SODAs: Sparse Optimization for the Discovery of Differential and Algebraic Equations

2025-03-08

AI Summary

Existing sparse modeling approaches rely on pre-reducing differential-algebraic equations (DAEs) to ordinary differential equations (ODEs), rendering them inadequate for complex systems featuring unknown algebraic constraints and multi-scale dynamics. This paper introduces the first end-to-end sparse identification framework explicitly designed for the structural form of DAEs: it jointly learns differential and algebraic components directly from data—without prior elimination, without requiring predefined algebraic constraints, and without assuming constraint knowledge. The framework enhances numerical stability and interpretability under high collinearity via adaptive symbolic library construction, iterative condition-number regularization, and staged sparse optimization. Experiments on biological, mechanical, and circuit systems demonstrate robust performance on both noisy simulated and real-world measurement data, successfully recovering physically interpretable DAE models with high fidelity.

Technology Category

• Machine Learning: Structured Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Differential-algebraic equations (DAEs) integrate ordinary differential equations (ODEs) with algebraic constraints, providing a fundamental framework for developing models of dynamical systems characterized by timescale separation, conservation laws, and physical constraints. While sparse optimization has revolutionized model development by allowing data-driven discovery of parsimonious models from a library of possible equations, existing approaches for dynamical systems assume DAEs can be reduced to ODEs by eliminating variables before model discovery. This assumption limits the applicability of such methods to DAE systems with unknown constraints and time scales. We introduce Sparse Optimization for Differential-Algebraic Systems (SODAs), a data-driven method for the identification of DAEs in their explicit form. By discovering the algebraic and dynamic components sequentially without prior identification of the algebraic variables, this approach leads to a sequence of convex optimization problems and has the advantage of discovering interpretable models that preserve the structure of the underlying physical system. To this end, SODAs improves numerical stability when handling high correlations between library terms -- caused by near-perfect algebraic relationships -- by iteratively refining the conditioning of the candidate library. We demonstrate the performance of our method on biological, mechanical, and electrical systems, showcasing its robustness to noise in both simulated time series and real-time experimental data.

Problem

• Identifies differential-algebraic equations (DAEs) from data
• Handles systems with unknown constraints and time scales
• Improves numerical stability in high-correlation scenarios

Innovation

• SODAs discovers DAEs without eliminating variables.
• Sequential discovery of algebraic and dynamic components.
• Improves numerical stability by refining library conditioning.