# Fast and Flexible Probabilistic Forecasting of Dynamical Systems using Flow Matching and Physical Perturbation

arXiv: https://arxiv.org/abs/2508.01101 · HTML: https://mlsanity.com/paper/2508.01101/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 21 of its 49 references that are in the map.
- First posted on arXiv 1 Aug 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
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- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
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- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Unsupervised Learning of Video Representations using LSTMs](https://mlsanity.com/paper/1502.04681/) (2015)
- [ClimaX: A foundation model for weather and climate](https://mlsanity.com/paper/2301.10343/) (2023)
- [WeatherBench: A benchmark dataset for data-driven weather forecasting](https://mlsanity.com/paper/2002.00469/) (2020)
- [Diagnosing and Enhancing VAE Models](https://mlsanity.com/paper/1903.05789/) (2019)
- [An Introduction to Deep Generative Modeling](https://mlsanity.com/paper/2103.05180/) (2021)
- [Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes](https://mlsanity.com/paper/2403.13724/) (2024)
- [FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scale](https://mlsanity.com/paper/2507.12144/) (2025)
- [Probabilistic Weather Forecasting with Hierarchical Graph Neural Networks](https://mlsanity.com/paper/2406.04759/) (2024)
- [CloudCast: A Satellite-Based Dataset and Baseline for Forecasting Clouds](https://mlsanity.com/paper/2007.07978/) (2020)
- [Advection Augmented Convolutional Neural Networks](https://mlsanity.com/paper/2406.19253/) (2024)

## Abstract (arXiv)

Learning dynamical systems from incomplete or noisy data is inherently ill-posed, as a single observation may correspond to multiple plausible futures. While physics-based ensemble forecasting relies on perturbing initial states to capture uncertainty, standard Gaussian or uniform perturbations often yield unphysical initial states in high-dimensional systems. Existing machine learning approaches address this via diffusion models, which rely on inference via computationally expensive stochastic differential equations (SDEs). We introduce a novel framework that decouples perturbation generation from propagation. First, we propose a flow matching-based generative approach to learn physically consistent perturbations of the initial conditions, avoiding artifacts caused by Gaussian noise. Second, we employ deterministic flow matching models with Ordinary Differential Equation (ODE) integrators for efficient ensemble propagation with fewer integration steps. We validate our method on nonlinear dynamical system benchmarks, including the Lotka-Volterra Predator-Prey system, MovingMNIST, and high-dimensional WeatherBench data (5.625^circ). Our approach achieves state-of-the-art probabilistic scoring, as measured by the Continuous Ranked Probability Score (CRPS), and physical consistency, while offering significantly faster inference than diffusion-based baselines.