## Publications

**Aug 02 2019**

#### Stress-Testing Memcomputing on Hard Combinatorial Optimization Problems

Memcomputing is a novel computing paradigm that employs time non-local dynamical systems to compute with and in memory. The digital version of these machines [digital memcomputing machines or (DMMs)] is scalable, and is particularly suited to solve combinatorial optimization problems. One of its possible realizations is by means of standard electronic circuits, with and without memory. Since these elements are non-quantum, they can be described by ordinary differential equations. Therefore, the circuit representation of DMMs can also be simulated efficiently on our traditional computers. We have indeed previously shown that these simulations only require time and memory resources that scale linearly with the problem size when applied to finding a good approximation to the optimum of hard instances of the maximum-satisfiability problem. The state-of-the-art algorithms, instead, require exponential resources for the same instances. However, in that work, we did not push the simulations to the limit of the processor used. Since linear scalability at smaller problem sizes cannot guarantee linear scalability at much larger sizes, we have extended these results in a stress-test up to 64×10⁶ variables (corresponding to about 1 billion literals), namely the largest case that we could fit on a single core of an Intel Xeon E5-2860 with 128 GB of dynamic random-access memory (DRAM). For this test, we have employed a commercial simulator, Falcon of MemComputing, Inc. We find that the simulations of DMMs still scale linearly in both time and memory up to these very large problem sizes versus the exponential requirements of the state-of-the-art solvers. These results further reinforce the advantages of the physics-based memcomputing approach compared with traditional ones.

**OCT 2019**

#### Digital memcomputing: From logic to dynamics to topology

##### Digital memcomputing machines (DMMs) are a class of computational machines designed to solve combinatorial optimization problems. A practical realization of DMMs can be accomplished via electrical circuits of highly non-linear, point-dissipative dynamical systems engineered so that periodic orbits and chaos can be avoided. A given logic problem is first mapped into this type of dynamical system whose point attractors represent the solutions of the original problem. A DMM then finds the solution via a succession of elementary instantons whose role is to eliminate solitonic configurations of logical inconsistency (“logical defects”) from the circuit. By employing a supersymmetric theory of dynamics, a DMM can be described by a cohomological field theory that allows for computation of certain topological matrix elements on instantons that have the mathematical meaning of intersection numbers on instantons. We discuss the “dynamical” meaning of these matrix elements, and argue that the number of elementary instantons needed to reach the solution cannot exceed the number of state variables of DMMs, which in turn can only grow at most polynomially with the size of the problem. These results shed further light on the relation between logic, dynamics and topology in digital memcomputing.

**MAR 21, 2019**

#### Aircraft Loading Optimization: MemComputing the 5th Airbus Problem

##### On the January 22th 2019, Airbus launched a quantum computing challenge to solve a set of problems relevant for the aircraft life cycle (Airbus challenge web-page). The challenge consists of a set of 5 problems that ranges from design to deployment of aircraft. This work addresses the 5th problem. The formulation exploits an Integer programming framework with a linear objective function and the solution relies on the MemComputing paradigm. It is discussed how to use current MemComputing software MemCPUTM to solve efficiently the proposed problem and assess scaling properties, which turns out to be polynomial for meaningful solutions of the problem at hand. Also discussed are possible formulations of the problem utilizing non-linear objective functions, allowing for different optimization schemes implementable in modified MemCPUTM software, potentially useful for field operation purposes.

**MAR 18, 2019**

#### Digital Memcomputing: from Logic to Dynamics to Topology

##### Digital memcomputing machines (DMMs) are a class of computational machines designed to solve combinatorial optimization problems. A practical realization of DMMs can be accomplished via electrical circuits of highly non-linear, point-dissipative dynamical systems engineered so that periodic orbits and chaos can be avoided. A given logic problem is first mapped into this type of dynamical system whose point attractors represent the solutions of the original problem. A DMM then finds the solution via a succession of elementary instantons whose role is to eliminate solitonic configurations of logical inconsistency (“logical defects”) from the circuit. By employing a supersymmetric theory of dynamics, a DMM can be described by a cohomological field theory that allows for computation of certain topological matrix elements on instantons that have the mathematical meaning of intersection numbers on instantons. We discuss the “dynamical” meaning of these matrix elements, and argue that the number of elementary instantons needed to reach the solution cannot exceed the number of state variables of DMMs, which in turn can only grow at most polynomially with the size of the problem. These results shed further light on the relation between logic, dynamics and topology in digital memcomputing.

**Nov 2018**

#### Accelerating deep learning with MemComputing

Restricted Boltzmann machines (RBMs) and their extensions, often called “deep-belief networks”, are powerful neural networks that have found applications in the fields of machine learning and artificial intelligence. The standard way to train these models resorts to an iterative unsupervised procedure based on Gibbssampling, called “contrastive divergence”, and additional supervised tuning via back-propagation. However, this procedure has been shown not to follow any gradient and can lead to suboptimal solutions. In this paper, we show an efficient alternative to contrastive divergence by means of simulations of digital memcomputing machines (DMMs) that compute the gradient of the log-likelihood involved in unsupervised training. We test our approach on pattern recognition using a modified version of the MNIST data set of hand-written numbers. DMMs sample effectively the vast phase space defined by the probability distribution of RBMs, and provide a good approximation close to the optimum. This efficient search significantly reduces the number of generative pretraining iterations necessary to achieve a given level of accuracy in the MNIST data set, as well as a total performance gain over the traditional approaches. In fact, the acceleration of the pretraining achieved by *simulating* DMMs is comparable to, in number of iterations, the recently reported *hardware* application of the quantum annealing method on the same network and data set. Notably, however, DMMs perform far better than the reported quantum annealing results in terms of *quality*of the training. Finally, we also compare our method to recent advances in supervised training, like batch-normalization and rectifiers, that seem to reduce the advantage of pretraining. We find that the memcomputing method still maintains a quality advantage ( in accuracy, corresponding to a 20% reduction in error rate) over these approaches, despite the network pretrained with memcomputing defines a more non-convex landscape using sigmoidal activation functions without batch-normalization. Our approach is agnostic about the connectivity of the network. Therefore, it can be extended to train full Boltzmann machines, and even deep networks at once.

**Oct 31 2018**

#### On the Universality of Memcomputing Machines

Universal memcomputing machines (UMMs) represent a novel computational model in which memory (time nonlocality) accomplishes both tasks of storing and processing of information. UMMs have been shown to be Turing-complete, namely, they can simulate any Turing machine. In this paper, we first introduce a novel set theory approach to compare different computational models and use it to recover the previous results on Turing-completeness of UMMs. We then relate UMMs directly to liquid-state machines (or “reservoir-computing”) and quantum machines (“quantum computing”). We show that UMMs can simulate both types of machines, hence they are both “liquid-” or “reservoir-complete” and “quantum-complete.” Of course, these statements pertain only to the type of problems these machines can solve and not to the amount of resources required for such simulations. Nonetheless, the set-theoretic method presented here provides a general framework which describes the relationship between any computational models.

**OCT 08, 2018**

#### Taming a non-convex landscape with dynamical long-range order: memcomputing the Ising spin-glass

##### Recent work on quantum annealing has emphasized the role of collective behavior in solving optimization problems. By enabling transitions of large clusters of variables, such solvers are able to navigate their state space and locate solutions efficiently despite having only local connections between elements. However, collective behavior is not exclusive to quantum annealers, and classical solvers that display collective dynamics should also possess an advantage in navigating a non-convex landscape. Here, we propose a simple model that demonstrates this effect, based on the recently suggested digital memcomputing machines (DMMs), which utilize a collection of dynamical components with memory connected to represent the structure of the underlying optimization problem. This model, when applied to finding the ground state of the Ising spin glass, undergoes a transient phase of avalanches which can span the entire lattice. We then show that a full implementation of a DMM exhibits superior scaling compared to other methods when tested on the same problem class. These results establish the advantages of computational approaches based on collective dynamics.

**AUG 29, 2018**

#### MemComputing Integer Linear Programming

##### Integer linear programming (ILP) encompasses a very important class of optimization problems that are of great interest to both academia and industry. Several algorithms are available that attempt to explore the solution space of this class efficiently, while requiring a reasonable compute time. However, although these algorithms have reached various degrees of success over the years, they still face considerable challenges when confronted with particularly hard problem instances, such as those of the MIPLIB 2010 library. In this work we propose a radically different non-algorithmic approach to ILP based on a novel physics-inspired computing paradigm: Memcomputing. This paradigm is based on digital (hence scalable) machines represented by appropriate electrical circuits with memory. These machines can be either built in hardware or, as we do here, their equations of motion can be efficiently simulated on our traditional computers. We first describe a new circuit architecture of memcomputing machines specifically designed to solve for the linear inequalities representing a general ILP problem. We call these self-organizing algebraic circuits, since they self-organize dynamically to satisfy the correct (algebraic) linear inequalities. We then show simulations of these machines using MATLAB running on a single core of a Xeon processor for several ILP benchmark problems taken from the MIPLIB 2010 library, and compare our results against a renowned commercial solver. We show that our approach is very efficient when dealing with these hard problems. In particular, we find within minutes feasible solutions for one of these hard problems (f2000 from MIPLIB 2010) whose feasibility, to the best of our knowledge, has remained unknown for the past eight years.

**JUN 30, 2018**

#### Stress-testing memcomputing on hard combinatorial optimization problems

##### Memcomputing is a novel paradigm of computation that utilizes dynamical elements with memory to both store and process information on the same physical location. Its building blocks can be fabricated in hardware with standard electronic circuits, thus offering a path to its practical realization. In addition, since memcomputing is based on non-quantum elements, the equations of motion describing these machines can be simulated efficiently on standard computers. In fact, it was recently realized that memcomputing, and in particular its digital (hence scalable) version, when simulated on a classical machine provides a significant speed-up over state-of-the-art algorithms on a variety of non-convex problems. …

**FEB 20, 2018**

#### Memcomputing: Leveraging memory and physics to compute efficiently

##### It is well known that physical phenomena may be of great help in computing some difficult problems efficiently. A typical example is prime factorization that may be solved in polynomial time by exploiting quantum entanglement on a quantum computer. There are, however, other types of (non-quantum) physical properties that one may leverage to compute efficiently a wide range of hard problems. In this perspective we discuss how to employ one such property, memory (time non-locality), in a novel physics-based approach to computation: Memcomputing. In particular, we focus on digital memcomputing machines (DMMs) that are scalable. DMMs can be realized with non-linear dynamical systems with memory…

**JAN 01, 2018**

#### Accelerating Deep Learning with Memcomputing

##### Restricted Boltzmann machines (RBMs) and their extensions, often called “deep-belief networks”, are very powerful neural networks that have found widespread applicability in the fields of machine learning and big data. The standard way to training these models resorts to an iterative unsupervised procedure based on Gibbs sampling, called “contrastive divergence”, and additional supervised tuning via back-propagation. However, this procedure has been shown not to follow any gradient and can lead to suboptimal solutions. In this paper, we show a very efficient alternative to contrastive divergence by means of simulations of digital memcomputing machines (DMMs). We test our approach on pattern recognition using the standard MNIST data set of hand-written numbers…

**DEC 23, 2017**

#### On the Universality of Memcomputing Machines

##### Universal memcomputing machines (UMMs) [IEEE Trans. Neural Netw. Learn. Syst. 26, 2702 (2015)] represent a novel computational model in which memory (time non-locality) accomplishes both tasks of storing and processing of information. UMMs have been shown to be Turing-complete, namely they can simulate any Turing machine. In this paper, using set theory and cardinality arguments, we compare them with liquid-state machines (or “reservoir computing”) and quantum machines (“quantum computing”). We show that UMMs can simulate both types of machines, hence they are both “liquid-” or “reservoir-complete” and “quantum-complete”. Of course, these statements pertain only to the type of problems these machines can solve, and not to the amount of resources required for such simulations. Nonetheless, the method presented here provides a general framework in which to describe the relation between UMMs and any other type of computational model.

**OCT 23, 2017**

#### Evidence of an exponential speed-up in the solution of hard optimization problems

##### Optimization problems pervade essentially every scientific discipline and industry. Many such problems require finding a solution that maximizes the number of constraints satisfied. Often, these problems are particularly difficult to solve because they belong to the NP-hard class, namely algorithms that always find a solution in polynomial time are not known. Over the past decades, research has focused on developing heuristic approaches that attempt to find an approximation to the solution. However, despite numerous research efforts, in many cases even approximations to the optimal solution are hard to find, as the computational time for further refining a candidate solution grows exponentially with input size. …

**SEP 2017**

#### Absence of periodic orbits in digital memcomputing machines with solutions

##### In Traversa and Di Ventra [Chaos 27, 023107 (2017)] we argued, without proof, that if the non-linear dynamical systems with memory describing the class of digital memcomputing machines (DMMs) have equilibrium points, then no periodic orbits can emerge. In fact, the proof of such a statement is a simple corollary of a theorem already demonstrated in Traversa and Di Ventra [Chaos 27, 023107 (2017)]. Here, we point out how to derive such a conclusion. Incidentally, the same demonstration implies absence of chaos, a result we have already demonstrated in Di Ventra and Traversa [Phys. Lett. A 381, 3255 (2017)] using topology. These results, together with those in Traversa and Di Ventra [Chaos 27, 023107 (2017)], guarantee that if the Boolean problem the DMMs are designed to solve has a solution, the system will always find it, irrespective of the initial conditions.

**AUG 29, 2017**

#### Instantons in Self-Organizing Logic Gates

##### Self-organizing logic is a recently suggested framework that allows the solution of Boolean truth tables “in reverse”; i.e., it is able to satisfy the logical proposition of gates regardless to which terminal(s) the truth value is assigned (“terminal-agnostic logic”). It can be realized if time nonlocality (memory) is present. A practical realization of self-organizing logic gates (SOLGs) can be done by combining circuit elements with and without memory. By employing one such realization, we show, numerically, that SOLGs exploit elementary instantons to reach equilibrium points. Instantons are classical trajectories of the nonlinear equations of motion describing SOLGs and connect topologically distinct critical points in the phase space. By linear analysis at those points, we show that these instantons connect the initial critical point of the dynamics, with at least one unstable direction, directly to the final fixed point. We also show that the memory content of these gates affects only the relaxation time to reach the logically consistent solution. Finally, we demonstrate, by solving the corresponding stochastic differential equations, that, since instantons connect critical points, noise and perturbations may change the instanton trajectory in the phase space but not the initial and final critical points. Therefore, even for extremely large noise levels, the gates self-organize to the correct solution. Our work provides a physical understanding of, and can serve as an inspiration for, models of bidirectional logic gates that are emerging as important tools in physics-inspired, unconventional computing.

**May 10 2017**

#### Memcomputing Numerical Inversion With Self-Organizing Logic Gates

We propose to use digital memcomputing machines (DMMs), implemented with self-organizing logic gates (SOLGs), to solve the problem of numerical inversion. Starting from fixed-point scalar inversion, we describe the generalization to solving linear systems and matrix inversion. This method, when realized in hardware, will output the result in only one computational step. As an example, we perform simulations of the scalar case using a 5-bit logic circuit made of SOLGs, and show that the circuit successfully performs the inversion. Our method can be extended efficiently to any level of precision, since we prove that producing n -bit precision in the output requires extending the circuit by at most n bits. This type of numerical inversion can be implemented by DMM units in hardware; it is scalable, and thus of great benefit to any real-time computing application.