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.

Foundation Inference Models for Markov Jump Processes

Markov jump processes are continuous-time stochastic processes which describe dynamical systems evolving in discrete state spaces. These processes find wide application in the natural sciences and machine learning, but their inference is known to be far from trivial. In this work we introduce a methodology for zero-shot inference of Markov jump processes (MJPs), on bounded state spaces, from noisy and sparse observations, which consists of two components. First, a broad probability distribution over families of MJPs, as well as over possible observation times and noise mechanisms, with which we simulate a synthetic dataset of hidden MJPs and their noisy observation process. Second, a neural network model that processes subsets of the simulated observations, and that is trained to output the initial condition and rate matrix of the target MJP in a supervised way. We empirically demonstrate that one and the same (pretrained) model can infer, in a zero-shot fashion, hidden MJPs evolving in state spaces of different dimensionalities. Specifically, we infer MJPs which describe (i) discrete flashing ratchet systems, which are a type of Brownian motors, and the conformational dynamics in (ii) molecular simulations, (iii) experimental ion channel data and (iv) simple protein folding models. What is more, we show that our model performs on par with state-of-the-art models which are finetuned to the target datasets.