Summary
It is hard to figure out the exact rules that govern a system just by looking at complex data like videos or pictures. The authors study how to find the hidden equations that describe how a system changes over time when you can only see high-dimensional data, like images, but not the actual variables. They introduce a new method called SPEED-AE that improves how we can extract variables from these images and then discover the equations that govern them, especially when those equations are polynomials. Tests on well-known systems, such as predator-prey populations and swinging pendulums, show that their method finds equations closer to the true ones and predicts future behavior better than previous methods.
What this means in practice
- •For control system engineers: Discover governing equations from high-dimensional sensor data such as images to design accurate controllers for dynamic systems.
- •For robotics developers: Identify interpretable motion and interaction models from visual data to improve robot prediction and planning capabilities.
Authors
Alessandro Trenta, Riccardo Massidda, Davide Bacciu, Sara Magliacane
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.