Identifying ODEs from Unstructured Data with Causal Representation Learning

2026-09-29

AI Summary

This study addresses the challenge of recovering dynamical ordinary differential equations (ODEs) from high-dimensional unstructured data, where existing methods lack theoretical guarantees and variables obtained via causal representation learning (CRL) are poorly suited for sparse discovery. To bridge this gap, we propose SPEED-AE, a framework that integrates component-wise autoencoders with sparse regression to constrain variable identifiability as monomial diffeomorphisms, thereby learning transformations amenable to sparse ODE discovery. Theoretically, this work reconciles CRL with equation discovery by formally bridging their compatibility gap. Empirically, experiments on systems such as Lotka-Volterra demonstrate that SPEED-AE significantly enhances disentanglement capabilities, accurately recovers ground-truth ODEs, and achieves state-of-the-art predictive performance.

Abstract

We study the problem of recovering the governing ODE of a dynamical system from unstructured, high-dimensional observations such as images. Existing methods for ODE discovery typically assume direct measurements of the variables, or do not provide theoretical guarantees on the learned variables and equations. While Causal Representation Learning (CRL) methods provide guarantees on identifying variables from high-dimensional observations up to component-wise diffeomorphisms, we show that in general these variables cannot be used directly as input to equation discovery methods, which typically assume that the variables will lead to sparse equations. So we introduce SParse Equivalent Equation Discovery AutoEncoder (SPEED-AE), a framework that combines a pretrained CRL method with a component-wise autoencoder that learns transformations of variables that are amenable to sparse ODE discovery. We show that for polynomial ODEs, this additional step allows us to restrict the identifiability of each variable from polynomial to monomial diffeomorphisms. Experiments on Lotka-Volterra, Lorenz, and a two-pendulum system show that SPEED-AE improves on the disentanglement of the CRL methods and that it recovers ODEs that are closest to the ground truth, while achieving state-of-the-art forecasting performance.

Problem

• Ordinary Differential Equations
• Unstructured Data
• Causal Representation Learning
• Equation Discovery
• Dynamical Systems

Innovation

• Causal Representation Learning
• ODE Discovery
• Sparse Equation
• Autoencoder
• Identifiability