## Mathematical modelling

NASCENCE aims to be able to predict and simulate the computational properties of a given material.For this we need to model the process governing the physics of the system on a computer.

This may circumvent doing time consuming and/or complicated experiments. In combination with the explorative methods that are being developed in work package 6, this may also unveil unforeseen computational abilities of the material.

Simulations in combination with analysis might also direct us to new types of materials with properties that are very well suited for particular computations.

In the first stages of NASCENCE we have focused on models and simulation tools for networks of nano-particles, like the one that is depicted here. Such networks have been produced at the University of Twente for the NASCENCE project.

The network has two input leads and one output lead. The remaining pins are used to manipulate the network in order to program a certain functionality, using a Genetic Algorithm. The network can be configured such that it exhibits any logic with two inputs and one output, all using the same IO-pins. The only things that change are the values of the voltages at the other leads.

These networks are being produced using a kind of controlled self-assembly process that involves trapping the nano-particles on a pre-produced electronic architecture and separating them by organic molecules that form tunnel barriers. The governing physical phenomena determining the particle interactions of such networks is pretty well-understood, whereas for general networks of other materials, like thin films of nanotubes, this is not so clear.

The transport of electrons is governed by the Coulomb blockade effect: transport is blocked, except at almost discrete energy levels; there exactly one electron can jump. The dynamics of such a system is governed by stochastic processes: electrons on all islands can tunnel through junctions with a certain probability. This makes that the Monte Carlo Method is the best candidate for our simulations.

A complete simulation code has been developed. In principle, it can handle an arbitrary network of any size. The code has been tested for a variety of networks that exhibit logic, as reported in the literature. Moreover, relative small sized networks are shown to be able to behave like logic gates by only changing the values of the lead/gate voltages. As an example, on a 4x4 grid we found logic such as the NAND-gate that is depicted in the figure below.

The figure shows a contour plot as a function of the two input signals. For the inputs (0,0), (1,0) and (0,1) the result is 1; for (1,1) the result is 0. Therefore this constitutes the logic NAND. It was found by simulation using a Monte Carlo method combined with a Genetic Algorithm to find the proper voltages at the vacant pins.

Combining the simulation tool with genetic algorithms that are produced in work package 3 will enable large scale simulations for searching for suitable candidate computational tasks. This will then form the scientific basis for predictions and for exploring the materials experimentally in a systematic way, using the hardware produced in work package 2 combined with the developed software in work package 3. Information gathered from the experiments will then feed back into the model and simulation tool, and might lead to adaptations.

We are now considering models and simulation tools for the other evolvable materials that are being used in NASCENCE.

### Virtual Material

For the purpose of testing the function extraction algorithms, as well as data-mining from the measured data, a simulator of a virtual material (VM) has been developed. VM simulates an arbitrarily complex function that transforms input signals to output signals. It encapsulates a Recurrent Neural Network (RNN) with a number of inputs, outputs and hidden nodes as parameters. The simulator is wrapped up in the NASCENCE API and,”as with the real material”, can be accessed through Thrift remotely.

*Virtual Material fully-connected reccurent neural network.*