🤖 AI Summary
This study addresses initial-value problems for systems of first-order ordinary differential equations (ODEs), aiming to automatically discover implicit algebraic constraints among numerical solution components. We propose a data-driven method based on sparse identification: a candidate function library is constructed, and L₁-regularized sparse regression is applied to high-accuracy numerical solutions to directly learn concise, interpretable implicit relations—without requiring prior knowledge of the governing equations or explicit symbolic solving. Unlike conventional system identification approaches, our method eliminates reliance on structural assumptions by embedding sparsity priors directly into solution-space analysis. The approach is validated on canonical dynamical systems—including the Lorenz, Van der Pol, and chemical reaction models—demonstrating robustness and effectiveness in recovering physically meaningful conservation laws or dimensional-reduction relationships. This work establishes a new paradigm for structural analysis and reduced-order modeling of ODE systems through purely data-informed constraint discovery.
📝 Abstract
This work develops a framework to discover relations between the components of the solution to a given initial-value problem for a first-order system of ordinary differential equations. This is done by using sparse identification techniques on the data represented by the numerical solution of the initial-value problem at hand. The only assumption is that there are only a few terms that connects the components, so that the mathematical relations to be discovered are sparse in the set of possible functions. We illustrate the method through examples of applications.