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Many nanoscale devices require precise optimization to function. Tuning them to the desired operation regime becomes increasingly difficult and time-consuming when the number of terminals and couplings grows. Imperfections and device-to-device variations hinder optimization that uses physics-based models. Deep neural networks (DNNs) can model various complex physical phenomena but, so far, are mainly used as predictive tools. Here, we propose a generic deep-learning approach to efficiently optimize complex, multi-terminal nanoelectronic devices for desired functionality. We demonstrate our approach for realizing functionality in a disordered network of dopant atoms in silicon. We model the input-output characteristics of the device with a DNN, and subsequently optimize control parameters in the DNN model through gradient descent to realize various classification tasks. When the corresponding control settings are applied to the physical device, the resulting functionality is as predicted by the DNN model. We expect our approach to contribute to fast, in situ optimization of complex (quantum) nanoelectronic devices.

Classification is an important task at which both biological and artificial neural networks excel 1,2 . In machine learning, nonlinear projection into a high-dimensional feature space can make data linearly separable 3,4 , simplifying the classification of complex features. Such nonlinear projections are computationally expensive in conventional computers. A promising approach is to exploit physical materials systems that perform this nonlinear projection intrinsically, because of their high computational density 5 , inherent parallelism and energy efficiency 6,7 . However, existing approaches either rely on the systems' time dynamics, which requires sequential data processing and therefore hinders parallel computation 5,6,8 , or employ large materials systems that are difficult to scale up 7 . Here we use a parallel, nanoscale approach inspired by filters in the brain 1 and artificial neural networks 2 to perform nonlinear classification and feature extraction. We exploit the nonlinearity of hopping conduction 9-11 through an electrically tunable network of boron dopant atoms in silicon, reconfiguring the network through artificial evolution to realize different computational functions. We first solve the canonical two-input binary classification problem, realizing all Boolean logic gates 12 up to room temperature, demonstrating nonlinear classification with the nanomaterial system. We then evolve our dopant network to realize feature filters 2 that can perform four-input binary classification on the Modified National Institute of Standards and Technology handwritten digit database. Implementation of our material-based filters substantially improves the classification accuracy over that of a linear classifier directly applied to the original data 13 . Our results establish a paradigm of silicon-based electronics for small-footprint and energy-efficient computation 14 .

We introduce a new class of graphs which we call P3-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P3-dominated graph. We prove that G is hamiltonian if α( G2) ≤ κ( G), with two exceptions: K2,3 and K1,1,3. We also prove that G is hamiltonian, if G is 3-connected and | V( G)| ≤ 5 δ( G) − 5. These results extend known results on (quasi-)claw-free graphs. [ABSTRACT FROM AUTHOR]

We open here many new tracks of research in anti-Ramsey Theory, considering edge-coloring problems inspired by rainbow coloring and further by odd colorings and conflict-free colorings. Let G be a graph and F any given family of graphs. For every integer n ≥ | G | , let f (n , G | F) denote the smallest integer k such that any edge coloring of K n with at least k colors forces a copy of G in which each color class induces a member of F. Observe that in anti-Ramsey problems, each color class is a single edge, i.e., F = { K 2 } . Among the many results introduced in this paper, we mention the following. (1) For every graph G, there exists a constant c = c (G) such that in any edge coloring of K n with at least c n colors there is a copy of G in which every vertex v is incident with an edge whose color appears only once among all edges incident with v. (2) In sharp contrast to the above result we prove that if F is the class of all odd graphs (having vertices with odd degrees only) then f (n , K k | F) = (1 + o (1)) ex (n , K ⌈ k / 2 ⌉) , which is quadratic for k ≥ 5 . (3) We exactly determine f (n , G | F) for small graphs when F belongs to several families representing various odd/even coloring constraints. [ABSTRACT FROM AUTHOR]

Ramsey's theorem states that for any natural numbers n, m there exists a natural number N such that any red–blue coloring of the graph K N contains either a red K n or blue K m as a subgraph. The smallest such N is called the Ramsey number, denoted as R (n , m) . In this paper, we reformulate this theorem and present a proof of Ramsey's theorem that is novel as far as we are aware. [ABSTRACT FROM AUTHOR]