Algebra, coalgebra, and minimization in polynomial differential equations

We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system structure based on the concept of Lie-derivative, thus inducing a notion of L-bisimulation. We prove that two states (variables) are L-bisimilar if and only if they correspond to the same solution in the ode's system. We then characterize L-bisimilarity algebraically, in terms of certain ideals in the polynomial ring that are invariant under Lie-derivation. This characterization allows us to develop a complete algorithm, based on building an ascending chain of ideals, for computing the largest L-bisimulation containing all valid identities that are instances of a user-specified template. A specific largest L-bisimulation can be used to build a reduced system of ode's, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations. A computationally less demanding approximate reduction and linearization technique is also proposed.Comment: 27 pages, extended and revised version of FOSSACS 2017 pape

On the Coalgebra of Partial Differential Equations

We note that the coalgebra of formal power series in commutative variables is final in a certain subclass of coalgebras. Moreover, a system Sigma of polynomial PDEs, under a coherence condition, naturally induces such a coalgebra over differential polynomial expressions. As a result, we obtain a clean coinductive proof of existence and uniqueness of solutions of initial value problems for PDEs. Based on this characterization, we give complete algorithms for checking equivalence of differential polynomial expressions, given Sigma

Formal lumping of polynomial differential equations through approximate equivalences

It is well known that exact notions of model abstraction and reduction for dynamical systems may not be robust enough in practice because they are highly sensitive to the specific choice of parameters. In this paper we consider this problem for nonlinear ordinary differential equations (ODEs) with polynomial derivatives. We introduce a model reduction technique based on approximate differential equivalence, i.e., a partition of the set of ODE variables that performs an aggregation when the variables are governed by nearby derivatives. We develop algorithms to (i) compute the largest approximate differential equivalence; (ii) construct an approximately reduced model from the original one via an appropriate perturbation of the coefficients of the polynomials; and (iii) provide a formal certificate on the quality of the approximation as an error bound, computed as an over-approximation of the reachable set of the reduced model. Finally, we apply approximate differential equivalences to case studies on electric circuits, biological models, and polymerization reaction networks

Algebra, coalgebra, and minimization in polynomial differential equations

We consider reasoning and minimization in systems of polynomial ordinary differential equations (ode's). The ring of multivariate polynomials is employed as a syntax for denoting system behaviours. We endow this set with a transition system structure based on the concept of Lie-derivative, thus inducing a notion of L-bisimulation. We prove that two states (variables) are L-bisimilar if and only if they correspond to the same solution in the ode's system. We then characterize L-bisimilarity algebraically, in terms of certain ideals in the polynomial ring that are invariant under Lie-derivation. This characterization allows us to develop a complete algorithm, based on building an ascending chain of ideals, for computing the largest L-bisimulation containing all valid identities that are instances of a user-specified template. A specific largest L-bisimulation can be used to build a reduced system of ode's, equivalent to the original one, but minimal among all those obtainable by linear aggregation of the original equations. A computationally less demanding approximate reduction and linearization technique is also proposed