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.
Linear aggregation and lumping of (polynomial) systems of 's are well known in the literature, se e.g. @cite_8 @cite_13 @cite_35 @cite_30 and references therein. However, as pointed out by @cite_32 , no general algorithms for computing the largest equivalence, hence the minimal reduction (in the sense of our Theorem ) was known.
@cite_30 @cite_35 @cite_8 @cite_32 @cite_13
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Let us consider the differential equation @math with an @math from @math to @math and suppose that there exists a transformation @math from @math to @math ( @math ) such that @math obeys a differential equation @math with some function @math ; then the first equation is said to be lumpable to the second by @math . Here mainly the case is investigated when the original differential equation has been induced by a complex chemical reaction. We provided a series of necessary and sufficient conditions for the existence of such functions @math and @math ; some of them are formulated in terms of @math and @math only. Beyond these conditions our main concern here is how lumping changes those properties of the solutions which are either interesting from the point of view of the qualitative theory of differential equations or from the point ... Abstract A general analysis of exact nonlinear lumping is presented. This analysis can be applied to the kinetics of any reaction system with n species described by a set of first-order ordinary differential equations d y d t = f ( y ), where y is an n -dimensional vector and f ( y ) is an arbitrary n -dimensional function vector. We consider lumping by means of n ( n ⩽ n )-dimensional arbitrary transformation ŷ = h ( y ). The lumped differential equation system is d y D t = y ( h (ŷ))f( h (ŷ)) , where h y (y) is teh Jacobian matrix of h(y) , h is a generalized inverse transformation of h satisfying the relation h( h ) = I n . Three necessary and sufficient conditions of the existence of exact nonlinear lumping schemes have been determined. The geometric and algebraic interpretations of these conditions are discussed. It is found that a system is exactly lumpable by h only if h(y) = 0 is its invariant manifold. A linear partial differential operator A = Σ n i =1 f i ( y )ϑ ϑ y i corresponding to d y d t = f(y ) is defined. Using the eigenfunctions and the generalized eigenfunctions of A , the operator can be transformed to Jordan or diagonal canonical forms which give the lumped differential equation systems without determination of h . These approaches are illustrated by a simple example. The results of this analysis serve as a theoretical basis for the development of approaches for approximate nonlinear lumping. Preface Part I. Introduction: 1. Introduction 2. Motivating examples Part II. Preliminaries: 3. Tools from matrix theory 4. Linear dynamical systems, Part 1 5. Linear dynamical systems, Part 2 6. Sylvester and Lyapunov equations Part III. SVD-based Approximation Methods: 7. Balancing and balanced approximations 8. Hankel-norm approximation 9. Special topics in SVD-based approximation methods Part IV. Krylov-based Approximation Methods: 10. Eigenvalue computations 11. Model reduction using Krylov methods Part V. SVD-Krylov Methods and Case Studies: 12. SVD-Krylov methods 13. Case studies 14. Epilogue 15. Problems Bibliography Index. Ordinary differential equations (ODEs) are widespread in many natural sciences including chemistry, ecology, and systems biology, and in disciplines such as control theory and electrical engineering. Building on the celebrated molecules-as-processes paradigm, they have become increasingly popular in computer science, with high-level languages and formal methods such as Petri nets, process algebra, and rule-based systems that are interpreted as ODEs. We consider the problem of comparing and minimizing ODEs automatically. Influenced by traditional approaches in the theory of programming, we propose differential equivalence relations. We study them for a basic intermediate language, for which we have decidability results, that can be targeted by a class of high-level specifications. An ODE implicitly represents an uncountable state space, hence reasoning techniques cannot be borrowed from established domains such as probabilistic programs with finite-state Markov chain semantics. We provide novel symbolic procedures to check an equivalence and compute the largest one via partition refinement algorithms that use satisfiability modulo theories. We illustrate the generality of our framework by showing that differential equivalences include (i) well-known notions for the minimization of continuous-time Markov chains (lumpability), (ii) bisimulations for chemical reaction networks recently proposed by , and (iii) behavioral relations for process algebra with ODE semantics. With a prototype implementation we are able to detect equivalences in biochemical models from the literature that cannot be reduced using competing automatic techniques. Detailed modeling of complex reaction systems is becoming increasingly important in the development, analysis, design, and control of chemical reaction processes. For industrial processes, complete incorporation of the chemistry into process models facilitates the minimization of byproduct and pollutant formation, increased efficiency, and improved product quality. Processes that involve complex reaction networks include a variety of noncatalytic and homogeneous or heterogeneous catalytic processes (such as fluid catalytic cracking, combustion, chemical vapor deposition, and alkylation). For some systems, large sets of relevant reactions have been identified for use in simulations.1-3 For others, the availability of advanced computing environments has enabled the automated generation of reaction networks and their models, based on computational descriptions of the reaction types occurring in the system.4-6 The use of such complex models is hindered by two obstacles. First, because of their sheer size and the presence of multiple time scales, these models are difficult to solve. Second, the models contain large numbers of uncertain (and sometimes unknown) kinetic parameters; regression to determine the parameters of complex nonlinear models is both difficult and unreliable, and the sensitivity of simulations to parameter uncertainties cannot be easily ascertained. Furthermore, for the purpose of gaining insights into the reaction system’s behavior, it is usually preferable to obtain simpler models that bring out the key features and components of the system. For these reasons, model simplification and order reduction are becoming central problems in the study of complex reaction systems. The simulation, monitoring, and control of a complex chemical process benefit from the derivation of accurate and reliable reduced models tailored to particular process modeling tasks. Model simplification is directly linked to identification of key reactions and sets of species that give valuable insights into the behavior of the network and how it may be influenced. Advanced control schemes such as model predictive control7 or multiple model adaptive control8 must be based on selecting appropriate reduced models and tracking key sets of species. Ideally, a model order reduction algorithm should have broad applicability, enable analysis at several levels of detail, and provide an assessment of the modeling error.