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Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes

Yifan Chen, Mark Goldstein, Mengjian Hua, Michael S. Albergo, Nicholas M. Boffi, Eric Vanden-Eijnden

TL;DR

The paper develops a principled framework for probabilistic forecasting of high-dimensional dynamical systems by transporting a point observation to the conditional forecast distribution $\rho_c(\cdot|x_0)$ using stochastic interpolants and SDEs. It shows how to learn the drift via square-loss regression and how to tune diffusion post-training to realize a Föllmer process through KL optimization, linking to Schrödinger bridge theory. The approach is validated on multi-modal synthetic dynamics, the stochastic Navier–Stokes equations, and video prediction tasks (KTH and CLEVRER), producing diverse, physically consistent ensembles and often outperforming deterministic baselines. The methodology enables fast, autoregressive forecasting in complex systems while preserving uncertainty quantification, with potential extensions to empirical weather data and physics-informed conditioning.

Abstract

We propose a framework for probabilistic forecasting of dynamical systems based on generative modeling. Given observations of the system state over time, we formulate the forecasting problem as sampling from the conditional distribution of the future system state given its current state. To this end, we leverage the framework of stochastic interpolants, which facilitates the construction of a generative model between an arbitrary base distribution and the target. We design a fictitious, non-physical stochastic dynamics that takes as initial condition the current system state and produces as output a sample from the target conditional distribution in finite time and without bias. This process therefore maps a point mass centered at the current state onto a probabilistic ensemble of forecasts. We prove that the drift coefficient entering the stochastic differential equation (SDE) achieving this task is non-singular, and that it can be learned efficiently by square loss regression over the time-series data. We show that the drift and the diffusion coefficients of this SDE can be adjusted after training, and that a specific choice that minimizes the impact of the estimation error gives a Föllmer process. We highlight the utility of our approach on several complex, high-dimensional forecasting problems, including stochastically forced Navier-Stokes and video prediction on the KTH and CLEVRER datasets.

Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes

TL;DR

The paper develops a principled framework for probabilistic forecasting of high-dimensional dynamical systems by transporting a point observation to the conditional forecast distribution using stochastic interpolants and SDEs. It shows how to learn the drift via square-loss regression and how to tune diffusion post-training to realize a Föllmer process through KL optimization, linking to Schrödinger bridge theory. The approach is validated on multi-modal synthetic dynamics, the stochastic Navier–Stokes equations, and video prediction tasks (KTH and CLEVRER), producing diverse, physically consistent ensembles and often outperforming deterministic baselines. The methodology enables fast, autoregressive forecasting in complex systems while preserving uncertainty quantification, with potential extensions to empirical weather data and physics-informed conditioning.

Abstract

We propose a framework for probabilistic forecasting of dynamical systems based on generative modeling. Given observations of the system state over time, we formulate the forecasting problem as sampling from the conditional distribution of the future system state given its current state. To this end, we leverage the framework of stochastic interpolants, which facilitates the construction of a generative model between an arbitrary base distribution and the target. We design a fictitious, non-physical stochastic dynamics that takes as initial condition the current system state and produces as output a sample from the target conditional distribution in finite time and without bias. This process therefore maps a point mass centered at the current state onto a probabilistic ensemble of forecasts. We prove that the drift coefficient entering the stochastic differential equation (SDE) achieving this task is non-singular, and that it can be learned efficiently by square loss regression over the time-series data. We show that the drift and the diffusion coefficients of this SDE can be adjusted after training, and that a specific choice that minimizes the impact of the estimation error gives a Föllmer process. We highlight the utility of our approach on several complex, high-dimensional forecasting problems, including stochastically forced Navier-Stokes and video prediction on the KTH and CLEVRER datasets.
Paper Structure (47 sections, 9 theorems, 114 equations, 10 figures, 3 tables, 3 algorithms)

This paper contains 47 sections, 9 theorems, 114 equations, 10 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Deterministic vs Probabilistic Forecasting.
  4. Probabilistic Forecasting with Generative Models.
  5. Föllmer processes.
  6. Setup and Main Results
  7. Conditional PDF
  8. Probabilistic forecasting.
  9. Signal recovery from corrupted data.
  10. Probabilistic forecasting from noisy observations.
  11. Generation with Stochastic Interpolants
  12. Generalizations with Tunable Diffusion
  13. KL Optimization and Föllmer Processes
  14. Implementation
  15. Numerical illustrations
  16. ...and 32 more sections

Key Result

Theorem 3.1

Let $b_s(x,x_0)$ be the unique minimizer over all $\hat{b}_s(x,x_0)$ of the objective where $\mathbb{E}$ denotes an expectation over $(x_0,x_1)\sim \rho$ and $W$ with $(x_0,x_1) \perp W$, $I_s$ is given in eq:stoch:int:def, and we definedHere and below the dot denotes derivative with respect to $s$. Then the solutions to the SDE are such that $\text{Law}(X_s) = \text{Law}(I_s|x_0)$ for all $(s,x

Figures (10)

  • Figure 1: Invariant PDF for the jump-diffusion process.
  • Figure 2: Forecasting comparison for the jump-diffusion process. (Left) Comparison of the truth to the forecasted prediction in the angular coordinates. (Middle) Ground truth KDEs at various lag times $\tau$. (Right) Forecasted KDEs at the same lag times $\tau$.
  • Figure 3: Temporal forecasting on stochastically-forced Navier Stokes. (Top left) Different forecasts from our method at lag $\tau =2$, for a fixed $\omega_t$. (Bottom left) Comparisons between the forecast sample mean and standard deviation for this $\omega_t$ against the truths. (Right) Enstrophy spectrum of the true and forecasted conditional distribution. Note: All the NS figures in this paper share the same color bars for the vorticity field (on a scale from $-5$ to $5$) and std (on a scale from $0$ to $3$) respectively.
  • Figure 4: Spatiotemporal forecasting on stochastically-forced Navier Stokes. (Left) Low resolution $\omega_t$ and three of our forecasted samples at $t+1$. (Right) Enstrophy spectrums of the low resolution $\omega_t$, the superresolution forecast of $\omega_{t+1}$, and the true $\omega_{t+1}$.
  • Figure 5: Video generation on the CLEVRER dataset. (Top row) Real trajectory. (Second row) Generated trajectory. A new, red cube enters the scene. (Third row) Real trajectory. (Fourth row) Generated trajectory. A new green cube enters the scene, and collision physics is respected (green ball hits red cube).
  • ...and 5 more figures

Theorems & Definitions (17)

  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Definition 2.1
  • Theorem 2.2
  • proof : Proof of Theorem \ref{['thm:1:b']}.
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof : Proof:
  • ...and 7 more