# Probabilistic Forecasting with Stochastic Interpolants and F"ollmer Processes

arXiv: https://arxiv.org/abs/2403.13724 · HTML: https://mlsanity.com/paper/2403.13724/

## Key facts

- Cited by 66 papers in the map (data through 14 Oct 2026).
- Builds on 42 of its 72 references that are in the map.
- First posted on arXiv 20 Mar 2024.

## Builds on (most cited first, 30 of 42)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Taming Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2012.09841/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Stable Video Diffusion: Scaling Latent Video Diffusion Models to Large Datasets](https://mlsanity.com/paper/2311.15127/) (2023)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Conditional Generative Adversarial Nets](https://mlsanity.com/paper/1411.1784/) (2014)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Towards Accurate Generative Models of Video: A New Metric & Challenges](https://mlsanity.com/paper/1812.01717/) (2018)
- [Cascaded Diffusion Models for High Fidelity Image Generation](https://mlsanity.com/paper/2106.15282/) (2021)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Time Series Forecasting With Deep Learning: A Survey](https://mlsanity.com/paper/2004.13408/) (2020)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Unsupervised Learning for Physical Interaction through Video Prediction](https://mlsanity.com/paper/1605.07157/) (2016)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Stochastic Adversarial Video Prediction](https://mlsanity.com/paper/1804.01523/) (2018)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [A Review on Deep Learning Techniques for Video Prediction](https://mlsanity.com/paper/2004.05214/) (2020)
- [DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting](https://mlsanity.com/paper/2306.01984/) (2023)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Theoretical guarantees for sampling and inference in generative models with latent diffusions](https://mlsanity.com/paper/1903.01608/) (2019)

## Built on by (most cited first, 30 of 66)

- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Schr"odinger Bridge Flow for Unpaired Data Translation](https://mlsanity.com/paper/2409.09347/) (2024)
- [Non-stationary Diffusion For Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2505.04278/) (2025)
- [Diffuse Everything: Multimodal Diffusion Models on Arbitrary State Spaces](https://mlsanity.com/paper/2506.07903/) (2025)
- [FlowDAS: A Stochastic Interpolant-based Framework for Data Assimilation](https://mlsanity.com/paper/2501.16642/) (2025)
- [Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach](https://mlsanity.com/paper/2506.03979/) (2025)
- [Lipschitz-Guided Design of Interpolation Schedules in Generative Models](https://mlsanity.com/paper/2509.01629/) (2025)
- [Demystifying Data-Driven Probabilistic Medium-Range Weather Forecasting](https://mlsanity.com/paper/2601.18111/) (2026)
- [Elucidating the Design Choice of Probability Paths in Flow Matching for Forecasting](https://mlsanity.com/paper/2410.03229/) (2024)
- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [Guiding diffusion models to reconstruct flow fields from sparse data](https://mlsanity.com/paper/2510.19971/) (2025)
- [Parametric model reduction of mean-field and stochastic systems via higher-order action matching](https://mlsanity.com/paper/2410.12000/) (2024)
- [Improving Flow Matching by Aligning Flow Divergence](https://mlsanity.com/paper/2602.00869/) (2026)
- [Plug-in estimation of Schr"odinger bridges](https://mlsanity.com/paper/2408.11686/) (2024)
- [Hierarchical Rectified Flow Matching with Mini-Batch Couplings](https://mlsanity.com/paper/2507.13350/) (2025)
- [Project and Generate: Divergence-Free Neural Operators for Incompressible Flows](https://mlsanity.com/paper/2603.24500/) (2026)
- [Probing forced responses and causality in data-driven climate emulators: conceptual limitations and the role of reduced-order models](https://mlsanity.com/paper/2506.22552/) (2025)
- [Generative Modeling from Black-box Corruptions via Self-Consistent Stochastic Interpolants](https://mlsanity.com/paper/2512.10857/) (2025)
- [Stochastic Flow Matching for Resolving Small-Scale Physics](https://mlsanity.com/paper/2410.19814/) (2024)
- [Scale-Adaptive Generative Flows for Multiscale Scientific Data](https://mlsanity.com/paper/2509.02971/) (2025)
- [DAISI: Data Assimilation with Inverse Sampling using Stochastic Interpolants](https://mlsanity.com/paper/2512.00252/) (2025)
- [Finite-Time Analysis of Discrete-Time Stochastic Interpolants](https://mlsanity.com/paper/2502.09130/) (2025)
- [Training-Free Bayesian Filtering with Generative Emulators](https://mlsanity.com/paper/2605.20028/) (2026)
- [An Efficient On-Policy Deep Learning Framework for Stochastic Optimal Control](https://mlsanity.com/paper/2410.05163/) (2024)
- [A Minimum Variance Path Principle for Accurate and Stable Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2602.00834/) (2026)
- [Is Flow Matching Just Trajectory Replay for Sequential Data?](https://mlsanity.com/paper/2602.08318/) (2026)
- [Recurrent Interpolants for Probabilistic Time Series Prediction](https://mlsanity.com/paper/2409.11684/) (2024)
- [Conditional Diffusion-Flow models for generating 3D cosmic density fields: applications to f(R) cosmologies](https://mlsanity.com/paper/2502.17087/) (2025)
- [A Jump-Diffusion Framework for Irregular Time Series Generation](https://mlsanity.com/paper/2505.23215/) (2025)

## Abstract (arXiv)

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 F"ollmer 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.