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

arXiv: https://arxiv.org/abs/2503.05993 · HTML: https://mlsanity.com/paper/2503.05993/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 91 references that are in the map.
- First posted on arXiv 8 Mar 2025.

## Builds on (most cited first, 24 of 24)

- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [A Unified Framework for Sparse Relaxed Regularized Regression: SR3](https://mlsanity.com/paper/1807.05411/) (2018)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [Latent Space Symmetry Discovery](https://mlsanity.com/paper/2310.00105/) (2023)
- [Active Learning in Robotics: A Review of Control Principles](https://mlsanity.com/paper/2106.13697/) (2021)
- [Discovering Sparse Interpretable Dynamics from Partial Observations](https://mlsanity.com/paper/2107.10879/) (2021)
- [Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants](https://mlsanity.com/paper/2211.10575/) (2022)
- [Numerical differentiation of noisy data: A unifying multi-objective optimization framework](https://mlsanity.com/paper/2009.01911/) (2020)
- [Physics-informed Spline Learning for Nonlinear Dynamics Discovery](https://mlsanity.com/paper/2105.02368/) (2021)
- [Asymptotic consistency of the WSINDy algorithm in the limit of continuum data](https://mlsanity.com/paper/2211.16000/) (2022)
- [Data-driven Discovery of Cyber-Physical Systems](https://mlsanity.com/paper/1810.00697/) (2018)
- [Inferring networks from time series: a neural approach](https://mlsanity.com/paper/2303.18059/) (2023)
- [High dimensional thresholded regression and shrinkage effect](https://mlsanity.com/paper/1605.03306/) (2016)
- [On Identification of Distribution Grids](https://mlsanity.com/paper/1711.01526/) (2017)
- [Discover governing differential equations from evolving systems](https://mlsanity.com/paper/2301.07863/) (2023)
- [On the design-dependent suboptimality of the Lasso](https://mlsanity.com/paper/2402.00382/) (2024)
- [Dynamic Modeling, Stability, and Control of Power Systems with Distributed Energy Resources](https://mlsanity.com/paper/1804.04933/) (2018)
- [Dynamic State Estimation for Power System Control and Protection](https://mlsanity.com/paper/2012.14927/) (2020)
- [On Data-Driven Modeling and Control in Modern Power Grids Stability: Survey and Perspective](https://mlsanity.com/paper/2308.03591/) (2023)
- [Identifiability of Differential-Algebraic Systems](https://mlsanity.com/paper/2405.13818/) (2024)

## Built on by (most cited first, 1 of 1)

- [Ill-Conditioning in Dictionary-Based Dynamic-Equation Learning: A Systems Biology Case Study](https://mlsanity.com/paper/2603.11330/) (2026)

## Abstract (arXiv)

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 for 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. It 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.