Coarse-graining of cellular automata, emergence, and the predictability of complex systems

We study the predictability of emergent phenomena in complex systems. Using nearest neighbor, one-dimensional Cellular Automata (CA) as an example, we show how to construct local coarse-grained descriptions of CA in all classes of Wolfram's classification. The resulting coarse-grained CA that we construct are capable of emulating the large-scale behavior of the original systems without accounting for small-scale details. Several CA that can be coarse-grained by this construction are known to be universal Turing machines; they can emulate any CA or other computing devices and are therefore undecidable. We thus show that because in practice one only seeks coarse-grained information, complex physical systems can be predictable and even decidable at some level of description. The renormalization group flows that we construct induce a hierarchy of CA rules. This hierarchy agrees well with apparent rule complexity and is therefore a good candidate for a complexity measure and a classification method. Finally we argue that the large scale dynamics of CA can be very simple, at least when measured by the Kolmogorov complexity of the large scale update rule, and moreover exhibits a novel scaling law. We show that because of this large-scale simplicity, the probability of finding a coarse-grained description of CA approaches unity as one goes to increasingly coarser scales. We interpret this large scale simplicity as a pattern formation mechanism in which large scale patterns are forced upon the system by the simplicity of the rules that govern the large scale dynamics.Comment: 18 pages, 9 figure

Equation-Free Multiscale Computational Analysis of Individual-Based Epidemic Dynamics on Networks

The surveillance, analysis and ultimately the efficient long-term prediction and control of epidemic dynamics appear to be one of the major challenges nowadays. Detailed atomistic mathematical models play an important role towards this aim. In this work it is shown how one can exploit the Equation Free approach and optimization methods such as Simulated Annealing to bridge detailed individual-based epidemic simulation with coarse-grained, systems-level, analysis. The methodology provides a systematic approach for analyzing the parametric behavior of complex/ multi-scale epidemic simulators much more efficiently than simply simulating forward in time. It is shown how steady state and (if required) time-dependent computations, stability computations, as well as continuation and numerical bifurcation analysis can be performed in a straightforward manner. The approach is illustrated through a simple individual-based epidemic model deploying on a random regular connected graph. Using the individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of Simulated Annealing I compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level under a pairwise representation perspective

Data driven modeling of self-similar dynamics

Multiscale modeling of complex systems is crucial for understanding their intricacies. Data-driven multiscale modeling has emerged as a promising approach to tackle challenges associated with complex systems. On the other hand, self-similarity is prevalent in complex systems, hinting that large-scale complex systems can be modeled at a reduced cost. In this paper, we introduce a multiscale neural network framework that incorporates self-similarity as prior knowledge, facilitating the modeling of self-similar dynamical systems. For deterministic dynamics, our framework can discern whether the dynamics are self-similar. For uncertain dynamics, it can compare and determine which parameter set is closer to self-similarity. The framework allows us to extract scale-invariant kernels from the dynamics for modeling at any scale. Moreover, our method can identify the power law exponents in self-similar systems. Preliminary tests on the Ising model yielded critical exponents consistent with theoretical expectations, providing valuable insights for addressing critical phase transitions in non-equilibrium systems.Comment: 11 pages,5 figures,1 tabl

Some Critical Thoughts on Computational Materials Science

1. A Model is a Model is a Model is a Model The title of this report is of course meant to provoke. Why? Because there always exists a menace of confusing models with reality. Does anyone now refer to “first principles simulations”? This point is well taken. However, practically all of the current predictions in this domain are based on simulating electron dynamics using local density functional theory. These simulations, though providing a deep insight into materials ground states, are not exact but approximate solutions of the Schrödinger equation, which - not to forget - is a model itself [1]. Does someone now refer to “finite element simulations”? This point is also well taken. However, also in this case one has to admit that approximate solutions to large sets of non-linear differential equations formulated for a (non-existing) continuum under idealized boundary conditions is what it is: a model of nature but not reality. But us let calm down and render the discussion a bit more serious: current methods of ground state calculations are definitely among the cutting-edge disciplines in computational materials science and the community has learnt much from it during the last years. Similar aspects apply for some continuum-based finite element simulations. After all this report is meant to attract readers into this exciting field and not to repulse them. And for this reason I feel obliged to first make a point in underscoring that any interpretation of a research result obtained by computer simulation should be accompanied by scrutinizing the model ingredients and boundary conditions of that calculation in the same critical way as an experimentalist would check his experimental set-up