Foundation Inference Models for Ordinary Differential Equations

Abstract

Ordinary differential equations (ODEs) are central to scientific modelling, but inferring their vector fields from noisy trajectories remains challenging. Current approaches such as symbolic regression, Gaussian process (GP) regression, and Neural ODEs often require complex training pipelines and substantial machine learning expertise, or they depend strongly on system-specific prior knowledge. We propose FIM-ODE, a pretrained Foundation Inference Model that *amortises* ODE inference by predicting the vector field directly from noisy trajectory data *in a single forward pass*. We pretrain FIM-ODE on a prior distribution over ODEs with low-degree polynomial vector fields and represent the target field with neural operators. FIM-ODE achieves strong *zero-shot* performance, matching and often improving upon ODEFormer, a recent pretrained symbolic baseline, across a range of regimes despite using a simpler training prior. Pretraining also provides a strong initialisation for finetuning, enabling fast and stable adaptation that outperforms modern neural and GP baselines without requiring machine learning expertise. We release pretrained weights with the supplementary material.