Optimal Monomial Quadratization for ODE Systems

Abstract

Quadratization is a transform of a system of ODEs with polynomial right-hand side into a system of ODEs with at most quadratic right-hand side via the introduction of new variables. Quadratization problem is, given a system of ODEs with polynomial right-hand side, transform the system to a system with quadratic right-hand side by introducing new variables. Such transformations have been used, for example, as a preprocessing step by model order reduction methods and for transforming chemical reaction networks.

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 in the original variables. The algorithm is guaranteed to produce an optimal transformation of this form (that is, the number of new variables is as small as possible), and it is the first algorithm with such a guarantee we are aware of. Its performance compares favorably with the existing software, and it is capable to tackle problems that were out of reach before.

The article was prepared within the framework of the HSE University Basic Research Program. GP was partially supported by NSF grants DMS-1853482, DMS-1760448, DMS-1853650, CCF-1564132, and CCF-1563942 and by the Paris Ile-de-France region. The authors are grateful to Mathieu Hemery, François Fages, and Sylvain Soliman for helpful discussions. The work has started when G. Pogudin worked at the Higher School of Economics, Moscow. The authors would like to thank the referees for their comments, which helped us improve the manuscript.

Appendix: Benchmark Systems

Most of the benchmark systems used in this paper (in Tables 2 and 3) are described in [13]. Here we show additional benchmarks we have introduced:

• 1. Cubic Cycle(n). For every integer $n > 1$, we define a system in variables $x_1, \ldots , x_n$ by

  $$ x_1' = x_2^3,\; x_2' = x_3^3,\;\ldots ,\; x_n' = x_1^3. $$

• 2. Cubic Bicycle(n). For every integer $n > 1$, we define a system in variables $x_1, \ldots , x_n$ by

  $$ x_1' = x_n^3 + x_2^3,\; x_2' = x_1^3 + x_3^3,\;\ldots ,\; x_n' = x_{n - 1}^3 + x_1^3. $$