Optimal monomial quadratization for ODE systems

Abstract

Transformation of a polynomial ODE system to a special quadratic form has been successfully used recently as a preprocessing step for model order reduction methods. However, to the best of our knowledge, there has been no practical algorithm for performing this step automatically with any optimality guarantees. We present an algorithm that, given a system of polynomial ODEs, finds a transformation into a quadratic ODE system by introducing new variables which are monomials of the original variables. The algorithm is guaranteed to produce an optimal transformation of this form. The algorithm is implemented, and we demonstrate it on examples from the literature.

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

“…On the benchmark presented here, our maxSAT implementation is sufficient but we also use by default a heuristic algorithm that trades optimality for better performance. It should also be noted that a new algorithm has been recently proposed for the global optimization problem in [3]. This work may be improved in several directions.…”

Compiling Elementary Mathematical Functions into Finite Chemical Reaction Networks via a Polynomialization Algorithm for ODEs

“…On the benchmark presented here, our maxSAT implementation is sufficient but we also use by default a heuristic algorithm that trades optimality for better performance. It should also be noted that a new algorithm has been recently proposed for the global optimization problem in [3].…”

“…That definition for input/output functions computed by a CRN is used in [11] to show the Turing completeness of continuous CRNs in the sense that any computable function over the real numbers can be computed by a CRN over a finite set of formal molecular species using at most bimolecular reactions with mass action law kinetics. The proof uses a previous result of Turing completeness for functions defined by polynomial ordinary differential equation initial value problems (PIVP) [2], the dual-rail encoding of real variables by the difference of concentration between two molecular species [21,18], and a back-end quadratization transformation to restrict to elementary reactions with at most two reactants [5,19,3]. This proof gives rise to a pipeline, implemented in BIOCHAM-4 2 , to compile any computable real function presented by a PIVP into a finite CRN.…”

Chemical Integration of ODEs using Idealized Abstract Solutions

“…Therefore, we may investigate if there exists a combinatorial algorithm that always returns the optimal reaction network with the least number of added variables after quadraticization (e.g. see [60]).…”