Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes

Yifan Chen, Mark Goldstein, Mengjian Hua, Michael Samuel Albergo, Nicholas Matthew Boffi, Eric Vanden-Eijnden
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:6728-6756, 2024.

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. The code is available at https://github.com/interpolants/forecasting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-chen24n, title = {Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes}, author = {Chen, Yifan and Goldstein, Mark and Hua, Mengjian and Albergo, Michael Samuel and Boffi, Nicholas Matthew and Vanden-Eijnden, Eric}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {6728--6756}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/chen24n/chen24n.pdf}, url = {https://proceedings.mlr.press/v235/chen24n.html}, 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. The code is available at https://github.com/interpolants/forecasting.} }
Endnote
%0 Conference Paper %T Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes %A Yifan Chen %A Mark Goldstein %A Mengjian Hua %A Michael Samuel Albergo %A Nicholas Matthew Boffi %A Eric Vanden-Eijnden %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-chen24n %I PMLR %P 6728--6756 %U https://proceedings.mlr.press/v235/chen24n.html %V 235 %X 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. The code is available at https://github.com/interpolants/forecasting.
APA
Chen, Y., Goldstein, M., Hua, M., Albergo, M.S., Boffi, N.M. & Vanden-Eijnden, E.. (2024). Probabilistic Forecasting with Stochastic Interpolants and Föllmer Processes. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:6728-6756 Available from https://proceedings.mlr.press/v235/chen24n.html.

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