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

Siddharth Rout, Eldad Haber, Stephane Gaudreault

TL;DR

This work proposes a flow matching-based generative approach to learn physically consistent perturbations of the initial conditions, avoiding artifacts caused by Gaussian noise, and employs deterministic flow matching models with Ordinary Differential Equation integrators for efficient ensemble propagation with fewer integration steps.

Abstract

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.

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

TL;DR

This work proposes a flow matching-based generative approach to learn physically consistent perturbations of the initial conditions, avoiding artifacts caused by Gaussian noise, and employs deterministic flow matching models with Ordinary Differential Equation integrators for efficient ensemble propagation with fewer integration steps.

Abstract

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). 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.

Paper Structure

This paper contains 49 sections, 2 theorems, 62 equations, 29 figures, 14 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Contributions
  4. Mathematical Foundation
  5. Probabilistic Forecasting and Flow Matching
  6. Sampling the Perturbed States
  7. Experiments
  8. Datasets
  9. Experiment and Results
  10. Metrics:
  11. Discussions
  12. Conclusion
  13. Definitions, Theorems and Proofs
  14. Algorithms
  15. Computational Cost: ODEs vs. SDEs
  16. ...and 34 more sections

Key Result

Lemma A.1

Let ${\cal Y}$ be the physical space for an n-dimensional dynamical system ${\cal D}$. Given an accurately trained Gaussifying flow matching function on the data in $\cal Y$, the encoder $\cal E$ and decoder $D$ are inverses of each other.

Figures (29)

  • Figure 1: A cartoonified graphical abstract of our proposition in the context of probabilistic forecasting of baseball trajectory after a hit using our technique. Notably, the physical perturbation is generated independently from the deterministic forecasting process.
  • Figure 2: Temporal trajectories for the predator-prey model. Note that the trajectories approach each other closely but do not intersect.
  • Figure 3: Left: The solution for ${\bf y}_1(0)=1$ and ${\bf y}_2(0) \sim U(0,1)$ at $t=200$. Right:The solution for ${\bf y}(0)=[0.1, 0.3]^{\top} + \epsilon$ where $\epsilon \sim N(0, 0.05 {\bf I})$ at $t=200$
  • Figure 4: Diagram to show the process for obtaining a physical perturbed state.
  • Figure 5: Comparison of the actual final distribution and that obtained using FM on the predator-prey model. Trajectories for transport learned by FM are in blue. Note that the trajectories are not physical.
  • ...and 24 more figures

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Example 1
  • Example 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 5 more