Numerical algebraic geometry for model selection and its application to the life sciences

Researchers working with mathematical models are often confronted by the related problems of parameter estimation, model validation, and model selection. These are all optimization problems, well-known to be challenging due to non-linearity, non-convexity and multiple local optima. Furthermore, the challenges are compounded when only partial data is available. Here, we consider polynomial models (e.g., mass-action chemical reaction networks at steady state) and describe a framework for their analysis based on optimization using numerical algebraic geometry. Specifically, we use probability-one polynomial homotopy continuation methods to compute all critical points of the objective function, then filter to recover the global optima. Our approach exploits the geometric structures relating models and data, and we demonstrate its utility on examples from cell signaling, synthetic biology, and epidemiology.

💡 Research Summary

The paper addresses three fundamental tasks that arise when working with mathematical models in the life sciences: model validation, model selection, and parameter estimation. All three are naturally cast as optimization problems, but the non‑linearity and non‑convexity of the underlying polynomial systems make local optimization methods unreliable, especially when data are sparse or noisy. The authors propose a unified framework based on Numerical Algebraic Geometry (NAG) that guarantees, with probability one, the discovery of every critical point of the objective function, thereby enabling exact global optimization.

The central idea is to treat a deterministic polynomial model (e.g., a mass‑action reaction network at steady state) as a real algebraic variety (V_M) defined by the steady‑state equations (f(a,x)=0) together with the output relations (y=g(x)). Experimental data define an affine linear variety (V_D) that fixes the observable components (y) to the measured values (\hat y). The distance between these two varieties, \