Model Creation and Equivalence Proofs of Cellular Automata and Artificial Neural Networks

Computational methods and mathematical models have invaded arguably every scientific discipline forming its own field of research called computational science. Mathematical models are the theoretical foundation of computational science. Since Newton's time, differential equations in mathematical models have been widely and successfully used to describe the macroscopic or global behaviour of systems. With spatially inhomogeneous, time-varying, local element-specific, and often non-linear interactions, the dynamics of complex systems is in contrast more efficiently described by local rules and thus in an algorithmic and local or microscopic manner. The theory of mathematical modelling taking into account these characteristics of complex systems has to be established still. We recently presented a so-called allagmatic method including a system metamodel to provide a framework for describing, modelling, simulating, and interpreting complex systems. Implementations of cellular automata and artificial neural networks were described and created with that method. Guidance from philosophy were helpful in these first studies focusing on programming and feasibility. A rigorous mathematical formalism, however, is still missing. This would not only more precisely describe and define the system metamodel, it would also further generalise it and with that extend its reach to formal treatment in applied mathematics and theoretical aspects of computational science as well as extend its applicability to other mathematical and computational models such as agent-based models. Here, a mathematical definition of the system metamodel is provided. Based on the presented formalism, model creation and equivalence of cellular automata and artificial neural networks are proved. It thus provides a formal approach for studying the creation of mathematical models as well as their structural and operational comparison.Comment: 13 pages, 1 tabl

Guaranteed methods based on constrained zonotopes for set-valued state estimation of nonlinear discrete-time systems

This paper presents new methods for set-valued state estimation of nonlinear discrete-time systems with unknown-but-bounded uncertainties. A single time step involves propagating an enclosure of the system states through the nonlinear dynamics (prediction), and then enclosing the intersection of this set with a bounded-error measurement (update). When these enclosures are represented by simple sets such as intervals, ellipsoids, parallelotopes, and zonotopes, certain set operations can be very conservative. Yet, using general convex polytopes is much more computationally demanding. To address this, this paper presents two new methods, a mean value extension and a first-order Taylor extension, for efficiently propagating constrained zonotopes through nonlinear mappings. These extend existing methods for zonotopes in a consistent way. Examples show that these extensions yield tighter prediction enclosures than zonotopic estimation methods, while largely retaining the computational benefits of zonotopes. Moreover, they enable tighter update enclosures because constrained zonotopes can represent intersections much more accurately than zonotopes.Comment: This includes the supplement "Supplementary material for: Guaranteed methods based on constrained zonotopes for set-valued state estimation of nonlinear discrete-time systems