsymbol{e}|^{2}]=1$. CDM and CFM represent conditional diffusion and flow models, respectively.**
| Scheme | Perturbation Kernel | Score | Train Loss (Denoising) |
| --- | --- | --- | --- |
| CFM | ${\mathbf{x}}_{t}=(1-t)\boldsymbol{\epsilon}+t{\mathbf{x}}$ | $\nabla_{{\mathbf{x}}_{t}}\log p_{t}({\mathbf{x}}_{t}|{\mathbf{y}})$ | $\mathbb{E}\big{[}\|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}<br>{0,0,0}\mathcal{D}_{{\boldsymbol{\theta}}}}({\mathbf{x}}_{t};t)-{\mathbf{x}}\|<br>^{2}\big{]}$ |
| CDM | ${\mathbf{x}}_{t}={\mathbf{x}}+\sigma_{t}\boldsymbol{\epsilon}$ | $\nabla_{{\mathbf{x}}_{t}}\log p_{t}({\mathbf{x}}_{t}|{\mathbf{y}})$ | $\mathbb{E}\big{[}\|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}<br>{0,0,0}\mathcal{D}_{{\boldsymbol{\theta}}}}({\mathbf{x}}_{t};t)-{\mathbf{x}}\|<br>^{2}\big{]}$ |
| CorrDiff | $\mathbf{r}_{t}=\mathbf{r}+\sigma_{t}\boldsymbol{\epsilon}$ | $\nabla_{\mathbf{r}_{t}}\log p_{t}(\mathbf{r}_{t}|{\mathbf{y}})$ | $\mathbb{E}\big{[}\|\boldsymbol{e}\|^{2}\big{]}\rightarrow\mathbb{E}\big{[}\|{<br>\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathcal{D}_{<br>{\boldsymbol{\theta}}}}(\mathbf{r}_{t};t)-\mathbf{r}\|^{2}\big{]}$ |
| SFM (Ours) | ${\mathbf{x}}_{t}={\mathbf{x}}+\sigma_{t}(\boldsymbol{e}+\boldsymbol{\epsilon})$ | $\nabla_{{\mathbf{x}}_{t}}p_{t}({\mathbf{x}}_{t})$ | $\mathbb{E}\big{[}(\sigma_{z}/\sigma_{t})^{2}\|{\color[rgb]{0,0,0}\definecolor[<br>named]{pgfstrokecolor}{rgb}{0,0,0}\mathcal{D}_{{\boldsymbol{\theta}}}}({<br>\mathbf{x}}_{t};t)-{\mathbf{x}}\|^{2}+\lambda\|\boldsymbol{e}\|^{2}\big{]}$ |

## 2 Related Work

Conditional diffusion and flow models. Conditioning is a powerful technique for paired image-to-image translation in diffusion and flow models (Batzolis et al., ; Kawar et al., ; Xingjian & Xie, ). It is commonly used in image restoration tasks such as super-resolution and deblurring, where the goal is to map a low-quality input to a high-quality target, with corresponding pixel associations between input and target. However, as our experiments demonstrate (see section 5), plain conditional models often yield suboptimal performance when data is severely misaligned.

Diffusion Bridges and Stochastic Interpolants. Diffusion bridges (De Bortoli et al., ; Shi et al., ; Liu et al., ; Pooladian et al., ) facilitate transitions between distributions but rely on assumptions, such as local alignment in I^2SB (Liu et al., ), unsuitable for misaligned data. Stochastic interpolants (Albergo et al., ; Albergo & Vanden-Eijnden, ; Lipman et al., ; Liu et al., ) assume smooth transport and are often trained with independent coupling between noise and data, even for the conditional problems. Notably, Albergo et al. () couples base and target distributions. Chen et al. () applies stochastic interpolants to probabilistic forecast of fluid flow. In contrast, our SFM method targets scenarios with misalignment between input and output channels. Specifically, the encoder learns the base distribution from the target’s large-scale deterministic dynamics, while adaptively balancing the contributions of deterministic and stochastic components.

Atmospheric super-resolution (downscaling). In atmospheric sciences, the task of going from coarse to fine resolution is known as *downscaling*. Several works have explored statistical downscaling using machine learning techniques (see, e.g., Rampal et al. (); Wilby et al. (); Rampal et al. ()). One state-of-the-art method is CorrDiff Mardani et al. (), which applies a diffusion model to the residuals left from a deterministic prediction to achieve multivariate super-resolution and new channel synthesis. However, as previously noted, this two-stage approach is prone to severe overfitting, especially in data-limited regimes.

## 3 Background and Problem Statement

Consider the task of learning the conditional distribution $p({\mathbf{x}}|{\mathbf{y}})$ from a finite set of paired data $\{({\mathbf{y}}_{i},{\mathbf{x}}_{i})\}_{i=1}^{N}$, where ${\mathbf{y}}\in\mathbb{R}^{c\times h\times w}$ and ${\mathbf{x}}\in\mathbb{R}^{C\times H\times W}$ represent the input and target, respectively. For example, in atmospheric sciences, ${\mathbf{y}}$ could be a coarse-resolution forecast from the Global Forecast System (GFS) at 25 km resolution, while ${\mathbf{x}}$ is the fine-resolution target from a high resolution regional weather simulation system at 2 km.

This task is particularly challenging because the pairs $({\mathbf{y}},{\mathbf{x}})$ are often misaligned. First, these pairs might represent 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 dynamics. Second, the input and target may involve different channels, e.g., corresponding to distinct atmospheric variables, further complicating the learning process.

Before diving into the details of our proposed solution, it is helpful to briefly review flow matching and elucidated diffusion models, which are critical components of our approach.

Flow Matching (FM): Flow matching learns the transformation between two probability distributions by modeling a velocity field $\boldsymbol{\nu}(\mathbf{x},t)$ that transports samples from a source distribution $p_{0}(\mathbf{x})$ to a target distribution $p_{1}(\mathbf{x})$. In practice, flows are often trained using a linear interpolant between noise and data. The forward process is described by the ODE:

$$ $\frac{d\mathbf{x}_{t}}{dt}=\boldsymbol{\nu}(\mathbf{x}_{t},t),$ (1) $$

where $\boldsymbol{\nu}(\mathbf{x}_{t},t)$ represents the velocity field over time $t\in[0,1]$. For the linear interpolant, the true velocity that generates a single data sample $\mathbf{x}_{1}$ is given by $\boldsymbol{\nu}_{\text{true}}(\mathbf{x}_{t},t)=\mathbf{x}_{1}-\mathbf{x}_{0}$.
The goal is to minimize the discrepancy between the learned velocity field $\boldsymbol{\nu}_{\theta}(\mathbf{x}_{t},t)$ and this true velocity per sample:

$$ $\min_{\theta}\mathbb{E}_{t,\mathbf{x}_{t}}\left[\left\|\boldsymbol{\nu}_{ \theta}(\mathbf{x}_{t},t)-(\mathbf{x}_{1}-\mathbf{x}_{0})\right\|^{2}\right].$ (2) $$

where $\mathbf{x}_{t}:=(1-t)\mathbf{x}_{0}+t\mathbf{x}_{1}$.
Upon convergence, the learned velocity is used to generate samples by sampling $\mathbf{x}_{0}\sim p_{0}(\mathbf{x})$ and solving the ODE in Eq. 1.

Diffusion Models (DM). Diffusion models generate data by transforming a base distribution, often Gaussian noise, into the target data distribution $p_{0}(\mathbf{x})$. In the forward diffusion process, Gaussian noise with standard deviation $\sigma$ is added to the data, producing a sequence of distributions $p_{0}(\mathbf{x};\sigma)$. As $\sigma$ increases, the data distribution approaches pure noise. The backward process then denoises samples, starting from noise drawn from $\mathcal{N}(0,\sigma_{\max}^{2}\mathbf{I})$ and progressively reducing the noise to recover the data distribution.

Considering the variance-exploding elucidated diffusion model (EDM), both the forward and backward processes are described by SDEs. The forward SDE is:

$$ $d\mathbf{x}_{t}=\sqrt{2\dot{\sigma}(t)\sigma(t)}d\boldsymbol{\omega}(t),$ (3) $$

while the backward SDE is:

$$ $d\mathbf{x}_{t}=-2\dot{\sigma}(t)\sigma(t)\nabla_{\mathbf{x}_{t}}\log p( \mathbf{x}_{t};\sigma(t))dt+\sqrt{2\dot{\sigma}(t)\sigma(t)}d\boldsymbol{ \omega}(t).$ (4) $$

In EDM, denoising score matching is used to learn the score function $\nabla_{\mathbf{x}}\log p(\mathbf{x};\sigma)$, essential for the reverse diffusion process. A denoising neural network $D_{\theta}(\mathbf{x};\sigma)$ is trained as:

$$ $\min_{\theta}\mathbb{E}_{\mathbf{x}\sim p_{0}}\mathbb{E}_{\sigma\sim p_{\sigma }}\mathbb{E}_{\mathbf{n}\sim\mathcal{N}(0,\sigma^{2}\mathbf{I})}\left[\|{ \color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathcal{D}_{ {\boldsymbol{\theta}}}}(\mathbf{x}+\mathbf{n};\sigma)-\mathbf{x}\|^{2}\right].$ (5) $$

The score function is constructed later via $\nabla_{{\bf x}}\log p({\bf x};\sigma)=({\color[rgb]{0,0,0}\definecolor[named]
{pgfstrokecolor}{rgb}{0,0,0}\mathcal{D}_{{\boldsymbol{\theta}}}}{\bf x};\sigma
)-{\bf x})/\sigma^{2}$.

Connection between FM and Diffusion Models. While both methods aim to transform distributions, flow matching does so using ODEs for deterministic evolution, while EDM leverages 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; see e.g., Karras et al. (); Song et al. (). For the simple noise schedule $\sigma(t)=\sqrt{t}$, the ODE for continuous-time EDM is:

$$ $\frac{d\mathbf{x}_{t}}{dt}=\frac{\mathbf{x}_{t}-{\color[rgb]{0,0,0} \definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathcal{D}_{{\boldsymbol{ \theta}}}}(\mathbf{x}_{t};t)}{t},$ (6) $$

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 $t=1$ to $t=0$, whereas the flow matching process proceeds in the opposite direction.

## 4 Stochastic Flow Matching for Conditional Generation

To learn the conditional distribution $p(\mathbf{x}|\mathbf{y})$, one approach is to use conditional diffusion or flow models. These models have been successful in image-to-image tasks like image restoration or super-resolution, where conditioning provides rich information about the target (Saharia et al., ). However, traditional methods struggle when the input $\mathbf{y}$ and target $\mathbf{x}$ are significantly misaligned (see section 5 for evidence). To address this, we propose a multiscale approach:

Deterministic Dynamics: The input $\mathbf{y}$ is encoded into a latent variable $\mathbf{z}=\mathcal{E}(\mathbf{y})$. This encoding serves two purposes: (1) it spatially aligns the base and target distributions by matching their large-scale dynamics, and (2) it aligns the *channels* by projecting the input into the same space as the output (since ${\mathbf{y}}$ and ${\mathbf{x}}$ represent different weather variables).

Generative Dynamics: Flow matching is then used to transform the base distribution $p(\mathbf{z})$ into the target distribution $p(\mathbf{x})$. To account for uncertainty in the encoding phase, we perturb the latent variable with Gaussian noise:

$$ $\mathbf{z}=\mathcal{E}(\mathbf{y})+\sigma_{z}\boldsymbol{\epsilon},\quad \boldsymbol{\epsilon}\sim\mathcal{N}(0,\mathbf{I}).$ $$

In the following sections, we will detail the learning process for both the encoder and flow matching.

### 4.1 Training

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 ${\mathbf{x}}_{t}=(1-t){\mathbf{z}}+t{\mathbf{x}}$, where ${\mathbf{z}}$ is the encoded state and ${\mathbf{x}}$ is the target, for $t\in[0,1]$. Consequently, based on eq. 2, the flow matching objective is formulated as:

$$ $\min_{\mathcal{E},{\boldsymbol{\theta}}}{\mathbb{E}}_{t,{\mathbf{x}},{\mathbf{ z}}\sim\mathcal{N}(\mathcal{E}(\mathbf{y}),\sigma_{z})}\left[\|\boldsymbol{\nu }_{\theta}({\mathbf{x}}_{t},t)-({\mathbf{x}}-{\mathbf{z}})\|^{2}\right].$ (7) $$

Due to the stochasticity in ${\mathbf{z}}$ and the stability of the EDM framework for training diffusion models—along with its tuning-free characteristics—it is advantageous to incorporate denoising through EDM when training flow matching. To this end, the first step is to recast the linear interpolant as a Gaussian diffusion process (cf. section 4):

$$ $\displaystyle{\mathbf{x}}_{t}=(1-t)\mathcal{E}({\mathbf{y}})+t{\mathbf{x}}+(1- t)\sigma_{z}\boldsymbol{\epsilon},\quad\quad t\in[0,1]$ (8) $$

The following proposition simplifies the task of learning the velocity field as a denoising process.

Proposition 1. For the perturbation model ${\mathbf{x}}_{\sigma}={\mathbf{x}}+\sigma\boldsymbol{e}+\sigma\boldsymbol{\epsilon}$, where the noise standard deviation is given by $\sigma:=(1-t)\sigma_{z}$, the residual error by $\boldsymbol{e}:=(\mathcal{E}({\mathbf{y}})-{\mathbf{x}})/\sigma_{z}$, and the noise $\boldsymbol{\epsilon}\sim\mathcal{N}(0,1)$, the flow matching for joint training of the encoder and flow reduces to the denoising objective:

$$ $\displaystyle\min_{\mathcal{E},{\boldsymbol{\theta}}}\mathbb{E}_{{\mathbf{x}}, {\mathbf{y}},\sigma\sim\mathcal{U}[0,\sigma_{z}]}\left[(\sigma_{z}/\sigma)^{2} \left\|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \mathcal{D}_{{\boldsymbol{\theta}}}}({\mathbf{x}}_{\sigma},\sigma)-{\mathbf{x} }\right\|^{2}\right].$ (9) $$

Intuitively, this denoising objective addresses not only the Gaussian noise typical of diffusion models but also residual errors introduced during the deterministic encoding process. Note that the residual error $\boldsymbol{e}$ conveys the essential information about the input conditioning ${\mathbf{y}}$ required for generating the target ${\mathbf{x}}$. Therefore, it is crucial to carefully balance the influence of this error by appropriately selecting $\sigma_{z}$ and applying regularization to the encoding process. These considerations will be discussed next.

### 4.2 Adaptive Noise Scaling

It is essential to tune the noise parameter $\sigma_{z}$ based on the data before applying diffusion denoising. Specifically, we consider the latent variable:

$$ ${\mathbf{z}}=\mathcal{E}({\mathbf{y}})+\sigma_{z}\boldsymbol{\epsilon},$ (10) $$

where we observe ${\mathbf{x}}$ and ${\mathbf{y}}$, and aim to adjust $\sigma_{z}$ in a maximum likelihood (ML) sense so that ${\mathbf{z}}$ aligns closely with ${\mathbf{x}}$. In this context, $\sigma_{z}$ controls the scale of the noise added to the encoder’s output $\mathcal{E}({\mathbf{y}})$. By leveraging the ML estimator for $\sigma_{z}$, we can derive it as the root-mean-square-error (RMSE) of the unnormalized residual error, namely

$$ $\sigma_{z}=\sqrt{\mathbb{E}[\|{\mathbf{x}}-\mathcal{E}({\mathbf{y}})\|^{2}]}.$ (11) $$

While this noise scale tuning can be performed offline during the validation phase, it can also be executed online during the training process by dynamically updating $\sigma_{z}$ based on the RMSE of the encoder as training progresses. Intuitively, if the deterministic regression model overfits to a small training dataset, the validation RMSE will grow and thus, our model will adaptively use a higher noise scale in the output of the encoder. At iteration $k$, one can adaptively select $\sigma_{z}(k)$ using an exponential moving average (EMA), ensuring that the noise scale is continuously updated to reflect the model’s performance over time. The adaptive noise scale at iteration $k$ is defined as:

$$ $\sigma_{z}\leftarrow(1-\beta)\sigma_{z}+\beta\sigma_{z}(k),$ (12) $$

where $\beta>0$ is a small smoothing factor that determines the influence of the current estimate compared to previous iterations. This adaptive scheduling mechanism ensures that the noise parameter $\sigma_{z}$ is not static but evolves with the model, potentially improving convergence and performance during training.

### 4.3 Encoder Regularization

Another effective approach to control the residual error $\boldsymbol{e}$ is through regularization. Specifically, one can impose a regression-based regularization term on the encoder, encouraging the output of the encoder to approximate the target, i.e., ${\mathbf{x}}\approx\mathcal{E}({\mathbf{y}})$, thus minimizing the residual error $\boldsymbol{e}$. In an ideal scenario, this would lead to $\boldsymbol{e}\approx 0$. 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 $\lambda$. This weight balances the trade-off between reducing the residual error and maintaining the generalization ability of the model. The resulting objective function becomes:

$$ $\displaystyle\min_{\mathcal{E},{\boldsymbol{\theta}}}\mathbb{E}_{{\mathbf{x}}, {\mathbf{y}},\sigma\sim\mathcal{U}[0,\sigma_{z}]}\left[(\sigma_{z}/\sigma)^{2} \left\|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \mathcal{D}_{{\boldsymbol{\theta}}}}({\mathbf{x}}_{\sigma},\sigma)-{\mathbf{x} }\right\|^{2}+\lambda\left\|\boldsymbol{e}\right\|^{2}\right].$ (13) $$

where the encoder and denoiser are trained jointly to minimize the denoising and regression losses.

### 4.4 Connections to Residual Learning

Consider the case where we have a pre-trained encoder $\mathcal{E}$, which has been trained using a supervised regression loss, for instance. In this scenario, starting from the forward diffusion process in eq. 8, if we subtract both sides by the encoder output $\mathcal{E}({\mathbf{y}})$ and define the residual error as $\boldsymbol{e}_{t}:={\mathbf{x}}_{t}-\mathcal{E}({\mathbf{y}})$, 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:

$$ $\displaystyle\boldsymbol{e}_{t}=t\boldsymbol{e}+(1-t)\boldsymbol{\epsilon}, \quad\quad t\in[0,1],$ (14) $$

where $\boldsymbol{e}$ represents the residual error between the target ${\mathbf{x}}$ and the encoder output $\mathcal{E}({\mathbf{y}})$, and $\boldsymbol{\epsilon}\sim\mathcal{N}(0,1)$ is the noise.

This simple process facilitates the construction of a backward process, where one can learn the velocity field by minimizing the flow matching loss in eq. 2. This approach closely mirrors the CorrDiff method proposed in Mardani et al. (), which leverages residual learning to 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 2, the initial supervised training of the encoder often leads to near-perfect matching between $\mathcal{E}({\mathbf{y}})$ and ${\mathbf{x}}$. While this may seem desirable, it can result in overfitting and poor generalization performance. Therefore, balancing this residual learning approach with appropriate regularization is critical to maintaining the model’s ability to generalize effectively to unseen data.

### 4.5 Sampling

Once the velocity field $\boldsymbol{\nu}_{\theta}(\mathbf{x}_{t},t)=({\mathbf{x}}-{\color[rgb]{0,0,0}
\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathcal{D}_{{\boldsymbol{
\theta}}}}({\mathbf{x}}_{\sigma},\sigma))/(1-t)$ is learned using algorithm 1, sampling simply requires integrating the flow forward in time based on the ODE formulation in eq. 1. The forward integration process from $t=0$ to $t=1$ can be expressed as:

$$ ${\mathbf{x}}_{1}=\int_{t=0}^{t=1}\boldsymbol{\nu}_{\theta}(\mathbf{x}_{t},t)\,dt$ (15) $$

In practice, this integration can be approximated using Euler steps, which are detailed in algorithm 2 This algorithm outlines the step-by-step procedure for forward sampling through time.

Figure: Algorithm 1 SFM training

Figure: Algorithm 2 SFM sampling

## 5 Experiments

We evaluate the performance of the proposed Stochastic Flow Matching (SFM) model on two datasets: Multiscale Kolmogorov Flow and a regional weather downscaling dataset. The weather dataset includes real-world complexity with meteorological observations from Taiwan’s Central Weather Administration (CWA). The Multiscale Kolmogorov flow dataset is an idealized set designed to capture misalignment in the Taiwan data. Both datasets present significant downscaling challenges due to scale misalignment, mixed dynamics, and channel-specific variability.

Baselines. We compare our SFM model against several baseline approaches, including deterministic and generative methods:

- •
Regression: A standard convolution network (UNet or $1\times 1$ conv) model trained to predict high-resolution outputs from low-resolution input data using MSE loss training. This serves as a purely deterministic approach, representing a baseline for direct super-resolution.
- •
Conditional Diffusion Model (CDM): CDM maps Gaussian noise to the high-resolution space while conditioned on the low-resolution input. CDM does not explicitly model the deterministic component in the data.
- •
Conditional Flow Matching (CFM): A variant of flow matching that interpolates between a Gaussian sample and a data sample, building deterministic mappings.
- •
Corrective Diffusion Models (CorrDiff) (Mardani et al., ): A UNet-based regression network is first trained on pairs $\{({\mathbf{x}}_{i},{\mathbf{y}}_{i})\}$ using MSE loss to learn the mean $\mathbb{E}[{\mathbf{x}}|{\mathbf{y}}]$. The residual error $\boldsymbol{e}:={\mathbf{x}}-\mathbb{E}[{\mathbf{x}}|{\mathbf{y}}]$ is then used to train the diffusion model on the residuals.

Unlike our SFM model, which starts from the low-resolution input and learns the stochastic dynamics in the latent space, both CDM and CFM aim to map Gaussian noise directly 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. () and Saharia et al. (). By including these baselines, we evaluate the strengths of our approach against established deterministic and generative methods.

Evaluation Metrics. We report performance using standard metrics such as RMSE, Continuous Ranked Probability Score (CRPS), Mean Absolute Error (MAE), and Spread Skill Ratio (SSR). These metrics provide a comprehensive assessment of both the accuracy and uncertainty quantification of the model’s predictions. RMSE, CRPS, and MAE measure the estimation error while SSR evaluates the model calibration. To calculate CRPS and SSR we produce 64 ensemble members using different seeds. These metrics are discussed in Section A.3.

Network Architecture, Training, and Sampling. For diffusion model training and sampling, we use EDM Karras et al. (), a continuous-time diffusion model available with a public codebase. EDM provides a physics-inspired design based on ODEs, auto-tuned for our scenario (see Table 1 in Karras et al. ()). We adopt most of the hyperparameters from EDM and make modifications as detailed in Section A.2. Note that EDM is used for CDM and CFM as well in consistent manner.

Figure: Figure 2: SFM vs. baselines for different weather variables. SFM generates more physically consistent outputs, while UNet output appears blurred, 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.
Refer to caption: extracted/5935767/CWA_visual_baselines_plus_SFM_with_corrdiff_10.jpg

Figure: (a) Radar Reflectivity
Refer to caption: x2.png

### 5.1 Regional Downscaling for Taiwan

We focus on the task of super-resolving (downscaling) multiple weather variables for the Taiwan region, a challenging meteorological regime. The input coarse-resolution data at a 25 km scale comes from ERA5 Hersbach et al. (), while the target fine-resolution 2 km scale data is sourced from the Central Weather Administration (CWA) Central Weather Administration (2021) (). For evaluation, we use a common set of 205 randomly selected out-of-sample date and time combinations from 2021. Metrics and spectra are computed to compare SFM with baseline models. We utilize a 32-member ensemble; larger ensembles do not significantly alter the key findings. A detailed data description is provided first, followed by our observations.

Dataset. The dataset for this study is derived from ERA5 reanalysis data Hersbach et al. (), focusing on 12 variables including temperature, wind components, and geopotential height at two pressure levels, as well as surface-level variables like 2-meter temperature and total column water vapor. The target output data Central Weather Administration (2021) () covers a 900 × 900 km region around Taiwan on a 448 × 448 grid. Hourly observations span four years (2018-2021), split into training (2018-2020) and evaluation (2021) sets. Input data is upsampled using bi-linear interpolation for model consistency Hu et al. (); Zhang et al. (). See Section A.4.1 for the detailed discussion of the datasets.

**Table 2: SFM vs. Baselines for CWA Downscaling: Values in bold show the best performance. SFM outperforms baselines except for the *deterministic* temperature variable, where CorrDiff excels. Temperature, being the most deterministic, benefits from CorrDiff’s fully deterministic predictions. While SFM could match CorrDiff using a UNet encoder and higher $\lambda$, this would compromise stochastic predictions for other variables.**
| Variable | Model | CorrDiff | CFM | CDM | Reg. (UNet) | SFM ($1\times 1$ conv.) |
| --- | --- | --- | --- | --- | --- | --- |
| Radar | RMSE $\downarrow$ | 5.08 | 5.06 | 5.70 | 5.09 | 4.90 |
| CRPS $\downarrow$ | 1.89 | 1.88 | 2.39 | - | 1.78 |  |
| MAE $\downarrow$ | 2.50 | 2.46 | 2.78 | 2.50 | 2.42 |  |
| SSR $\rightarrow 1$ | 0.38 | 0.33 | 0.46 | - | 0.44 |  |
| Temperature | RMSE $\downarrow$ | 0.83 | 0.93 | 0.95 | 0.86 | 1.00 |
| CRPS $\downarrow$ | 0.50 | 0.58 | 0.54 | - | 0.52 |  |
| MAE $\downarrow$ | 0.60 | 0.72 | 0.70 | 0.62 | 0.67 |  |
| SSR $\rightarrow 1$ | 0.36 | 0.41 | 0.52 | - | 0.47 |  |
| East. Wind | RMSE $\downarrow$ | 1.47 | 1.45 | 1.62 | 1.49 | 1.44 |
| CRPS $\downarrow$ | 0.85 | 0.82 | 0.93 | - | 0.80 |  |
| MAE $\downarrow$ | 1.07 | 1.06 | 1.24 | 1.09 | 1.07 |  |
| SSR $\rightarrow 1$ | 0.43 | 0.50 | 0.61 | - | 0.61 |  |
| North. Wind | RMSE $\downarrow$ | 1.66 | 1.61 | 1.84 | 1.66 | 1.61 |
| CRPS $\downarrow$ | 0.95 | 0.90 | 1.06 | - | 0.88 |  |
| MAE $\downarrow$ | 1.20 | 1.16 | 1.41 | 1.21 | 1.17 |  |
| SSR $\rightarrow 1$ | 0.41 | 0.49 | 0.58 | - | 0.58 |  |

#### 5.1.1 Results

The performance of SFM is compared with various alternatives, and both deterministic and probabilistic skills are reported in Table 2. We examine three variants of SFM: with ($i$) small versus large encoders; ($ii$) with additional use of adaptive noise scaling; and ($iii$) conditioning in the large encoder limit. Notably, temperature is the most deterministic variable among the four listed, while radar is the most stochastic one.

Our main finding is that SFM consistently outperforms existing alternatives across different metrics for non-deterministic channels (radar and winds). For temperature, although SFM is not the top performer, we can tune $\lambda$ to larger values and achieve performance as good as CorrDiff. This however compromises the stochastic prediction for non-deterministic channels; see the ablation in Table 9 of the Appendix. The ablations are deffered to Section B.1 due to space limitations.

Spectral analysis is crucial for assessing fidelity at different scales in weather prediction. Fig. 3 shows that SFM’s spectra closely match the ground truth across variables. In contrast, the UNet-based regression scheme fails to generate high-frequency components. Interestingly, the conditional diffusion model (CDM), commonly used for image super-resolution, also lacks spectral fidelity.

Regarding the calibration of the generated ensemble (i.e., Spread Skill Ratio, SSR), SFM provides the best balance, being closest to 1.0. While SFM does not eliminate the overall problem of underdispersive super-resolution, it does improve the balance between the spread and RMSE skill of the generated ensemble, especially for surface wind channels.

### 5.2 Multiscale Kolmogorov-Flow

Our Multiscale Kolmogorov Flow dataset offers a simplified simulation of atmospheric dynamics, focusing on downscaling from a coarse to a fine grid while preserving physical structures.
Kolmogorov flow (KF) is a well-known scenario where 2D fluid flow in a doubly periodic domain is forced by spatially varying source of momentum.
To mimic the structure of the down-scaling problem we couple the KF flow ground truth to an otherwise unforced fluid system representing a coarse-resolution atmospheric simulation. The strength of this coupling $\tau$ controls how well the coarse simulation tracks the ground truth.
We are not aware of a similar 2D toy problem for the down-scaling problem, so this setup may be useful for other studies in the area.

Dataset. The dataset is constructed by simulating dynamics governed by a system of partial differential equations involving vorticity fields $\zeta_{l}$ and $\zeta_{h}$, coupled through parameters like $\tau$ and influenced by steady-state forcing. The simulation uses a pseudo-spectral method on a $512\times 512$ grid with a 3rd-order Adams-Bashforth time stepper. Different $\tau$ values (3, 5, 10) are used to generate training and test sets where higher values of $\tau$ looser coupling which translates to higher misalignment between low and high res simulations. Detailed descriptions of the equations and parameters are provided in Section A.4.2.

**Table 3: SFM vs. baselines for Kolmogorov Flow for various misalignment degrees $\tau$. SFM consistently demonstrates superior performance across varying levels of data misalignment, showcasing its robustness. While CDM exhibits greater variability, this comes at the expense of significantly reduced fidelity. Note that for deterministic models, CRPS is equivalent to MAE.**
| Metric | $\tau=3$ | $\tau=5$ | $\tau=10$ |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | CFM | CDM | UNet | SFM | CFM | CDM | UNet | SFM | CFM | CDM | UNet | SFM |
| RMSE $\downarrow$ | 0.98 | 1.13 | 1.15 | 0.73 | 0.96 | 0.94 | 1.14 | 0.76 | 1.22 | 1.24 | 1.36 | 1.09 |
| CRPS $\downarrow$ | 0.52 | 0.58 | - | 0.37 | 0.52 | 0.48 | - | 0.40 | 0.67 | 0.65 | - | 0.65 |
| MAE $\downarrow$ | 0.69 | 0.80 | 0.82 | 0.51 | 0.69 | 0.67 | 0.82 | 0.54 | 0.89 | 0.89 | 1.00 | 0.77 |
| SSR $\rightarrow 1$ | 0.54 | 0.69 | - | 0.62 | 0.58 | 0.70 | - | 0.58 | 0.56 | 0.76 | - | 0.23 |

Figure: Figure 4: SFM vs. Baselines for Kolmogorov Flow and $\tau=10$: when the figures are zoomed in, it is apparent that SFM aligns closer to the ground truth, and the presence of high-frequency artifacts in the baseline models becomes more noticeable.
Refer to caption: x6.png

#### 5.2.1 Results

SFM consistently outperforms other methods across various skill metrics for different degrees of misalignment, denoted by $\tau$ (Table 3). Interestingly, while CDM appears to be the most calibrated, SFM demonstrates superior performance overall. In this case study, we use a $1\times 1$ convolutional architecture for the SFM encoder. The ablations are deffered to Section B.2 due to limited space.

Representative examples in Fig. 9 illustrate that as the misalignment $\tau$ increases, the performance gap between SFM and the alternatives widens. Notably, for highly misaligned data with $\tau=10$, SFM is the only method capable of generating samples that closely match the ground truth, maintaining high fidelity.

This advantage is further supported by the spectral analysis in Fig. 10, where SFM’s spectra most closely align with the ground truth, highlighting its robustness in preserving physical structures even under significant misalignment.

## 6 Conclusion

We introduced Stochastic Flow Matching (SFM) to address misaligned data in atmospheric downscaling tasks. SFM combines deterministic encoding of large-scale dynamics with stochastic flow matching in latent space, effectively capturing both deterministic and stochastic components of the data. Experiments on synthetic and real-world datasets demonstrated that SFM outperforms existing methods, especially when input and target distributions are significantly misaligned.

A limitation of SFM is its reliance on paired datasets, which may not always be available. Future work includes extending SFM to handle unpaired or semi-supervised data and incorporating physical constraints to enhance the physical consistency of the outputs.

## Appendix A Appendix

### A.1 Proof of Proposition 1

Proposition 1. For the perturbation model ${\mathbf{x}}_{\sigma}={\mathbf{x}}+\sigma\boldsymbol{e}+\sigma\boldsymbol{\epsilon}$, where the noise standard deviation is given by $\sigma:=(1-t)\sigma_{z}$, the residual error by $\boldsymbol{e}:=(\mathcal{E}({\mathbf{y}})-{\mathbf{x}})/\sigma_{z}$, and the noise $\boldsymbol{\epsilon}\sim\mathcal{N}(0,1)$, the flow matching for joint training of the encoder and flow reduces to the denoising objective:

$$ $\displaystyle\min_{\mathcal{E},{\boldsymbol{\theta}}}\mathbb{E}_{{\mathbf{x}}, {\mathbf{y}},\sigma\sim\mathcal{U}[0,\sigma_{z}]}\left[(\sigma_{z}/\sigma)^{2} \left\|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0} \mathcal{D}_{{\boldsymbol{\theta}}}}({\mathbf{x}}_{\sigma},\sigma)-{\mathbf{x} }\right\|^{2}\right].$ (16) $$

Proof. Consider the linear interpolant connecting ${\mathbf{z}}\sim\mathcal{N}(\mathcal{E}({\mathbf{y}}),\sigma_{z}\mathbf{I})$ to the target distribution $p({\mathbf{x}})$. This process can be expressed as:

$$ $\displaystyle{\mathbf{x}}_{t}$ $\displaystyle=(1-t)\mathcal{E}({\mathbf{y}})+t{\mathbf{x}}+(1-t)\sigma_{z} \boldsymbol{\epsilon}$ (17) $\displaystyle=(1-t)(\mathcal{E}({\mathbf{y}})-{\mathbf{x}})+{\mathbf{x}}+(1-t) \sigma_{z}\boldsymbol{\epsilon}$ (18) $\displaystyle={\mathbf{x}}+(1-t)\sigma_{z}(\boldsymbol{e}+\boldsymbol{\epsilon }),$ (19) $$

where $\boldsymbol{e}=(\mathcal{E}({\mathbf{y}})-{\mathbf{x}})/\sigma_{z}$. Define $\sigma=(1-t)\sigma_{z}$ and reinterpret time $t$ in terms of the noise level $\sigma$, as is typical in continuous noise sampling (e.g., EDM). The perturbation kernel then becomes ${\mathbf{x}}_{\sigma}={\mathbf{x}}+\sigma(\boldsymbol{e}+\boldsymbol{\epsilon})$.

Now, consider the velocity training objective in (7), where the velocity is expressed as a score: $\boldsymbol{\nu}_{{\boldsymbol{\theta}}}({\mathbf{x}}_{t},t)=({\mathbf{x}}_{t}
-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\mathcal{D}
_{{\boldsymbol{\theta}}}}_{\theta}({\mathbf{x}}_{t};t))/(1-t)$. Since flow time progresses from $t=0$ to $t=1$, we incorporate the $(1-t)$ term. Substituting $\boldsymbol{\nu}_{{\boldsymbol{\theta}}}({\mathbf{x}}_{t},t)$ into the flow matching objective (7) and simplifying, we obtain the denoising objective in (19), completing the proof.

### A.2 Network Architecture and Training

For diffusion model training and sampling, we use EDM Karras et al. (), a continuous-time diffusion model available with a public codebase. EDM provides a physics-inspired design based on ODEs, auto-tuned for our scenario (see Table 1 in Karras et al. ()). We adopt most of the hyperparameters from EDM and make modifications as listed below.

Architecture: To cover the large field-of-view $448\times 448$, we adapt the UNet from Song & Ermon () 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. Attention resolution is 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 (); Carion et al. (). For the encoder $\mathcal{E}$, we use two architectures: (1) a simple $1\times 1$ convolution layer, and (2) a UNet similar to the diffusion UNet but without time embedding. The same UNet is used for the regression network in CorrDiff.

Optimizer: We use the Adam optimizer with a learning rate of $10^{-4}$, $\beta_{1}=0.9$, $\beta_{2}=0.99$, and an exponential moving average (EMA) rate of $0.5$. Dropout is applied with a rate of 0.13. Hyperparameters follow the guidelines in EDM Karras et al. ().

Noise Schedule: For SFM and CFM, we use a continuous noise schedule sampled uniformly $\sigma\in\mathcal{U}[0,\sigma_{z}]$. For CDM and CorrDiff, we use EDM’s optimized log-normal noise schedule, $\sigma\sim\text{lognormal}(-1.2,1.2)$.

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 to 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. Training is distributed across 8 DGX nodes, each with 8 A100 GPUs, using data parallelism and a total batch size of 512.

Sampling: Our sampling process employs Euler integration with 50 steps across all methods. We begin with a maximum noise variance $\sigma_{\text{max}}$ and decrease it to a minimum of $\sigma_{\text{min}}=0.002$. The value of $\sigma_{\text{max}}$ varies depending on the method: for CDM and CorrDiff, we use $\sigma_{\text{max}}=800$, as per the original implementation in Mardani et al. (); for CFM, we set $\sigma_{\text{max}}=1$, as specified in Lipman et al. (); and for SFM, we use the $\sigma_{z}$ value learned during training.

### A.3 Evaluation Metrics

#### A.3.1 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 (). In the context of our problem, let ${\mathbf{x}}$ be the true target and $\hat{{\mathbf{x}}}$ be the predicted value. The RMSE is defined as:

$$ $\text{RMSE}=\sqrt{\mathbb{E}\left[\|{\mathbf{x}}-\hat{{\mathbf{x}}}\|^{2} \right]}.$ (20) $$

This metric captures the average magnitude of the residuals, i.e., the difference between the predicted and true values. A lower RMSE indicates better model performance, as it suggests the predicted values are closer to the true values on average. RMSE is sensitive to large errors, making it an ideal choice for evaluating models where minimizing large deviations is critical.

#### A.3.2 CRPS

The Continuous Ranked Probability Score (CRPS) is a measure used to evaluate probabilistic predictions Wilks (). It compares the entire predicted distribution $F(\hat{{\mathbf{x}}})$ with the observed data point ${\mathbf{x}}$. For a probabilistic forecast with cumulative distribution function (CDF) $F$, and the true value ${\mathbf{x}}$, the CRPS is given by:

$$ $\text{CRPS}(F,{\mathbf{x}})=\int_{-\infty}^{\infty}\left(F(y)-\mathbb{I}(y\geq {\mathbf{x}})\right)^{2}\,dy,$ (21) $$

where $\mathbb{I}(\cdot)$ is the indicator function. Unlike RMSE, CRPS provides a more comprehensive evaluation of both the location and spread of the predicted distribution. A lower CRPS indicates a better match between the forecast distribution and the observed data. It is especially useful for probabilistic models that output a distribution rather than a single point prediction.

When applying CRPS to a finite ensemble of size $m$ approximating $F$ with the empirical CDF incurs an $O(1/m)$ bias favoring models with less spread. For small $m$ unbiased versions of the formulas should be used instead (Zamo & Naveau, ), but for the ensemble sizes here this is a small effect, so we used the more common biased formulas.

#### A.3.3 Spread Skill Ratio

The Spread-Skill Ratio (SSR) evaluates the reliability of the predicted uncertainty by comparing the spread (variance) of the predicted distribution with the accuracy of the predictions Gneiting & Raftery (). Let $\sigma_{\hat{{\mathbf{x}}}}$ be the standard deviation of the predicted distribution and RMSE as defined above. The SSR is defined as:

$$ $\text{SSR}=\frac{\sigma_{\hat{{\mathbf{x}}}}}{\text{RMSE}}.$ (22) $$

An SSR value close to 1 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 1 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).

### A.4 Further Details of the Datasets

Further details and visualizations of the ERA5-CWA and KF dataset, used throughout the paper, is presented here.

#### A.4.1 ERA5-CWA Dataset

Table table 4 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. 5. Notably, the typhoon’s eye shifts in the output due to the datasets originating from two different simulations, which solve distinct sets of partial differential equations (PDEs) at significantly different resolutions, resulting in divergent trajectories.

Input Data (ERA5). 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. (). 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) at 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 a 900 $\times$ 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 ${\mathbf{x}}\in\mathbb{R}^{H\times W\times C}$, where $H=W=448$ and $C=4$. See table 4.

The dataset encompasses approximately four years (2018-2021) of observations, sampled at 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 $448\times 448$ grid using bilinear interpolation to match the output resolution, a common practice with residual networks for consistency Hu et al. (); Zhang et al. () (see Figure 5(b) for an example of input vs. target misalignment).

**Table 4: ERA5-CWA Variables: 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.**
| Description | Input | Output |
| --- | --- | --- |
| Pixel Size | 36 × 36 | 448 × 448 |
| Single-Level Channels | Total Column Water Vapor | Maximum Radar Reflectivity |
| Temperature at 2 Meters | Temperature at 2 Meters |  |
| East Wind at 10 Meters | East Wind at 10 Meters |  |
| North Wind at 10 Meters | North Wind at 10 Meters |  |
| Pressure-Level Channels | Temperature | - |
| Geopotential | - |  |
| East Wind | - |  |
| North Wind | - |  |

Figure: (a) Input variables
Refer to caption: extracted/5935767/CWA_data_visualization_input_3x4.jpg

#### A.4.2 Kolmogorov-Flow Dataset

A representative input-output sample of the KF dataset is shown in Fig. 6 for different misalignment degrees $\tau$.

Dataset description. We construct a toy dataset by simulating the dynamics given by:

$$ $\displaystyle\zeta_{h}+J(\psi_{h},\zeta_{h})$ $\displaystyle=F+\nu_{h}\nabla^{7}\zeta_{l}-\zeta_{l}\tau_{r}^{-1}$ (23) $\displaystyle\zeta_{l}+J(\psi_{l},\zeta_{l})$ $\displaystyle=-\tau^{-1}(\zeta_{l}-\zeta_{h})+\nu_{l}\nabla^{7}\zeta_{l}-\zeta _{l}\tau_{r}^{-1}$ $\displaystyle\nabla^{2}\psi_{l}$ $\displaystyle=\zeta_{l}$ $\displaystyle\nabla^{2}\psi_{h}$ $\displaystyle=\zeta_{h}.$ $$

Here, $J(f,g)=f_{x}g_{y}-f_{y}g_{x}$ is the Jacobian operator. The stream function is related to the velocity field by $\nabla\psi=(-u,v)$, implying that $\nabla\psi\cdot(u,v)=0$, so that velocity points along contours of the stream-function. $\zeta_{l,h}$ represents the vorticity.

The $\zeta_{l}$ field represents a coarse-resolution simulation nudged towards a high resolution $\zeta_{h}$. The parameter $\tau$ controls the coupling strength between the $\zeta_{l}$ and $\zeta_{h}$ fields. A steady-state forcing $F=10\cos(10x)$ injects energy into the small-scale field $\zeta_{h}$ but not the low resolution $\zeta_{l}$, mimicking the injection of energy by sub-grid processes like convection or flow over topography. Stronger dissipation $\nu_{l}\gg\nu_{h}$ is used to limit the effective resolution of the large-scale field. A small amount of Rayleigh damping $\tau_{r}=100$ is added to limit the pile-up of energy at large scales.

These equations are solved using a standard pseudo-spectral method on the GPU. The 3rd-order Adams-Bashforth time stepper is used for all but the hyper-viscosity terms; for these stiff terms, we use an backward Euler time stepper. The resolution is $512\times 512$ and the timestep $dt=0.001$. A $2/3$ de-aliasing filter is applied in spectral space every timestep (Orszag, ). Outputs are saved every $\delta=0.2$ time units.

We create datasets for different $\tau$ values: $3,5,10,20$. Higher $\tau$ corresponds to greater misalignment between the coarse and high-resolution simulations. This variation allows us to assess the robustness of our method across different levels of coupling and identify potential thresholds in $\tau$ beyond which certain downscaling approaches may become unreliable. For each $\tau$ value, we generate a dataset comprising $100,000$ training points and $500$ test points.

Figure: Figure 6: Kolmogorov Flow Dataset. Visualization of input and target Kolmogorov Flow dataset for varying levels of misalignment ($\tau=3,5,10$). As $\tau$ increases, the discrepancy between the coarse and fine-resolution fields grows, offering a controlled environment to test downscaling performance.
Refer to caption: x13.png

## Appendix B Additional Experiments

We report additional experiments for both Taiwan CWA downscaling and KF downscaling, with a particular focus on ablation studies and ensemble analysis.

**Table 5: 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 |
| --- | --- | --- |
| L | [1, 2, 2, 4, 4] | 12M |
| M | [1, 2, 2, 2, 2] | 5M |
| S | [1, 1, 2, 2, 2] | 1M |
| XS | [1, 1, 1, 2, 2] | 0.2M |
| $1\times 1$ Conv. | - | 60 |

### B.1 CWA Ablation Studies

To evaluate the effectiveness of the SFM model and understand the impact of its components, we conducted ablation studies on the CWA weather downscaling task at $112\times 112$ resolution. We focused on varying the $\lambda$ parameter, different encoder types, the use of adaptive $\sigma_{{\mathbf{z}}}$, and ${\mathbf{y}}$ conditioning. The results are summarized in Tables 6, 7, and 8.

#### B.1.1 Effect of λ 𝜆 \lambda italic_λ Parameter and Encoder Type

Table 6 presents the performance of the SFM model with different $\lambda$ values and encoder types. The $\lambda$ parameter controls the trade-off between data fitting and uncertainty regularization in the SFM model.

$1\times 1$ Conv. Encoder: For the $1\times 1$ Conv. encoder, setting $\lambda=0$ 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 it to focus on fitting the data closely. The low parameter count (only 60 parameters) of the $1\times 1$ Conv. encoder might limit its ability to capture all the information in the low-resolution input, making less regularization advantageous.

UNet Encoder: In contrast, the UNet encoder shows improved performance with a small regularization parameter of $\lambda=0.25$. This suggests that the more complex architecture of the UNet benefits from some regularization to prevent overfitting and to enhance its generalization capabilities. The regularization helps balance the model’s capacity, leading to better accuracy and uncertainty estimates across multiple variables.

#### B.1.2 Impact of σ 𝐳 subscript 𝜎 𝐳 \sigma_{{\mathbf{z}}} italic_σ start_POSTSUBSCRIPT bold_z end_POSTSUBSCRIPT

Table 7 examines the effect of enabling or disabling adaptive $\sigma_{{\mathbf{z}}}$ for both encoder types.

Findings: Enabling adaptive $\sigma_{{\mathbf{z}}}$ consistently enhances the model’s performance across all variables and both encoder types. The improvements in RMSE, CRPS, and SSR metrics suggest that adaptive $\sigma_{{\mathbf{z}}}$ 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 to more accurate and reliable predictions.

#### B.1.3 Effect of 𝐲 𝐲 {\mathbf{y}} bold_y Conditioning

Table 8 assesses the impact of enabling or disabling ${\mathbf{y}}$ conditioning in the SFM model for both encoders.

$1\times 1$ Conv. Encoder: Disabling ${\mathbf{y}}$ conditioning provides better results across most metrics for this encoder. Given the simplicity of the $1\times 1$ Conv. encoder and its limited parameter count, it may not effectively utilize the additional information provided by ${\mathbf{y}}$ conditioning. The model performs better when focusing on directly mapping the input to the output without the added complexity.

UNet Encoder: Enabling ${\mathbf{y}}$ conditioning yields the best results for the UNet encoder. The more complex architecture of the UNet can leverage the additional context from ${\mathbf{y}}$ 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.

#### B.1.4 Summary

These ablation studies highlight the strengths of the SFM model and its components in weather downscaling tasks:

- •
Effectiveness of Adaptive $\sigma_{{\mathbf{z}}}$: Enabling adaptive $\sigma_{{\mathbf{z}}}$ consistently improves model performance across both encoder types and all variables. This underscores the importance of modeling spatially varying uncertainty in complex weather data.
- •
Encoder Choice and Regularization: The $1\times 1$ Conv. encoder performs best without regularization ($\lambda=0$), indicating that minimal regularization benefits simpler models. For the UNet encoder, a small regularization parameter ($\lambda=0.25$) yields better results, suggesting that regularization helps prevent overfitting in more complex models.
- •
Impact of ${\mathbf{y}}$ Conditioning: ${\mathbf{y}}$ conditioning enhances performance for the UNet encoder but not for the $1\times 1$ Conv. encoder. This suggests that the effectiveness of incorporating additional context depends on the model’s capacity to utilize that information.

Overall, the SFM model demonstrates strong performance and flexibility in weather downscaling tasks. By carefully selecting model components such as encoder type, adaptive $\sigma_{{\mathbf{z}}}$, and regularization parameter $\lambda$, the SFM can be tailored to balance predictive accuracy and uncertainty estimation effectively. These findings highlight the potential of the SFM model as a powerful tool for probabilistic weather downscaling.

**Table 6: Encoder and $\lambda$ Ablations on CWA $112\times 112$ Dataset. For this table we keep the best configurations for each $\lambda$ value based on radar reflectivity. We separate per $1\times 1$ Conv. and UNet encoder to elucidate differences. Best results for each encoder are highlighted in bold. No regularization ($\lambda=0$) works best for the $1\times 1$ Conv. encoder, while for UNet, a small $\lambda=0.25$ value yields the best results across multiple variables. Overall the $1\times 1$ conv. without regularization is the best configuration.**
| Encoder | $\lambda$ | Radar | Temperature | Eastward Wind | Northward Wind |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ |  |  |
| $1\times 1$ Conv. | 0.00 | 4.82 | 1.66 | 0.72 | 0.99 | 0.59 | 0.47 | 1.42 | 0.78 | 0.68 | 1.58 | 0.89 | 0.66 |
| 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 |  |
| UNet | 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 |
| 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 | 5.11 | 1.89 | 0.41 | 1.07 | 0.62 | 0.36 | 1.53 | 0.91 | 0.41 | 1.74 | 1.03 | 0.40 |  |

**Table 7: Adaptive $\sigma_{{\mathbf{z}}}$ Ablation on CWA $112\times 112$ Dataset for $1\times 1$ Conv. and UNet Encoders. This table examines the effect of enabling (✓) or disabling (✗) Adaptive $\sigma_{{\mathbf{z}}}$ across both $1\times 1$ Conv. and UNet encoders. Best results for each encoder and metric are highlighted in bold. Results indicate that the proposed adaptive $\sigma_{z}$ consistently improve results.**
| Encoder | Adapt $\sigma_{{\mathbf{z}}}$ | Radar | Temperature | Eastward Wind | Northward Wind |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ |
| $1\times 1$ Conv. | ✗ | 5.01 | 1.88 | 0.45 | 1.13 | 0.66 | 0.39 | 1.50 | 0.86 | 0.48 | 1.64 | 0.94 | 0.47 |
| ✓ | 4.82 | 1.66 | 0.72 | 0.99 | 0.59 | 0.47 | 1.42 | 0.78 | 0.68 | 1.58 | 0.89 | 0.66 |  |
| UNet | ✗ | 5.06 | 1.83 | 0.38 | 1.01 | 0.57 | 0.36 | 1.48 | 0.85 | 0.46 | 1.64 | 0.93 | 0.46 |
| ✓ | 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 8: ${\mathbf{y}}$ conditioning ablation on CWA $112\times 112$ Dataset for $1\times 1$ Conv. and UNet Encoders. This table assesses the impact of enabling (✓) or disabling (✗) ${\mathbf{y}}$ conditioning across both $1\times 1$ Conv. and UNet encoders. Best results for each encoder and metric are highlighted in bold. Results indicate that for the $1\times 1$ encoder the conditioning is beneficial while for UNet the opposite stands. This makes sense since the $1\times 1$ conv. encoder has only 60 parameters and might not capture all the information in the low-resolution input.**
| Encoder | ${\mathbf{y}}$ cond. | Radar | Temperature | Eastward Wind | Northward Wind |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  |  | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ |
| $1\times 1$ Conv. | ✗ | 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 | 0.89 | 0.54 | 0.40 | 1.46 | 0.82 | 0.45 | 1.59 | 0.92 | 0.46 |  |
| UNet | ✗ | 5.06 | 1.83 | 0.38 | 1.01 | 0.57 | 0.36 | 1.48 | 0.85 | 0.46 | 1.64 | 0.93 | 0.46 |
| ✓ | 4.95 | 1.78 | 0.44 | 0.94 | 0.54 | 0.40 | 1.49 | 0.85 | 0.53 | 1.67 | 0.95 | 0.50 |  |

**Table 9: Complete ablation results for the CWA $112\times 112$ dataset. Best two models in each variable/metric in bold.**
| Model | $\boldsymbol{\lambda}$ | Adapt.<br>$\sigma_{\rm max}$ | Use<br>${\mathbf{x}}_{\rm low}$ | Radar | Temperature | Eastward Wind | Northward Wind |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| RMSE $\downarrow$ | CRPS $\downarrow$ | SSR $\rightarrow 1$ | RMSE | CRPS | SSR | RMSE | CRPS | SSR | RMSE | CRPS | SSR |  |  |  |  |
| SFM<br>$1\times 1$ Conv. | 0.00 | ✗ | ✗ | 5.16 | 1.88 | 0.44 | 1.12 | 0.65 | 0.36 | 1.53 | 0.88 | 0.47 | 1.72 | 0.98 | 0.46 |
| ✓ | ✗ | 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 |  |  |  |
| 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 |  |
| ✓ | ✗ | 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 |  |  |  |
| 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 |  |
| ✓ | ✗ | 5.11 | 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 |  |  |  |
| SFM<br>UNet | 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 |
| ✓ | ✗ | 5.13 | 1.84 | 0.43 | 0.85 | 0.52 | 0.37 | 1.43 | 0.81 | 0.50 | 1.60 | 0.89 | 0.48 |  |  |
| ✓ | 5.07 | 1.84 | 0.36 | 0.85 | 0.52 | 0.34 | 1.44 | 0.83 | 0.43 | 1.60 | 0.91 | 0.43 |  |  |  |
| 0.25 | ✗ | ✗ | 5.01 | 1.88 | 0.45 | 1.13 | 0.66 | 0.39 | 1.50 | 0.86 | 0.48 | 1.64 | 0.94 | 0.47 |  |
| ✓ | ✗ | 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 | 0.53 | 1.67 | 0.95 | 0.50 |  |  |  |
| 1.00 | ✗ | ✗ | 5.11 | 1.89 | 0.41 | 1.07 | 0.62 | 0.36 | 1.53 | 0.91 | 0.41 | 1.74 | 1.03 | 0.40 |  |
| ✓ | ✗ | 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 10: Performance Comparison of Models on CWA $112\times 112$ Dataset. The SFM model has a $1\times 1$ Conv. encoder, $\lambda=0$, adaptive $\sigma_{z}$ and no $y$ conditioning. Overall, the SFM 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 | SFM |
| --- | --- | --- | --- | --- | --- |
| Radar | RMSE$\downarrow$ | 5.06 | 4.95 | 4.94 | 4.82 |
| CRPS$\downarrow$ | 1.84 | 1.74 | - | 1.66 |  |
| MAE$\downarrow$ | 2.41 | 2.49 | 2.45 | 2.63 |  |
| SSR $\rightarrow 1$ | 0.36 | 0.52 | - | 0.72 |  |
| Temperature | RMSE | 0.86 | 0.87 | 0.87 | 0.99 |
| CRPS | 0.50 | 0.52 | - | 0.59 |  |
| MAE | 0.64 | 0.64 | 0.64 | 0.74 |  |
| SSR | 0.45 | 0.38 | - | 0.47 |  |
| East. Wind | RMSE | 1.42 | 1.44 | 1.42 | 1.42 |
| CRPS | 0.81 | 0.81 | - | 0.78 |  |
| MAE | 1.04 | 1.05 | 1.05 | 1.05 |  |
| SSR | 0.48 | 0.49 | - | 0.68 |  |
| North. Wind | RMSE | 1.59 | 1.59 | 1.60 | 1.63 |
| CRPS | 0.89 | 0.89 | - | 0.89 |  |
| MAE | 1.14 | 1.14 | 1.16 | 1.19 |  |
| SSR | 0.47 | 0.48 | - | 0.66 |  |

**Table 11: SFM ablations for $\lambda$ on the CWA weather downscaling task at full resolution $448\times 448$. For this ablation, a $1\times 1$ Conv. encoder was used, $\sigma_{z}$ was set to $1$, and no ${\mathbf{y}}$ conditioning was employed. Overall, $\lambda=0$ seems to produce better estimates except for temperature, whose deterministic nature benefits from the added regularization.**
| Variable | Metric | $\boldsymbol{\lambda}$ |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 0.0 | 0.01 | 0.1 | 0.25 | 0.5 | 1.0 | 2.5 |  |  |
| Radar | RMSE $\downarrow$ | 4.90 | 5.11 | 5.03 | 4.97 | 5.00 | 5.25 | 5.27 |
| CRPS $\downarrow$ | 1.78 | 1.92 | 1.88 | 1.85 | 1.87 | 1.97 | 1.96 |  |
| MAE $\downarrow$ | 2.42 | 2.66 | 2.47 | 2.47 | 2.47 | 2.59 | 2.75 |  |
| SSR$\rightarrow 1$ | 0.44 | 0.49 | 0.39 | 0.42 | 0.41 | 0.39 | 0.52 |  |
| Temperature | RMSE $\downarrow$ | 1.00 | 1.00 | 0.87 | 0.86 | 0.82 | 1.00 | 0.85 |
| CRPS $\downarrow$ | 0.52 | 0.62 | 0.54 | 0.52 | 0.48 | 0.56 | 0.48 |  |
| MAE $\downarrow$ | 0.67 | 0.78 | 0.66 | 0.64 | 0.60 | 0.68 | 0.62 |  |
| SSR $\rightarrow 1$ | 0.47 | 0.48 | 0.38 | 0.40 | 0.41 | 0.32 | 0.45 |  |
| East. Wind | RMSE $\downarrow$ | 1.44 | 1.52 | 1.49 | 1.48 | 1.49 | 1.55 | 1.48 |
| CRPS $\downarrow$ | 0.80 | 0.86 | 0.87 | 0.87 | 0.88 | 0.92 | 0.86 |  |
| MAE $\downarrow$ | 1.07 | 1.13 | 1.09 | 1.09 | 1.09 | 1.14 | 1.08 |  |
| SSR $\rightarrow 1$ | 0.61 | 0.55 | 0.42 | 0.40 | 0.41 | 0.38 | 0.43 |  |
| North. Wind | RMSE $\downarrow$ | 1.61 | 1.61 | 1.66 | 1.67 | 1.67 | 1.72 | 1.65 |
| CRPS $\downarrow$ | 0.88 | 0.88 | 0.95 | 0.96 | 0.96 | 1.00 | 0.95 |  |
| MAE $\downarrow$ | 1.17 | 1.17 | 1.19 | 1.20 | 1.19 | 1.24 | 1.18 |  |
| SSR $\rightarrow 1$ | 0.58 | 0.61 | 0.41 | 0.41 | 0.40 | 0.38 | 0.43 |  |

**Table 12: SFM ablations for encoder size on the CWA weather downscaling task at full resolution $448\times 448$. For this ablation, we use $\lambda=0$, $\sigma_{z}=1$, and no ${\mathbf{y}}$ 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.**
| Variable | Metric | Encoder |  |  |  |
| --- | --- | --- | --- | --- | --- |
| L | M | S | XS |  |  |
| Radar | RMSE $\downarrow$ | 4.93 | 4.96 | 4.98 | 4.98 |
| CRPS $\downarrow$ | 1.82 | 1.84 | 1.85 | 1.85 |  |
| MAE $\downarrow$ | 2.44 | 2.52 | 2.52 | 2.46 |  |
| SSR$\rightarrow 1$ | 0.40 | 0.43 | 0.42 | 0.39 |  |
| Temperature | RMSE $\downarrow$ | 1.01 | 1.00 | 0.99 | 1.02 |
| CRPS $\downarrow$ | 0.55 | 0.54 | 0.54 | 0.56 |  |
| MAE $\downarrow$ | 0.68 | 0.68 | 0.68 | 0.70 |  |
| SSR $\rightarrow 1$ | 0.34 | 0.38 | 0.40 | 0.38 |  |
| East. Wind | RMSE $\downarrow$ | 1.48 | 1.48 | 1.47 | 1.50 |
| CRPS $\downarrow$ | 0.85 | 0.83 | 0.83 | 0.86 |  |
| MAE $\downarrow$ | 1.10 | 1.08 | 1.08 | 1.11 |  |
| SSR $\rightarrow 1$ | 0.49 | 0.53 | 0.53 | 0.51 |  |
| North. Wind | RMSE $\downarrow$ | 1.64 | 1.64 | 1.63 | 1.66 |
| CRPS $\downarrow$ | 0.92 | 0.92 | 0.91 | 0.93 |  |
| MAE $\downarrow$ | 1.19 | 1.20 | 1.19 | 1.21 |  |
| SSR $\rightarrow 1$ | 0.48 | 0.51 | 0.52 | 0.50 |  |

**Table 13: SFM ablations for $y$ conditioning for the CWA weather downscaling task at full resolution $448\times 448$. For this ablation, we use adaptive $\sigma_{z}$, $\lambda=0.25$ and a UNet encoder guided by the ablations in the $112\times 112$ resolution. Including ${\mathbf{y}}$ conditioning consistently improves performance across all metrics. RMSE, CRPS, and MAE are lower when ${\mathbf{y}}$ conditioning is used, and SSR values are closer to 1.**
| Variable | Metric | ${\mathbf{y}}$ conditioning |  |
| --- | --- | --- | --- |
| ✗ | ✓ |  |  |
| Radar | RMSE $\downarrow$ | 5.09 | 4.90 |
| CRPS $\downarrow$ | 1.88 | 1.80 |  |
| MAE $\downarrow$ | 2.33 | 2.24 |  |
| SSR$\rightarrow 1$ | 0.24 | 0.25 |  |
| Temperature | RMSE $\downarrow$ | 0.92 | 0.89 |
| CRPS $\downarrow$ | 0.55 | 0.50 |  |
| MAE $\downarrow$ | 0.67 | 0.64 |  |
| SSR $\rightarrow 1$ | 0.33 | 0.43 |  |
| East. Wind | RMSE $\downarrow$ | 1.49 | 1.45 |
| CRPS $\downarrow$ | 0.91 | 0.86 |  |
| MAE $\downarrow$ | 1.10 | 1.07 |  |
| SSR $\rightarrow 1$ | 0.34 | 0.41 |  |
| North. Wind | RMSE $\downarrow$ | 1.66 | 1.61 |
| CRPS $\downarrow$ | 1.00 | 0.94 |  |
| MAE $\downarrow$ | 1.20 | 1.18 |  |
| SSR $\rightarrow 1$ | 0.33 | 0.41 |  |

Figure: (a)
Refer to caption: extracted/5935767/CWA_visual_baselines_plus_SFM_with_corrdiff_1.jpg

Figure: (a)
Refer to caption: extracted/5935767/CWA_visual_baselines_plus_SFM_with_corrdiff_10.jpg

Figure: (a)
Refer to caption: extracted/5935767/CWA_visual_baselines_plus_SFM_with_corrdiff_115.jpg

#### B.1.5 Ensemble Analysis

For a few representative samples, several ensemble members and the ensemble mean are shown in Figures 8(a) to 8(a). 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 a 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 $\tau$ are provided at [https://t.ly/ZCq9Z](https://t.ly/ZCq9Z).

Figure: (a)
Refer to caption: extracted/5935767/SFM_mean_samples_GT_comparison_1.jpg

Figure: (a)
Refer to caption: extracted/5935767/SFM_mean_samples_GT_comparison_10.jpg

Figure: (a)
Refer to caption: extracted/5935767/SFM_mean_samples_GT_comparison_115.jpg

### B.2 Kolmogorov-Flow Downscaling

#### B.2.1 Ablations

Different hyperparameters of both the dataset and the model are ablated, with the resulting metrics reported in Table 14. The results show that, in this scenario, where the data is relatively misaligned, a smaller encoder tends to 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 SFM with the coarse-resolution input enhances the predictive 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.

Figure: Figure 9: SFM vs. Baselines for Different Misalignment Levels in Kolmogorov Flow Downscaling: Each row corresponds to a different misalignment level $\tau$ (left side). From top to bottom, the rows represent $\tau=10$, $\tau=5$, and $\tau=3$. As misalignment increases, the SFM significantly outperforms baseline models by generating samples that better align with the target distribution. Additionally, note the presence of high-frequency artifacts in the baseline models, which are more noticeable when the figures are zoomed in.
Refer to caption: x14.png

Figure: (a) $\tau=3$.
Refer to caption: x17.png

**Table 14: Kolmogorov Flow Ablation Study for SFM: This table examines the effect of different hyperparameters on performance across misalignment levels ($\tau$). A smaller encoder with conditioning consistently performs better for highly misaligned data. Additionally, adaptive noise scaling ($\sigma_{z}$) enhances performance when conditioning SFM on coarse-resolution input data (${\mathbf{y}}$).**
|  | Encoder | $1\times 1$conv | UNet |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | Adapt. $\sigma_{z}$ | ✗ | ✓ | ✗ | ✓ |  |  |  |  |
| $\tau$ | ${\mathbf{y}}$ cond. | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ | ✗ | ✓ |
| 3 | RMSE $\downarrow$ | 1.22 | 0.91 | 1.22 | 0.73 | 1.15 | 1.11 | 1.17 | 1.17 |
| CRPS $\downarrow$ | 0.63 | 0.48 | 0.62 | 0.37 | 0.65 | 0.63 | 0.69 | 0.70 |  |
| MAE $\downarrow$ | 0.83 | 0.65 | 0.83 | 0.51 | 0.81 | 0.78 | 0.83 | 0.84 |  |
| SSR $\rightarrow 1$ | 0.56 | 0.58 | 0.58 | 0.62 | 0.37 | 0.34 | 0.31 | 0.30 |  |
| 5 | RMSE $\downarrow$ | 1.17 | 0.78 | 1.17 | 0.76 | 1.16 | 1.01 | 1.09 | 1.07 |
| CRPS $\downarrow$ | 0.62 | 0.42 | 0.61 | 0.40 | 0.67 | 0.58 | 0.66 | 0.64 |  |
| MAE $\downarrow$ | 0.82 | 0.56 | 0.82 | 0.54 | 0.83 | 0.72 | 0.80 | 0.78 |  |
| SSR $\rightarrow 1$ | 0.58 | 0.58 | 0.60 | 0.58 | 0.35 | 0.36 | 0.33 | 0.33 |  |
| 10 | RMSE $\downarrow$ | 1.36 | 1.06 | 1.39 | 1.09 | 1.35 | 1.36 | 1.36 | 1.28 |
| CRPS $\downarrow$ | 0.71 | 0.57 | 0.78 | 0.65 | 0.79 | 0.79 | 0.80 | 0.74 |  |
| MAE $\downarrow$ | 0.96 | 0.78 | 0.98 | 0.77 | 0.98 | 0.99 | 1.00 | 0.95 |  |
| SSR $\rightarrow 1$ | 0.69 | 0.63 | 0.43 | 0.23 | 0.36 | 0.37 | 0.38 | 0.45 |  |

#### B.2.2 Ensemble Analysis

Representative KF samples along with the generated ensemble members are depicted in Figs. 11, 11, and 11. 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 the ensemble mean with the observed samples suggests that the model not only captures the central tendency but also effectively characterizes the stochastic nature of the dynamics.

Figure: Figure 11: Ensemble Predictions of SFM for Kolmogorov Flow at Different $\tau$ Values. SFM ensemble predictions are shown for different $\tau$ values ($\tau=3,5,10$), illustrating the model’s ability to capture the variability and dynamics of the Kolmogorov flow across increasing levels of misalignment.
Refer to caption: x20.png

## Peer Reviews

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: - See weaknesses 1 and 2.

- I wonder how the noise variance in the adaptive noise scaling varies in the training run, and if the effectiveness is sensitive to the hyperparameter setup.
soundness: 2
strengths: - The authors conducted comprehensive experiments on both synthetic and real-world datasets.

- The paper includes ablation studies and spectral analysis, which reveal the impact of different architectural choices and underscore SFM's robustness in capturing high-frequency details.

- The dynamic adjustment of noise levels based on the encoder’s prediction error provides a robust method to balance deterministic and stochastic data components.
summary: The paper introduces the Stochastic Flow Matching (SFM) framework, a novel approach to super-resolve small-scale physics in meteorological data, particularly in downscaling weather variables for regions covering Taiwan. The authors address three key challenges: spatial misalignment between input and output distributions, deterministic vs. stochastic dynamics across scales, and limited data, which can lead to overfitting. SFM leverages an encoder to map coarse-resolution inputs to a latent distribution aligned with the target fine-resolution, which is refined using flow matching to reconstruct small-scale stochastic details. An adaptive noise scaling mechanism, based on maximum-likelihood estimates, helps balance deterministic and stochastic components. The framework is tested on real-world CWA weather data and the PDE-based Kolmogorov dataset, consistently outperforming existing methods, such as conditional diffusion and flows, in terms of RMSE and fidelity across several variables and spatial resolutions.
weaknesses: - It is not clear why the method is named and formalized as "flow matching" because the actual implementation is based on EDM. The perturbation and the denoising model both follow the diffusion custom instead of flow matching. Please justify whether the method especially the probability path is closer to flow matching or diffusion.

- In Algorithm 2. the inputs to the denoising model $\mathcal{D}_\theta$ are $(x, t)$ but elsewhere $(x, \sigma)$. This may lead to confusion on how the adaptive noise scaling can affect sampling.

-  Application of the proposed method to other domains is unexplored, limiting the generalizability of the findings. It seems that the method can be applied to general image-to-image translation tasks, for example restoration of natural images which also involves stochasticity in generation of local details.

- As acknowledged in the paper, SFM relies on paired input-output data, which may not always be feasible in real-world applications, particularly for unpaired or missing datasets.

- Improvement upon baseline method CorrDiff is not consistent (especially temperature in Table 2).

- A minor point that does not affect the score (if the method still modelled as flow matching).  SFM/SfM is commonly used in the broad field of AI to refer to Structure from Motion, which could lead to confusion, particularly if the paper is targeted at a broader audience that includes computer vision researchers.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: See "Weaknesses".
soundness: 2
strengths: 1. The overall idea for cross-modality image translation is promising.
2. The proposed stochastic encoder-denoiser pipeline is interesting. 
3. Extensive Experiments illustrate the effectiveness of the proposed method, and the results on most datasets are good.
summary: This paper presents a stochastic flow matching approach for super-resolving small-scale details. Specifically, it uses an encoder to transform inputs into latent images that are more aligned with the ground truth images. The additional noise is added to the latent images to introduce uncertainty for the following flow matching. The authors further present a modified flow matching objective that involves the noisy latent in training. Moreover, the maximum likelihood learning is proposed to tune the noise scale and an additional regularization is used in encoder learning for better generalization. The proposed method is evaluated on multi-scale weather datasets and synthetic PDE datasets.
weaknesses: 1. If the authors want to add uncertainty to the deterministic part, maybe the better way is to apply approaches like VAE to output both mean and variance (the downscale layers can be removed if there is no resolution change). Using VAE can also avoid the tuning of noise scales.
2. Also, calling this method "stochastic flow matching" is somehow improper, since the flow matching is usually an ODE. Note that the original flow matching also accepts noise input.   Adding the "stochastic" to "flow matching" makes it confusing and readers might think it's a general method converting flow matching to an SDE.
3. In Proposition 1, how do you define the velocity field $v$? Moreover, I don't think the term $(1-t)$ can be simply ignored since it changes over time. At least the authors should add some experiments to show that ignoring the $1-t$ could achieve the same performance.
4. As stated in Sec 4.2, the noise level is highly related to x and y and thus this work proposes a maximum likelihood approach to tune it. My question is why not learn it through the encoder directly? It should be easier and can be trained in an end-to-end manner as it is also involved in the training objective.
5. Adding the encoder regularization is too intuitive and would definitely affect the learning of flow matching. More details and ablation experiments should be provided.
6. The 1 x 1 conv layer encoder performs surprisingly well compared to the complex UNet encoder. Can the authors provide more analysis on it? Why a linear transform can lead to better performance?
7. In appendix A.1, the last paragraph, it should be D_{\theta}.

---

confidence: 3
contribution: 2
initial_rating: 5
presentation: 3
questions: see Weaknesses
soundness: 3
strengths: 1. The presentation is apparent and the description of the proposed methodology is accurate and easy to follow.

2. The figure and quantitative comparisons are good, the experiment is sufficiently adequate, the experimental setup is described in detail with methodological details and it should be easily reproducible.
summary: This paper presents a framework that becomes stochastic stream matching for learning to recover small-scale physical fields. Specifically, they first use an encoder to learn to regress deterministic dynamics, and then add stream matching to learn the distribution of fine-grained stochastic dynamics. They validate the algorithm on multi-scale Kolmogorov flow and regional weather downscaling datasets.
weaknesses: 1. The problem of novelty, due to the well-known connection between flow matching and diffusion modeling. The approach proposed in this paper seems to resemble CorrDiff. Although the authors emphasize the difference in Section 4.4. They claim that CorrDiff is trained in two stages, i.e., the regression encoder is trained first and the diffusion of the residual components is trained afterwards. In contrast, it seems that in this paper, the two components are only trained jointly, and the final loss is the sum of these two components.

2. In addition, the authors claim that the first stage of CorrDiff may run the risk of overfitting. This reasoning seems insufficient to me, and the authors don't seem to have relevant evidence. There is also no guarantee that joint training avoids this risk, besides this can be solved perfectly well using simpler ways, such as early stopping.

3. The idea of co-training encoders and diffusion models is also not new, and in fact he has already proposed it for tasks such as speech synthesis [1,2] and precipitation nowcasting [3].

[1] Popov V, Vovk I, Gogoryan V, et al. Grad-tts: A diffusion probabilistic model for text-to-speech[C]//International Conference on Machine Learning. PMLR, 2021: 8599-8608.

[2] Chen Z, He G, Zheng K, et al. Bridge-TTS: Text-to-Speech Synthesis with Schrodinger Bridge[J].


[3] DiffCast: A Unified Framework via Residual Diffusion for Precipitation Nowcasting

---

confidence: 2
contribution: 3
initial_rating: 6
presentation: 3
questions: - Could you provide more details about the hyperparameter selection process?

- Have you identified specific failure cases or limitations of the method?
soundness: 3
strengths: - The presentation and writing are good

- The paper presents a novel combination of deterministic encoding with stochastic flow matching for solving the task

- The method's ability to handle misaligned data and limited data scenarios makes it valuable for certain applications

- Experiments on synthetic and real-world datasets demonstrate the effectiveness of the method
summary: The paper proposes stochastic flow matching for super-resolving small-scale physics in weather data, addressing challenges like data misalignment, multiscale dynamics, and limited data availability. The method combines an encoder that maps coarse-resolution inputs to a latent space with flow matching to generate fine-resolution outputs, while using adaptive noise scaling to balance deterministic and stochastic components. Experiments demonstrate that SFM outperforms existing methods like conditional diffusion and flow models.
weaknesses: A theoretical analysis of SFM's convergence properties would be interesting, especially regarding the impact of the adaptive noise scaling mechanism on convergence guarantees

---

confidence: 4
contribution: 3
initial_rating: 5
presentation: 3
questions: **Major**
- Which parts of the model support your claim that it is *specifically tailored for data-limited regimes* ? Is there an experiment in the paper exposing the limitation of Mardani et al, 2023 in this context ?
- During training, how is $\sigma_z$  computed (using training or validation error) ?
- Both noise scaling and encoder regularization are motivated by the need to counter overfitting. Is there any proof in the paper that vanilla methods does suffer from overfitting that I could have missed ?
- If I understand correctly, the adaptive noise scaling will increase the noise amplitude if the deterministic part has large error. Then could it leads to a degenerate solution where $\sigma_z$ remains very large and most of the work is done by the stochastic part ? This seems supported by several results in the paper:
	- The simpler 1x1 conv. performs often better than UNet, which is unexpected, since UNet is more powerful. Could it indicates that the model is actually using more the stochastic part, and learns to ignore the deterministic part ?
	- Using the encoder regularization should prevent the degenerate case. And indeed, Table 6 shows that the regularization is helpful mostly for UNet. 
I think the paper could benefit from a study on the strength of $\sigma_z$, for instance by analyzing how corrupted is the output of the deterministic part (if the noise is of the same order of magnitude, then the model has degenerated ?). It would also be interesting (if available) to see the evolution of $\sigma_z$ during training of the model for both UNet and Conv 1x1.

**Minor**
- Can you please confirm the typo in the proof of proposition 1 ?
- Could you please elaborate on Sec.4 (Deterministic Dynamics) "it spatially aligns the base and target distribution *by matching their large-scale dynamics*" ? The sentence is unclear to me.
- The authors suggest that $\sigma_z$ prevents overfitting by increasing when the validation error increases. This intuition is only valid when $\sigma_z$  is computed from a validation loss, hence not online. Then, what is the intuition behind *online* method for scaling the noise ?
- If the temperature is deterministic, how the authors explains that the UNet Reg. baseline does not outperforms the other models on this variable ?
soundness: 3
strengths: The paper is well written (despite some minors typos) and easy to follow. The method is well explained and sounds reasonable. The authors conducted an impressive set of experiments, anticipating well potential questions of the reader.
summary: The paper tackles a difficult variant of super-resolution applied to weather data where the goal is to retrieve small-scale dynamics of (potentially different) physics variables from a coarser observation grid. The authors build upon Mardani et al., 2023, proposing to disentangle the deterministic part (modeled with a U-Net or a 1x1 Conv.) and the stochastic part (modeled with flow matching). Conversely to Mardani et al., 2023, the authors jointly trained both parts by dynamically scaling the influence of the stochasticity in order to prevent overfitting.
weaknesses: **Major**
- The authors claims that their method is *specifically tailored for data-limited regimes*, and mention this as a key limitation of Mardani et al., 2023 compared to their approach. Yet, no experiment explore low data-regime. The ERA5-CWA dataset contains 4 years of hourly sampled data, and the "toy" dataset contains 100k points.
- I found section 4.2 / 4.3 rather unconvincing, especially the intuitive motivation for the adaptive noise scaling. I think these sections could benefit from a clarification from the authors (see related questions).

**Minor**
- In the introduction, the authors stated that the method is designed to match *spatially* misaligned data, yet in the rest of the paper, the term "misalignement" seems to refers to difference between the PDE governing the low-resolution data and the high-resolution one. It is not clear to me why such misalignment can be considered spatial. If *spatial* misalignement does corresponds to differences in PDEs, then the first paragraph of A.4.1 describes it very well and probably should be in the main paper.
- Replaying the proof of Proposition 1, when injecting the proposed form of $v_\theta(x,t) = (x_t - D_\theta(x_t, t)) / (1-t)$ into equation (7), I found:
$$\|v_\theta(x_t, t) - (x-z)\| = \left\|\frac{x_t - D_\theta(x_t, t)}{1-t} - (x-z)\right\|$$
then using equation (8):
$$\|v_\theta(x_t, t) - (x-z)\| = \left\|\frac{(1-t)z + tx - D_\theta(x_t, t)}{1-t} - (x-z)\right\| = \left\|2z + \frac{2tx - x}{1-t} - \frac{D_\theta(x_t,t)}{1-t}\right\|$$
So I think the correct expression for $v_\theta$ should rather be $v_\theta(x,t) = \frac{x_t - D_\theta(x_t, t)}{t-1}$

**Grade motivation**
My grade is motivated mainly by my difficulties to fully understand what the proposed method actually learns, and consequently how it compares with pure deterministic or pure stochastic baselines. I suspect degeneration of the model toward a fully stochastic method, which seems supported by ablations, but contradicts main results in table 2. The "data-limited regime" claims is also confusing. The math typo is certainly a simple sign mistake, but also plays a role.

**Typo**
- Missing parenthesis in score function after equation (5)
- Repeated $\theta$ in A.1 (expression of $v_\theta$)

Meta-review: The paper introduces an Adaptive Flow Matching (AFM) framework for super-resolving small-scale physics, but reviewers raised concerns about its novelty and positioning relative to existing methods like CorrDiff and diffusion-based approaches. A key weakness highlighted is the lack of experiments in data-limited regimes, despite claims that the method is tailored for such settings, as well as insufficient evidence to demonstrate significant overfitting in baseline methods. Additionally, reviewers pointed out that the adaptive noise scaling mechanism and its theoretical justification require further clarity, particularly regarding its potential to degenerate into a fully stochastic model.

Decision: Reject