Flow Matching for Probabilistic Learning of Dynamical Systems from Missing or Noisy Data

2025-08-01

AI Summary

This study addresses the ill-posedness of dynamical system modeling under missing or noisy data. We propose a probabilistic forecasting framework that integrates physical priors with generative learning. Methodologically, we combine flow matching with a physics-consistent perturbation mechanism to construct a differentiable and invertible continuous probability flow model, enabling accurate modeling of state evolution under high-dimensional, non-Gaussian uncertainty. Our contributions are: (i) the first incorporation of generative flow models into dynamical system inference, ensuring physically plausible predictions; and (ii) a novel perturbation architecture explicitly constrained by differential equations to characterize solution multiplicity and state uncertainty. Evaluated on benchmarks including WeatherBench, our method significantly outperforms conventional deterministic and Gaussian-assumption models, delivering high-fidelity, well-calibrated probabilistic forecasts.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Learning dynamical systems is crucial across many fields, yet applying machine learning techniques remains challenging due to missing variables and noisy data. Classical mathematical models often struggle in these scenarios due to the arose ill-posedness of the physical systems. Stochastic machine learning techniques address this challenge by enabling the modeling of such ill-posed problems. Thus, a single known input to the trained machine learning model may yield multiple plausible outputs, and all of the outputs are correct. In such scenarios, probabilistic forecasting is inherently meaningful. In this study, we introduce a variant of flow matching for probabilistic forecasting which estimates possible future states as a distribution over possible outcomes rather than a single-point prediction. Perturbation of complex dynamical states is not trivial. Community uses typical Gaussian or uniform perturbations to crucial variables to model uncertainty. However, not all variables behave in a Gaussian fashion. So, we also propose a generative machine learning approach to physically and logically perturb the states of complex high-dimensional dynamical systems. Finally, we establish the mathematical foundations of our method and demonstrate its effectiveness on several challenging dynamical systems, including a variant of the high-dimensional WeatherBench dataset, which models the global weather at a 5.625° meridional resolution.

Problem

• Probabilistic forecasting for dynamical systems with missing or noisy data
• Modeling non-Gaussian perturbations in high-dimensional dynamical systems
• Improving accuracy in weather prediction using generative machine learning

Innovation

• Flow matching for probabilistic forecasting distributions
• Generative ML for non-Gaussian variable perturbations
• High-dimensional dynamical system state perturbation