Efficient Stochastic Simulation of Systems with Multiple Time Scales via Statistical Abstraction

Abstract

Abstract. Stiffness in chemical reaction systems is a frequently encountered computational problem, arising when different reactions in the system take place at different time-scales. Computational savings can be obtained under time-scale separation. Assuming that the system can be partitioned into slow-and fast-equilibrating subsystems, it is then possible to efficiently simulate the slow subsystem only, provided that the corresponding kinetic laws have been modified so that they reflect their dependency on the fast system. We show that the rate expectation with respect to the fast subsystem's steady-state is a continuous function of the state of the slow system. We exploit this result to construct an analytic representation of the modified rate functions via statistical modelling, which can be used to simulate the slow system in isolation. The computational savings of our approach are demonstrated in a number of non-trivial examples of stiff systems.

Machine Learning Methods in Statistical Model Checking and System Design – Tutorial

“…Another area of formal methods in which machine learning methods have a large potential is that of abstraction, recasted in a statistical sense. At the moment, these ideas have been used to speed up simulation of systems with multiple time scales [40] and systems where only a small portion is simulated explicitly, while the rest of the system is abstracted by a Gaussian Process [41]. They have also been used for modular decomposition of systems in parameter estimation tasks [42] Finally, the integration of machine learning and formal methods is happening also at the level of modelling languages.…”

Statistical Abstraction for Multi-scale Spatio-Temporal Systems

“…Clearly, this iterative procedure, while asymptotically exact (in the limit of small time discretisation), is computationally very demanding. This has motivated several lines of research in recent years [1,8,13,10].…”

“…We achieve this by learning a parameters-to-behaviours regression map using Gaussian Processes (GPs), a powerful class of non-parametric Bayesian regression models. Our work is motivated by earlier work on using GPs to learn effective characterisations of system behaviour [2,1,11].…”

Property-Driven State-Space Coarsening for Continuous Time Markov Chains

“…Example 1. 5 We illustrate and evaluate the quality of the coarsened trajectories with respect to the original ones on the SIRS example. In particular, we examine the probability distribution over the macro-states at different times in the evolution of the system.…”

“…Broadly speaking, state-space reduction can be achieved by either model simplification, usually by abstracting some system behaviours into a simpler system, or state aggregation, often by exploiting symmetries or approximate invariances. A prime example of model simplification is the technique of time-scale separation, which replaces a large system with multiple weakly dependent sub-systems [5]. Most aggregation methods, instead, are based on grouping different states with similar behaviour with respect to their transition probabilities.…”