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 Cardelli et al., 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.
Mean field approximation is a powerful tool to study the performance of large stochastic systems that is known to be exact as the system's size N goes to infinity. Recently, it has been shown that, when one wants to compute expected performance metric in steady-state, mean field approximation can be made more accurate by adding a term in 1/N to the original approximation. This is called the refined mean field approximation in . In this paper, we show how to obtain the same result for the transient regime and we provide a further refinement by expanding the term in 1/N2 (both for transient and steady-state regime). Our derivations are inspired by moment-closure approximation. We provide a number of examples that show this new approximation is usable in practice for systems with up to a few tens of dimensions.
Mean-field models are an established method to analyze large stochastic systems with N interacting objects by means of simple deterministic equations that are asymptotically correct when N tends to infinity. For finite N, mean-field equations provide an approximation whose accuracy is model- and parameter-dependent. Recent research has focused on refining the approximation by computing suitable quantities associated with expansions of order $1/N$ and $1/N^2$ to the mean-field equation. In this paper we present a new method for refining mean-field approximations. It couples the master equation governing the evolution of the probability distribution of a truncation of the original state space with a mean-field approximation of a time-inhomogeneous population process that dynamically shifts the truncation across the whole state space. We provide a result of asymptotic correctness in the limit when the truncation covers the state space; for finite truncations, the equations give a correction of the mean-field approximation. We apply our method to examples from the literature to show that, even with modest truncations, it is effective in models that cannot be refined using existing techniques due to non-differentiable drifts, and that it can outperform the state of the art in challenging models that cause instability due orbit cycles in their mean-field equations.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
334 Leonard St
Brooklyn, NY 11211
Copyright © 2023 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.