Stochastic Flow Matching for Resolving Small-Scale Physics (2410.19814v1)

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) misalignment between input and output distributions (i.e., solutions to distinct partial differential equations (PDEs) follow different trajectories), (ii) multi-scale dynamics, deterministic dynamics at large scales vs. stochastic at small scales, and (iii) 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 output using an adaptive noise scaling mechanism, which is dynamically adjusted based on maximum-likelihood estimates of the encoder 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.

• The paper introduces a novel framework that integrates encoder-decoder architecture with flow matching to correct data misalignments in super-resolution tasks.
• The paper employs an adaptive noise scaling mechanism to dynamically balance deterministic and stochastic components for high-fidelity output.
• The paper demonstrates significant improvements in high-frequency fidelity over traditional methods on both real-world and synthetic datasets.

Stochastic Flow Matching for Resolving Small-Scale Physics

The paper presents a novel approach, termed Stochastic Flow Matching (SFM), to address challenges in super-resolving small-scale physics, specifically in fields like atmospheric science where data sets are often sparse and misaligned. Traditional methods have struggled due to misalignment between input and output distributions, multi-scale dynamics, and limited data availability, which contribute to overfitting risks. This work proposes an innovative solution that significantly enhances super-resolution performance by introducing a unique combination of flow and diffusion models.

Key Methodological Advances

The SFM framework integrates an encoder-decoder architecture with flow matching principles. The encoder converts the input data into a latent representation closely resembling the target distribution and corrects spatial misalignments. This design addresses the misalignment challenge by aligning channels and repositioning critical features, such as the eye of a typhoon, using flow matching to generate small-scale features.

1. Latent Encoding and Flow Matching: The method involves mapping coarse-resolution inputs to a latent space, capturing deterministic components. Flow matching then reforms this information into high-resolution outputs by stochastically infusing small-scale physics details.
2. Adaptive Noise Scaling: To handle model uncertainty, particularly in the deterministic component, the paper introduces an adaptive noise scaling mechanism. This mechanism dynamically adjusts noise levels based on the encoder’s performance, as determined by maximum likelihood estimates.
3. End-to-End Learning: Unlike traditional multi-stage learning approaches, SFM facilitates an end-to-end learning process in which the deterministic and stochastic components of the data are jointly learned.

Experimental Evaluation

The authors conducted extensive experiments on both real-world datasets, such as the CWA weather dataset, and synthetic datasets, like the PDE-based Kolmogorov flow dataset. The CWA dataset challenges models to super-resolve weather variables from a 25 km scale down to a 2 km scale, an area known for complex atmospheric phenomena.

1. Performance Metrics: The paper utilized metrics like RMSE, CRPS, and SSR to evaluate performance. The results show that SFM consistently outperformed baselines like conditional diffusion and flow models, particularly in high-frequency fidelity and stochastic accuracy.
2. Spectral Fidelity: SFM displayed superior spectral fidelity, maintaining the integrity of high-frequency content in weather data, crucial for capturing small-scale atmospheric phenomena.

Implications and Future Directions

The implications of this research extend to any field dealing with sparse, misaligned datasets across resolutions or scales— a common scenario in physical sciences. By effectively integrating stochastic and deterministic elements, SFM improves upon traditional super-resolution techniques, providing more reliable outcomes particularly in data-constrained environments. Moreover, it sets the groundwork for further exploration into unsupervised or semi-supervised scenarios, where labeled data is even more limited, by potentially incorporating physical constraints into its framework to enhance model robustness.

Conclusion

This paper contributes significantly to methodologies for resolution enhancement in physical sciences. The introduction of SFM with adaptive noise scaling presents a meaningful advance over existing approaches, offering robust performance across scales and data regimes. Given the challenges addressed, the paper opens new avenues for research in multi-scale modeling and deterministic-stochastic dynamics integration, offering a blueprint for future innovations in high-fidelity modeling of complex systems.