# Ill-Conditioning in Dictionary-Based Dynamic-Equation Learning: A Systems Biology Case Study

arXiv: https://arxiv.org/abs/2603.11330 · HTML: https://mlsanity.com/paper/2603.11330/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 60 references that are in the map.
- First posted on arXiv 11 Mar 2026.

## Builds on (most cited first, 8 of 8)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [A unified sparse optimization framework to learn parsimonious physics-informed models from data](https://mlsanity.com/paper/1906.10612/) (2019)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Analysis of sloppiness in model simulations: unveiling parameter uncertainty when mathematical models are fitted to data](https://mlsanity.com/paper/2203.15184/) (2022)
- [Stabilizing black-box model selection with the inflated argmax](https://mlsanity.com/paper/2410.18268/) (2024)
- [Information theory and discriminative sampling for model discovery](https://mlsanity.com/paper/2512.16000/) (2025)
- [SODAs: Sparse Optimization for the Discovery of Differential and Algebraic Equations](https://mlsanity.com/paper/2503.05993/) (2025)

## Built on by (most cited first, 2 of 2)

- [Self-explainable Operator Learning for Discovering Spatial Patterns in Functional Data](https://mlsanity.com/paper/2607.02203/) (2026)
- [Sparse Orthogonal Regression Technique: A Spectral Framework for Equation Discovery, Approximation, and Integration](https://mlsanity.com/paper/2608.13504/) (2026)

## Abstract (arXiv)

Data-driven discovery of governing equations from time-series data provides a powerful framework for understanding complex biological systems. Library-based approaches that use sparse regression over candidate functions have shown considerable promise, but they face a critical challenge when candidate functions become strongly correlated: numerical ill-conditioning. Poor or restricted sampling, together with particular choices of candidate libraries, can produce strong multicollinearity and numerical instability. In such cases, measurement noise may lead to widely different recovered models, obscuring the true underlying dynamics and hindering accurate system identification. Although sparse regularization promotes parsimonious solutions and can partially mitigate conditioning issues, strong correlations may persist, regularization may bias the recovered models, and the regression problem may remain highly sensitive to small perturbations in the data. We present a systematic analysis of how ill-conditioning affects sparse identification of biological dynamics using benchmark models from systems biology. We show that combinations involving as few as two or three terms can already exhibit strong multicollinearity and extremely large condition numbers. We further show that orthogonal polynomial bases do not consistently resolve ill-conditioning and can perform worse than monomial libraries when the data distribution deviates from the weight function associated with the orthogonal basis. Finally, we demonstrate that when data are sampled from distributions aligned with the appropriate weight functions corresponding to the orthogonal basis, numerical conditioning improves, and orthogonal polynomial bases can yield improved model recovery accuracy across two baseline models.