Foundation Inference Models for Ordinary Differential Equations

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 low-dimensional {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 pretraining prior distribution. 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.