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.

Towards Fast Coarse-graining and Equation Discovery with Foundation Inference Models

High-dimensional recordings of dynamical processes are often characterized by a much smaller set of effective variables, evolving on low-dimensional manifolds. Identifying these latent dynamics requires solving two intertwined problems: discovering appropriate coarse-grained variables and simultaneously fitting the governing equations. Most machine learning approaches tackle these tasks jointly by training autoencoders together with models that enforce dynamical consistency. We propose to decouple the two problems by leveraging the recently introduced Foundation Inference Models (FIMs). FIMs are pretrained models that estimate the infinitesimal generators of dynamical systems (e.g., the drift and diffusion of a stochastic differential equation) in zero-shot mode. By amortizing the inference of the dynamics through a FIM with frozen weights, and training only the encoder-decoder map, we define a simple, simulation-consistent loss that stabilizes representation learning. A proof of concept on a stochastic double-well system with semicircle diffusion, embedded into synthetic video data, illustrates the potential of this approach for fast and reusable coarse-graining pipelines.

Towards Foundation Inference Models that Learn ODEs In-Context

Ordinary differential equations (ODEs) describe dynamical systems evolving deterministically in continuous time. Accurate data-driven modeling of systems as ODEs, a central problem across the natural sciences, remains challenging, especially if the data is sparse or noisy. We introduce FIM-ODE (Foundation Inference Model for ODEs), a pretrained neural model designed to estimate ODEs zero-shot (i.e., in context) from sparse and noisy observations. Trained on synthetic data, the model utilizes a flexible neural operator for robust ODE inference, even from corrupted data. We empirically verify that FIM-ODE provides accurate estimates, on par with a neural state-of-the-art method, and qualitatively compare the structure of their estimated vector fields.