Foundation Inference Models for Stochastic Differential Equations: A Transformer-based Approach for Zero-shot Function Estimation

Stochastic differential equations (SDEs) describe dynamical systems where deterministic flows, governed by a drift function, are superimposed with random fluctuations dictated by a diffusion function. The accurate estimation (or discovery) of these functions from data is a central problem in machine learning, with wide application across natural and social sciences alike. Yet current solutions are brittle, and typically rely on symbolic regression or Bayesian non-parametrics. In this work, we introduce FIM-SDE (Foundation Inference Model for SDEs), a transformer-based recognition model capable of performing accurate zero-shot estimation of the drift and diffusion functions of SDEs, from noisy and sparse observations on empirical processes of different dimensionalities. Leveraging concepts from amortized inference and neural operators, we train FIM-SDE in a supervised fashion, to map a large set of noisy and discretely observed SDE paths to their corresponding drift and diffusion functions. We demonstrate that one and the same (pretrained) FIM-SDE achieves robust zero-shot function estimation (i.e. without any parameter fine-tuning) across a wide range of synthetic and real-world processes, from canonical SDE systems (e.g. double-well dynamics or weakly perturbed Hopf bifurcations) to human motion recordings and oil price and wind speed fluctuations.