Abstract: We present a symbolic algorithmic approach that allows to compute invariant manifolds and corresponding reduced systems for differential equations modeling biological networks which comprise chemical reaction networks for cellular biochemistry, and compartmental models for pharmacology, epidemiology and ecology. Multiple time scales of a given network are obtained by scaling, based on tropical geometry. Our reduction is mathematically justified within a singular perturbation setting using a recent result by Cardin and Teixeira. The existence of invariant manifolds is subject to hyperbolicity conditions, which we test algorithmically using Hurwitz criteria. We finally obtain a sequence of nested invariant manifolds and respective reduced systems on those manifolds. Our theoretical results are generally accompanied by rigorous algorithmic descriptions suitable for direct implementation based on existing off-the-shelf software systems, specifically symbolic computation libraries and Satisfiability Modulo Theories solvers. We present computational examples taken from the well-known BioModels database using our own prototypical implementations.
(with Niclas Kruff, Ovidiu Radulescu, Thomas Sturm and Sebastian Walcher)
The paper has been submitted to a journal and a preprint is available on arXiv.
I posted an updated version of my paper with faster timings and better text.
Abstract: “A novel way to use SMT (Satisfiability Modulo Theories) solvers to compute the tropical prevariety (resp. equilibrium) of a polynomial system is presented. The new method is benchmarked against a naive approach that uses purely polyhedral methods. It turns out that the SMT approach is faster than the polyhedral approach for models that would otherwise take more than one minute to compute, in many cases by a factor of 60 or more, and in the worst case is only slower by a factor of two. Furthermore, the new approach is an anytime algorithm, thus offering a way to compute parts of the solution when the polyhedral approach is infeasible. ”
Paper on arXiv: https://arxiv.org/abs/2004.07058
Source code: https://gitlab.com/cxxl/smtcut
Abstract: “I am presenting a novel way to use SMT (Satisfiability Modulo Theory) to compute the tropical prevariety (resp. equilibrium) of a polynomial system. The new method is benchmarked against a naive approach that uses purely polyhedral methods. It turns out that the SMT approach is faster than the polyhedral approach for models that would otherwise take more than one minute to compute, in many cases by a factor of 25 or more. Furthermore, the new approach offers a way to compute at least parts of the solution if the polyhedral approach is infeasible.”
The paper is available on arXiv: https://arxiv.org/abs/2004.07058
The source code is available at https://gitlab.com/cxxl/smtcut.