Institute of for Industry Kyushu University 九州大学マ ス・フ ォ ア・イ ン ダ スト リ 研 究 所

Adaptive Network Theory with Organism New viewpoints of mathematics via “rigorous numerics” Atsushi TERO Kaname MATSUE Degree: PhD (Science)(Hokkaido University) Degree: Ph.D.（Science）（Kyoto University） Research Interests: Mathematical Modeling, Network Theory Research Interests: Dynamical systems, , Topology

During an organism’s long-term struggle for existence, it evolves many Toshiyuki Nakagaki (Future University-Hakodate). However, how the Mathematics is applied to various fields for explaining events such as natural Figure 2. Ideas of validated numerics. refined techniques for life phenomena. I transcribe such life phenomena to slime mold solves the network problem without a brain or global informa- phenomena, there are still many problems which abstract mathematical arguments cannot be applied directly. numerical formulas in my work. I also extract the techniques from the tion remains unanswered. organisms and apply them industrially. I reproduced this phenomenon by describing it with numerical For example, “singular perturbation method” for (1) provides the There are various types of transportation networks, such as railway equations (Figs. 2d–f). The parameter to solve the maze was found to be existence of solutions for “sufficiently small parameterε ”. However, this mathematics does not tell us how small or large such ε can be treated. On the other hand, we can networks, ant trails, blood vessel networks, and leaf veins. Commonly the boundary of the network topology. When the growth rate of a thick path treat this problems by numerical simulations with concrete ε. However, we cannot used paths develop in these networks, while paths that are not frequently is strong, it is the only path that remains (Fig. 3a) because a thick path at get any information if such ε is included in a category of “singular perturbations”. In used degenerate. These networks are called "adaptive networks". The the initial state easily grows and further growth also becomes easy. particular, there is an unavoidable gap between mathematical results and numerical network topology of an adaptive network varies (such as capillaries and the Conversely, if the maintenance cost is high for a thick path, the path cannot ones. aorta). The purpose of my study is to understand the formation of such an maintain its thickness, and the other paths remain (Fig. 3c). The parameter Top : represent all “numbers” by intervals, and do all operations among the category of There is also a case that the special structure of numerical solutions cannot be intervals. “Rigorous numerics” can be applied on computers in the sense that the true adaptive network. for solving the maze is the boundary of these two types of network validated, like blow-up solutions of differential equations. We cannot treat “infinity” value is always contained in intervals. Bottom：Reduce all problems to several simple formation(Fig. 3b). rigorously on computers, and hence we often regard large solutions in a suitable ones so that rigorous numerics can be applied. Many of them are reduced to zero-finding situation as blow-up solutions (Figure 1). However, are they truly blow-up solutions Fig.1 I applied this common theory of adaptive networks to a real railway problem (left), or inequalities (right). ? Ordinary numerics can never answer the question. Since such special structures are network (Figs. 4a–c). I will also apply it for various adaptive networks. In assumed in advance in , then arguments with unclear assumptions Recently, I study computer assisted proofs of systems with singularities, such addition, I study action control using a variety of rhythms and voluntary may cause wrong results. as singular perturbation problems, blow-up solutions and shock waves containing morphosis of organisms. discontinuities. These objects possess completely different features and difficult issues Figure 1. from both mathematical and numerical viewpoints. I have succeeded the reduction Fig.3 of existence problems for these objects to topological equations called covering （a） （b） （c） relations and several inequalities by mathematical theories such as geometric singular , compactification and desingularizations, and reduction of systems. Then “rigorous numerics” of solutions of (1) with explicit range of ε, blow-up solutions and discontinuous solutions can be achieved.

A true slime mold, Physarum polycephalum, was used for this study Figure 3. Rigorous numerics to dynamical systems. (Fig. 1). This slime mold is a unicellular organism, but it has the collective One Path Shortest Path All Paths property of containing many nuclei. For example, if it is cut into pieces, （by intial condition） (1) A differential equation with singular perturbation structure, and a numerical example each piece can live as an individual. However, the pieces can also fuse and for some system with ε=0.005. Is the valueε=0.005 “sufficiently small” so that singular become one living individual. The mold has an adaptive network to Fig.4 perturbation theory can be applied ? transport nutrients. It is a superior specimen for understanding adaptive (2) A numerical solution of a differential equation. We can see that this solution “blows up” at a finite time, but is this solution truly a “blow-up solution” ? networks because it can be cut and handled in this way. The transportation network of this slime mold is a product of its solving of mazes(Figs. 2a-c) “Validated numerics”, which bridge the above unavoidable gaps between and is an optimal network(Fig. 3b), according to my co-worker Prof. mathematical analysis and numerics, are recently well-studied. Validated numerics are one of applications of “”, which regards intervals as fundamental units Top：Represent trajectories by “covering relations”: graphs involving zero-finding Fig.2 instead of numbers, and do all operations among intervals. All numerical errors (such problems. Middle：Reduce “asymptotic behavior” to positive definiteness of a matrix. as truncation and rounding errors) can be included in intervals, and hence “rigorous Bottom：Compactification embedding the phase space into a compact manifold. “The numerics” can be realized on computers. Studies of validated numerics have begun infinity” corresponds to boundary of the manifold. Using these tools, we can validate (a) Real Railway network (b) Physarum Network (c) Simulation results with linear algebra or solvers of nonlinear equations, and nowadays there are a lot of singular perturbed solutions or blow-up solutions on computers. applications to differential equations, dynamical systems and so on. The key idea of applications is reduction of problems so that rigorous numerics can be actually applied Rigorous numerics has a potential which extends applicability of mathematics to within suitable processes. Many of problems are reduced to zero-finding problems or numerics in various concrete problems extensively. On the other hand, the methodology inequality arguments (Figure 2). relies on “reductions of problems” for practical computations, and we have to study the “root” of targets deeply so that we can make the reduction in a natural way.

I thus believe that rigorous numerics is the which requires not only numerical computations but also a lot of mathematical considerations, and which will develop new mathematical issues through numerics.

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