Search

The Forward-Forward (FF) algorithm is a recent, purely forward-mode learning method, that updates weights locally and layer-wise and supports supervised as well as unsupervised learning. These features make it ideal for applications such as brain-inspired learning, low-power hardware neural networks, and distributed learning in large models. However, while FF has shown promise on written digit recognition tasks, its performance on natural images and time-series remains a challenge. A key limitation is the need to generate high-quality negative examples for contrastive learning, especially in unsupervised tasks, where versatile solutions are currently lacking. To address this, we introduce the Self-Contrastive Forward-Forward (SCFF) method, inspired by self-supervised contrastive learning. SCFF generates positive and negative examples applicable across different datasets, surpassing existing local forward algorithms for unsupervised classification accuracy on MNIST (MLP: 98.7%), CIFAR-10 (CNN: 80.75%), and STL-10 (CNN: 77.3%). Additionally, SCFF is the first to enable FF training of recurrent neural networks, opening the door to more complex tasks and continuous-time video and text processing.

The ability to process time-series at low energy cost is critical for many applications. Recurrent neural network, which can perform such tasks, are computationally expensive when implementing in software on conventional computers. Here we propose to implement a recurrent neural network in hardware using spintronic oscillators as dynamical neurons. Using numerical simulations, we build a multi-layer network and demonstrate that we can use backpropagation through time (BPTT) and standard machine learning tools to train this network. Leveraging the transient dynamics of the spintronic oscillators, we solve the sequential digits classification task with $89.83\pm2.91~\%$ accuracy, as good as the equivalent software network. We devise guidelines on how to choose the time constant of the oscillators as well as hyper-parameters of the network to adapt to different input time scales., Comment: 7 pages, 4 figures

A possible spintronic route to hardware implementation for decision making involves injecting a domain wall into a bifurcated magnetic nanostrip resembling a Y-shaped junction. A decision is made when the domain wall chooses a particular path through the bifurcation. Recently, it was shown that a structure like a nanomagnetic Galton Board, which is essentially an array of interconnected Y-shaped junctions, produces outcomes that are stochastic and therefore relevant to artificial neural networks. However, the exact mechanism leading to the robust nature of randomness is unknown. Here, we directly image the decision-making process in nanomagnetic Galton Boards using Lorentz transmission electron microscopy. We identify that the stochasticity in nanomagnetic Galton Boards arises as a culmination of: (1) topology of the injected domain wall, (2) local disorder, and (3) strength of the applied field. Our results pave the way to a detailed understanding of stochasticity in nanomagnetic networks.

Physical neural networks (PNNs) are a class of neural-like networks that leverage the properties of physical systems to perform computation. While PNNs are so far a niche research area with small-scale laboratory demonstrations, they are arguably one of the most underappreciated important opportunities in modern AI. Could we train AI models 1000x larger than current ones? Could we do this and also have them perform inference locally and privately on edge devices, such as smartphones or sensors? Research over the past few years has shown that the answer to all these questions is likely "yes, with enough research": PNNs could one day radically change what is possible and practical for AI systems. To do this will however require rethinking both how AI models work, and how they are trained - primarily by considering the problems through the constraints of the underlying hardware physics. To train PNNs at large scale, many methods including backpropagation-based and backpropagation-free approaches are now being explored. These methods have various trade-offs, and so far no method has been shown to scale to the same scale and performance as the backpropagation algorithm widely used in deep learning today. However, this is rapidly changing, and a diverse ecosystem of training techniques provides clues for how PNNs may one day be utilized to create both more efficient realizations of current-scale AI models, and to enable unprecedented-scale models., Comment: 29 pages, 4 figures

Designing algorithms for versatile AI hardware that can learn on the edge using both labeled and unlabeled data is challenging. Deep end-to-end training methods incorporating phases of self-supervised and supervised learning are accurate and adaptable to input data but self-supervised learning requires even more computational and memory resources than supervised learning, too high for current embedded hardware. Conversely, unsupervised layer-by-layer training, such as Hebbian learning, is more compatible with existing hardware but does not integrate well with supervised learning. To address this, we propose a method enabling networks or hardware designed for end-to-end supervised learning to also perform high-performance unsupervised learning by adding two simple elements to the output layer: Winner-Take-All (WTA) selectivity and homeostasis regularization. These mechanisms introduce a "self-defined target" for unlabeled data, allowing purely unsupervised training for both fully-connected and convolutional layers using backpropagation or equilibrium propagation on datasets like MNIST (up to 99.2%), Fashion-MNIST (up to 90.3%), and SVHN (up to 81.5%). We extend this method to semi-supervised learning, adjusting targets based on data type, achieving 96.6% accuracy with only 600 labeled MNIST samples in a multi-layer perceptron. Our results show that this approach can effectively enable networks and hardware initially dedicated to supervised learning to also perform unsupervised learning, adapting to varying availability of labeled data.

Integrating magnetic skyrmion properties into neuromorphic computing promises advancements in hardware efficiency and computational power. However, a scalable implementation of the weighted sum of neuron signals, a core operation in neural networks, has yet to be demonstrated. In this study, we exploit the non-volatile and particle-like characteristics of magnetic skyrmions, akin to synaptic vesicles and neurotransmitters, to perform this weighted sum operation in a compact, biologically-inspired manner. To this aim, skyrmions are electrically generated in numbers proportional to the input with an efficiency given by a non-volatile weight. These chiral particles are then directed using localized current injections to a location where their presence is quantified through non-perturbative electrical measurements. Our experimental demonstration, currently with two inputs, can be scaled to accommodate multiple inputs and outputs using a crossbar array design, potentially nearing the energy efficiency observed in biological systems., Comment: 12 pages, 5 figures

Controlling magnetic properties through the application of an electric field is a significant challenge in modern nanomagnetism. In this study, we investigate the magneto-ionic control of magnetic anisotropy in the topmost Co layer in Ta/Pt/[Co/Al/Pt]$_n$/Co/Al/AlO$_\text{x}$ multilayer stacks comprising $n +1$ Co layers and its impact on the magnetic properties of the multilayers. We demonstrate that the perpendicular magnetic anisotropy can be reversibly quenched through gate-driven oxidation of the intermediary Al layer between Co and AlO$_\text{x}$, enabling dynamic control of the magnetic layers contributing to the out-of-plane remanence - varying between $n$ and $n +1$. For multilayer configurations with $n = 2$ and $n = 4$, we observe reversible and non-volatile additions of 1/3 and 1/5, respectively, to the anomalous Hall effect amplitude based on the applied gate voltage. Magnetic imaging reveals that the gate-induced spin-reorientation transition occurs through the propagation of a single 90$^{\circ}$ magnetic domain wall separating the perpendicular and in-plane anisotropy states. In the 5-repetition multilayer, the modification leads to a doubling of the period of the magnetic domains at remanence. These results demonstrate that the magneto-ionic control of the anisotropy of a single magnetic layer can be used to control the magnetic properties of coupled multilayer systems, extending beyond the gating effects on a single magnetic layer., Comment: 9 pages + 3 pages of supplementary materials, 5 main figures + 6 supplementary figures

Spintronic devices have recently attracted a lot of attention in the field of unconventional computing due to their non-volatility for short and long term memory, non-linear fast response and relatively small footprint. Here we report how voltage driven magnetization dynamics of dual free layer perpendicular magnetic tunnel junctions enable to emulate spiking neurons in hardware. The output spiking rate was controlled by varying the dc bias voltage across the device. The field-free operation of this two terminal device and its robustness against an externally applied magnetic field make it a suitable candidate to mimic neuron response in a dense Neural Network (NN). The small energy consumption of the device (4-16 pJ/spike) and its scalability are important benefits for embedded applications. This compact perpendicular magnetic tunnel junction structure could finally bring spiking neural networks (SNN) to sub-100nm size elements.

Ising machines, which are hardware implementations of the Ising model of coupled spins, have been influential in the development of unsupervised learning algorithms at the origins of Artificial Intelligence (AI). However, their application to AI has been limited due to the complexities in matching supervised training methods with Ising machine physics, even though these methods are essential for achieving high accuracy. In this study, we demonstrate a novel approach to train Ising machines in a supervised way through the Equilibrium Propagation algorithm, achieving comparable results to software-based implementations. We employ the quantum annealing procedure of the D-Wave Ising machine to train a fully-connected neural network on the MNIST dataset. Furthermore, we demonstrate that the machine's connectivity supports convolution operations, enabling the training of a compact convolutional network with minimal spins per neuron. Our findings establish Ising machines as a promising trainable hardware platform for AI, with the potential to enhance machine learning applications.

In the "Beyond Moore's Law" era, with increasing edge intelligence, domain-specific computing embracing unconventional approaches will become increasingly prevalent. At the same time, adopting a variety of nanotechnologies will offer benefits in energy cost, computational speed, reduced footprint, cyber resilience, and processing power. The time is ripe for a roadmap for unconventional computing with nanotechnologies to guide future research, and this collection aims to fill that need. The authors provide a comprehensive roadmap for neuromorphic computing using electron spins, memristive devices, two-dimensional nanomaterials, nanomagnets, and various dynamical systems. They also address other paradigms such as Ising machines, Bayesian inference engines, probabilistic computing with p-bits, processing in memory, quantum memories and algorithms, computing with skyrmions and spin waves, and brain-inspired computing for incremental learning and problem-solving in severely resource-constrained environments. These approaches have advantages over traditional Boolean computing based on von Neumann architecture. As the computational requirements for artificial intelligence grow 50 times faster than Moore's Law for electronics, more unconventional approaches to computing and signal processing will appear on the horizon, and this roadmap will help identify future needs and challenges. In a very fertile field, experts in the field aim to present some of the dominant and most promising technologies for unconventional computing that will be around for some time to come. Within a holistic approach, the goal is to provide pathways for solidifying the field and guiding future impactful discoveries., Comment: 80 pages accepted in Nano Futures

Spintronic nano-oscillators with reduced non-linearity could offer key benefits for realizing neuromorphic applications such as spike-based neurons and frequency multiplexing in neural networks. Here, we experimentally demonstrate the reduction in non-linearity of a spin-Hall nano-oscillator (SHNO) by compensation of its effective magnetic anisotropy. The study involves optimization of Co/Ni multilayer growth to achieve the compensation, followed by spin diode measurements on patterned microstrips to quantify their anisotropy. The relation between the second ($H_{k2}$ = 0.47 mT) and the first order ($H_{k1}^{eff}$ = $-$0.8 mT) anisotropy fields reveals the existence of an easy cone, thereby validating the presence of compensation. Furthermore, we demonstrate a synapse based on the compensated spin diode which has a fixed frequency when the input power is varied. We then study the current-induced auto-oscillation properties of SHNOs on compensated films by patterning nano-constrictions of widths 200 and 100 nm. The invariance of the resonance frequency and linewidth of the compensated SHNO with applied dc current indicates the absence of non-linearity. This independence is maintained irrespective of the applied external fields and its orientations. The compensated SHNO obtained has a linewidth of 1.1 MHz and a peak output power of up to 1 pW/MHz emulating a nano-neuron with a low linewidth and a fixed frequency.

Spintronic nano-synapses and nano-neurons perform complex cognitive computations with high accuracy thanks to their rich, reproducible and controllable magnetization dynamics. These dynamical nanodevices could transform artificial intelligence hardware, provided that they implement state-of-the art deep neural networks. However, there is today no scalable way to connect them in multilayers. Here we show that the flagship nano-components of spintronics, magnetic tunnel junctions, can be connected into multilayer neural networks where they implement both synapses and neurons thanks to their magnetization dynamics, and communicate by processing, transmitting and receiving radio frequency (RF) signals. We build a hardware spintronic neural network composed of nine magnetic tunnel junctions connected in two layers, and show that it natively classifies nonlinearly-separable RF inputs with an accuracy of 97.7%. Using physical simulations, we demonstrate that a large network of nanoscale junctions can achieve state-of the-art identification of drones from their RF transmissions, without digitization, and consuming only a few milliwatts, which is a gain of more than four orders of magnitude in power consumption compared to currently used techniques. This study lays the foundation for deep, dynamical, spintronic neural networks.

Extracting information from radiofrequency (RF) signals using artificial neural networks at low energy cost is a critical need for a wide range of applications from radars to health. These RF inputs are composed of multiples frequencies. Here we show that magnetic tunnel junctions can process analogue RF inputs with multiple frequencies in parallel and perform synaptic operations. Using a backpropagation-free method called extreme learning, we classify noisy images encoded by RF signals, using experimental data from magnetic tunnel junctions functioning as both synapses and neurons. We achieve the same accuracy as an equivalent software neural network. These results are a key step for embedded radiofrequency artificial intelligence., Comment: 9 pages, 5 figures

Quantum reservoir computing is a promising approach for quantum neural networks, capable of solving hard learning tasks on both classical and quantum input data. However, current approaches with qubits suffer from limited connectivity. We propose an implementation for quantum reservoir that obtains a large number of densely connected neurons by using parametrically coupled quantum oscillators instead of physically coupled qubits. We analyse a specific hardware implementation based on superconducting circuits: with just two coupled quantum oscillators, we create a quantum reservoir comprising up to 81 neurons. We obtain state-of-the-art accuracy of 99 % on benchmark tasks that otherwise require at least 24 classical oscillators to be solved. Our results give the coupling and dissipation requirements in the system and show how they affect the performance of the quantum reservoir. Beyond quantum reservoir computing, the use of parametrically coupled bosonic modes holds promise for realizing large quantum neural network architectures, with billions of neurons implemented with only 10 coupled quantum oscillators.

Hardware spiking neural networks hold the promise of realizing artificial intelligence with high energy efficiency. In this context, solid-state and scalable memristors can be used to mimic biological neuron characteristics. However, these devices show limited neuronal behaviors and have to be integrated in more complex circuits to implement the rich dynamics of biological neurons. Here we studied a NbOx memristor neuron that is capable of emulating numerous neuronal dynamics, including tonic spiking, stochastic spiking, leaky-integrate-and-fire features, spike latency, temporal integration. The device also exhibits phasic bursting, a property that has scarcely been observed and studied in solid-state nano-neurons. We show that we can reproduce and understand this particular response through simulations using non-linear dynamics. These results show that a single NbOx device is sufficient to emulate a collection of rich neuronal dynamics that paves a path forward for realizing scalable and energy-efficient neuromorphic computing paradigms.

We analyze the properties of a quantum system composed of two coherently coupled quantum oscillators and show through simulations that it fulfills the two properties required for reservoir computing: non-linearity and fading memory. We first show that the basis states of this system apply a set of nonlinear transformations on the input signals and thus can implement neurons. We then show that the system exhibits a fading memory that can be controlled by its dissipation rates. Finally we show that a strong coupling between the oscillators is important in order to ensure complex dynamics and to populate a number of basis state neurons that is exponential in the number of physical devices.

Neuromorphic computing approaches become increasingly important as we address future needs for efficiently processing massive amounts of data. The unique attributes of quantum materials can help address these needs by enabling new energy-efficient device concepts that implement neuromorphic ideas at the hardware level. In particular, strong correlations give rise to highly non-linear responses, such as conductive phase transitions that can be harnessed for short and long-term plasticity. Similarly, magnetization dynamics are strongly non-linear and can be utilized for data classification. This paper discusses select examples of these approaches, and provides a perspective for the current opportunities and challenges for assembling quantum-material-based devices for neuromorphic functionalities into larger emergent complex network systems.

Convolutional neural networks are state-of-the-art and ubiquitous in modern signal processing and machine vision. Nowadays, hardware solutions based on emerging nanodevices are designed to reduce the power consumption of these networks. Spintronics devices are promising for information processing because of the various neural and synaptic functionalities they offer. However, due to their low OFF/ON ratio, performing all the multiplications required for convolutions in a single step with a crossbar array of spintronic memories would cause sneak-path currents. Here we present an architecture where synaptic communications have a frequency selectivity that prevents crosstalk caused by sneak-path currents. We first demonstrate how a chain of spintronic resonators can function as synapses and make convolutions by sequentially rectifying radio-frequency signals encoding consecutive sets of inputs. We show that a parallel implementation is possible with multiple chains of spintronic resonators to avoid storing intermediate computational steps in memory. We propose two different spatial arrangements for these chains. For each of them, we explain how to tune many artificial synapses simultaneously, exploiting the synaptic weight sharing specific to convolutions. We show how information can be transmitted between convolutional layers by using spintronic oscillators as artificial microwave neurons. Finally, we simulate a network of these radio-frequency resonators and spintronic oscillators to solve the MNIST handwritten digits dataset, and obtain results comparable to software convolutional neural networks. Since it can run convolutional neural networks fully in parallel in a single step with nano devices, the architecture proposed in this paper is promising for embedded applications requiring machine vision, such as autonomous driving.

We show analytically using a macrospin approximation that easy-plane spin Hall nano-oscillators excited by a spin-current polarized perpendicularly to the easy-plane have phase dynamics analogous to that of Josephson junctions. Similarly to Josephson junctions, they can reproduce the spiking behavior of biological neurons that is appropriate for neuromorphic computing. We perform micromagnetic simulations of such oscillators realized in the nano-constriction geometry and show that the easy-plane spiking dynamics is preserved in an experimentally feasible architecture. Finally we simulate two elementary neural network blocks that implement operations essential for neuromorphic computing. First, we show that output spikes energies from two neurons can be summed and injected into a following layer neuron and second, we demonstrate that outputs can be multiplied by synaptic weights implemented by locally modifying the anisotropy., Comment: 9 pages, 11 figures

Deep learning has an increasing impact to assist research, allowing, for example, the discovery of novel materials. Until now, however, these artificial intelligence techniques have fallen short of discovering the full differential equation of an experimental physical system. Here we show that a dynamical neural network, trained on a minimal amount of data, can predict the behavior of spintronic devices with high accuracy and an extremely efficient simulation time, compared to the micromagnetic simulations that are usually employed to model them. For this purpose, we re-frame the formalism of Neural Ordinary Differential Equations (ODEs) to the constraints of spintronics: few measured outputs, multiple inputs and internal parameters. We demonstrate with Spin-Neural ODEs an acceleration factor over 200 compared to micromagnetic simulations for a complex problem -- the simulation of a reservoir computer made of magnetic skyrmions (20 minutes compared to three days). In a second realization, we show that we can predict the noisy response of experimental spintronic nano-oscillators to varying inputs after training Spin-Neural ODEs on five milliseconds of their measured response to different excitations. Spin-Neural ODE is a disruptive tool for developing spintronic applications in complement to micromagnetic simulations, which are time-consuming and cannot fit experiments when noise or imperfections are present. Spin-Neural ODE can also be generalized to other electronic devices involving dynamics., Comment: 16 pages, 4 figures

Here we review various themes of current research within mesoscopic magnetic systems., Comment: APL Special Topic Guest Editorial

Modern computation based on the von Neumann architecture is today a mature cutting-edge science. In the Von Neumann architecture, processing and memory units are implemented as separate blocks interchanging data intensively and continuously. This data transfer is responsible for a large part of the power consumption. The next generation computer technology is expected to solve problems at the exascale with 1018 calculations each second. Even though these future computers will be incredibly powerful, if they are based on von Neumann type architectures, they will consume between 20 and 30 megawatts of power and will not have intrinsic physically built-in capabilities to learn or deal with complex data as our brain does. These needs can be addressed by neuromorphic computing systems which are inspired by the biological concepts of the human brain. This new generation of computers has the potential to be used for the storage and processing of large amounts of digital information with much lower power consumption than conventional processors. Among their potential future applications, an important niche is moving the control from data centers to edge devices. The aim of this Roadmap is to present a snapshot of the present state of neuromorphic technology and provide an opinion on the challenges and opportunities that the future holds in the major areas of neuromorphic technology, namely materials, devices, neuromorphic circuits, neuromorphic algorithms, applications, and ethics. The Roadmap is a collection of perspectives where leading researchers in the neuromorphic community provide their own view about the current state and the future challenges. We hope that this Roadmap will be a useful resource to readers outside this field, for those who are just entering the field, and for those who are well established in the neuromorphic community. https://doi.org/10.1088/2634-4386/ac4a83

Vortex based spin torque nano oscillators (STVOs) can present more complex dynamics than the spin torque induced gyrotropic (G) motion of the vortex core. The respective dynamic modes and the transition between them can be controlled by experimental parameters such as the applied dc current. An interesting behavior is the stochastic transition from the G- to a dynamic C-state occurring for large current densities. Moreover, the C-state oscillations exhibit a constant active magnetic volume. We present noise measurements in the different dynamic states that allow accessing specific properties of the stochastic transition, such as the characteristic state transition frequency. Furthermore,we confirm, as theoretically predicted, an increase of flicker noise with $I_{dc}^2$ when the oscillation volume remains constant with the current. These results bring insight into the potential optimization of noise properties sought for many potential rf applications with spin torque oscillators. Furthermore, the investigated stochastic characteristics open up new potentialities, for instance in the emerging field of neuromorphic computing schemes.

Artificial neural networks are a valuable tool for radio-frequency (RF) signal classification in many applications, but digitization of analog signals and the use of general purpose hardware non-optimized for training make the process slow and energetically costly. Recent theoretical work has proposed to use nano-devices called magnetic tunnel junctions, which exhibit intrinsic RF dynamics, to implement in hardware the Multiply and Accumulate (MAC) operation, a key building block of neural networks, directly using analogue RF signals. In this article, we experimentally demonstrate that a magnetic tunnel junction can perform multiplication of RF powers, with tunable positive and negative synaptic weights. Using two magnetic tunnel junctions connected in series we demonstrate the MAC operation and use it for classification of RF signals. These results open the path to embedded systems capable of analyzing RF signals with neural networks directly after the antenna, at low power cost and high speed., Comment: 10 pages, 4 figures

Equilibrium Propagation (EP) is an algorithm intrinsically adapted to the training of physical networks, thanks to the local updates of weights given by the internal dynamics of the system. However, the construction of such a hardware requires to make the algorithm compatible with existing neuromorphic CMOS technologies, which generally exploit digital communication between neurons and offer a limited amount of local memory. In this work, we demonstrate that EP can train dynamical networks with binary activations and weights. We first train systems with binary weights and full-precision activations, achieving an accuracy equivalent to that of full-precision models trained by standard EP on MNIST, and losing only 1.9% accuracy on CIFAR-10 with equal architecture. We then extend our method to the training of models with binary activations and weights on MNIST, achieving an accuracy within 1% of the full-precision reference for fully connected architectures and reaching the full-precision accuracy for convolutional architectures. Our extension of EP to binary networks opens new solutions for on-chip learning and provides a compact framework for training BNNs end-to-end with the same circuitry as for inference.

Equilibrium Propagation (EP) is a biologically-inspired counterpart of Backpropagation Through Time (BPTT) which, owing to its strong theoretical guarantees and the locality in space of its learning rule, fosters the design of energy-efficient hardware dedicated to learning. In practice, however, EP does not scale to visual tasks harder than MNIST. In this work, we show that a bias in the gradient estimate of EP, inherent in the use of finite nudging, is responsible for this phenomenon and that cancelling it allows training deep ConvNets by EP, including architectures with distinct forward and backward connections. These results highlight EP as a scalable approach to compute error gradients in deep neural networks, thereby motivating its hardware implementation., Comment: NeurIPS 2020 Workshop : "Beyond Backpropagation Novel Ideas for Training Neural Architectures". arXiv admin note: substantial text overlap with arXiv:2006.03824

Metamaterials present the possibility of artificially generating advanced functionalities through engineering of their internal structure. Artificial spin networks, in which a large number of nanoscale magnetic elements are coupled together, are promising metamaterial candidates that enable the control of collective magnetic behavior through tuning of the local interaction between elements. In this work, the motion of magnetic domain-walls in an artificial spin network leads to a tunable stochastic response of the metamaterial, which can be tailored through an external magnetic field and local lattice modifications. This type of tunable stochastic network produces a controllable random response exploiting intrinsic stochasticity within magnetic domain-wall motion at the nanoscale. An iconic demonstration used to illustrate the control of randomness is the Galton board. In this system, multiple balls fall into an array of pegs to generate a bell-shaped curve that can be modified via the array spacing or the tilt of the board. A nanoscale recreation of this experiment using an artificial spin network is employed to demonstrate tunable stochasticity. This type of tunable stochastic network opens new paths towards post-Von Neumann computing architectures such as Bayesian sensing or random neural networks, in which stochasticity is harnessed to efficiently perform complex computational tasks., Comment: 24 pages, 10 figures

Finding spike-based learning algorithms that can be implemented within the local constraints of neuromorphic systems, while achieving high accuracy, remains a formidable challenge. Equilibrium Propagation is a promising alternative to backpropagation as it only involves local computations, but hardware-oriented studies have so far focused on rate-based networks. In this work, we develop a spiking neural network algorithm called EqSpike, compatible with neuromorphic systems, which learns by Equilibrium Propagation. Through simulations, we obtain a test recognition accuracy of 97.6% on MNIST, similar to rate-based Equilibrium Propagation, and comparing favourably to alternative learning techniques for spiking neural networks. We show that EqSpike implemented in silicon neuromorphic technology could reduce the energy consumption of inference and training respectively by three orders and two orders of magnitude compared to GPUs. Finally, we also show that during learning, EqSpike weight updates exhibit a form of Spike Timing Dependent Plasticity, highlighting a possible connection with biology.

In the present study, we investigate a dynamical mode beyond the gyrotropic (G) motion of a magnetic vortex core in a confined magnetic disk of a nano-pillar spin torque nano oscillator. It is characterized by the in-plane circular precession associated to a C-shaped magnetization distribution. We show a transition between G and C-state mode which is found to be purely stochastic in a current-controllable range. Supporting our experimental findings with micromagnetic simulations, we believe that the results provide novel opportunities for the dynamic and stochastic control of STOs, which could be interesting to be implemented for example in neuromorphic networks.

Quantum neuromorphic computing physically implements neural networks in brain-inspired quantum hardware to speed up their computation. In this perspective article, we show that this emerging paradigm could make the best use of the existing and near future intermediate size quantum computers. Some approaches are based on parametrized quantum circuits, and use neural network-inspired algorithms to train them. Other approaches, closer to classical neuromorphic computing, take advantage of the physical properties of quantum oscillator assemblies to mimic neurons and compute. We discuss the different implementations of quantum neuromorphic networks with digital and analog circuits, highlight their respective advantages, and review exciting recent experimental results.

Equilibrium Propagation (EP) is a biologically-inspired algorithm for convergent RNNs with a local learning rule that comes with strong theoretical guarantees. The parameter updates of the neural network during the credit assignment phase have been shown mathematically to approach the gradients provided by Backpropagation Through Time (BPTT) when the network is infinitesimally nudged toward its target. In practice, however, training a network with the gradient estimates provided by EP does not scale to visual tasks harder than MNIST. In this work, we show that a bias in the gradient estimate of EP, inherent in the use of finite nudging, is responsible for this phenomenon and that cancelling it allows training deep ConvNets by EP. We show that this bias can be greatly reduced by using symmetric nudging (a positive nudging and a negative one). We also generalize previous EP equations to the case of cross-entropy loss (by opposition to squared error). As a result of these advances, we are able to achieve a test error of 11.7% on CIFAR-10 by EP, which approaches the one achieved by BPTT and provides a major improvement with respect to the standard EP approach with same-sign nudging that gives 86% test error. We also apply these techniques to train an architecture with asymmetric forward and backward connections, yielding a 13.2% test error. These results highlight EP as a compelling biologically-plausible approach to compute error gradients in deep neural networks.

Equilibrium Propagation (EP) is a biologically inspired alternative algorithm to backpropagation (BP) for training neural networks. It applies to RNNs fed by a static input x that settle to a steady state, such as Hopfield networks. EP is similar to BP in that in the second phase of training, an error signal propagates backwards in the layers of the network, but contrary to BP, the learning rule of EP is spatially local. Nonetheless, EP suffers from two major limitations. On the one hand, due to its formulation in terms of real-time dynamics, EP entails long simulation times, which limits its applicability to practical tasks. On the other hand, the biological plausibility of EP is limited by the fact that its learning rule is not local in time: the synapse update is performed after the dynamics of the second phase have converged and requires information of the first phase that is no longer available physically. Our work addresses these two issues and aims at widening the spectrum of EP from standard machine learning models to more bio-realistic neural networks. First, we propose a discrete-time formulation of EP which enables to simplify equations, speed up training and extend EP to CNNs. Our CNN model achieves the best performance ever reported on MNIST with EP. Using the same discrete-time formulation, we introduce Continual Equilibrium Propagation (C-EP): the weights of the network are adjusted continually in the second phase of training using local information in space and time. We show that in the limit of slow changes of synaptic strengths and small nudging, C-EP is equivalent to BPTT (Theorem 1). We numerically demonstrate Theorem 1 and C-EP training on MNIST and generalize it to the bio-realistic situation of a neural network with asymmetric connections between neurons.

Equilibrium Propagation (EP) is a learning algorithm that bridges Machine Learning and Neuroscience, by computing gradients closely matching those of Backpropagation Through Time (BPTT), but with a learning rule local in space. Given an input $x$ and associated target $y$, EP proceeds in two phases: in the first phase neurons evolve freely towards a first steady state; in the second phase output neurons are nudged towards $y$ until they reach a second steady state. However, in existing implementations of EP, the learning rule is not local in time: the weight update is performed after the dynamics of the second phase have converged and requires information of the first phase that is no longer available physically. In this work, we propose a version of EP named Continual Equilibrium Propagation (C-EP) where neuron and synapse dynamics occur simultaneously throughout the second phase, so that the weight update becomes local in time. Such a learning rule local both in space and time opens the possibility of an extremely energy efficient hardware implementation of EP. We prove theoretically that, provided the learning rates are sufficiently small, at each time step of the second phase the dynamics of neurons and synapses follow the gradients of the loss given by BPTT (Theorem 1). We demonstrate training with C-EP on MNIST and generalize C-EP to neural networks where neurons are connected by asymmetric connections. We show through experiments that the more the network updates follows the gradients of BPTT, the best it performs in terms of training. These results bring EP a step closer to biology by better complying with hardware constraints while maintaining its intimate link with backpropagation.

Neuromorphic computing takes inspiration from the brain to create energy efficient hardware for information processing, capable of highly sophisticated tasks. In this article, we make the case that building this new hardware necessitates reinventing electronics. We show that research in physics and material science will be key to create artificial nano-neurons and synapses, to connect them together in huge numbers, to organize them in complex systems, and to compute with them efficiently. We describe how some researchers choose to take inspiration from artificial intelligence to move forward in this direction, whereas others prefer taking inspiration from neuroscience, and we highlight recent striking results obtained with these two approaches. Finally, we discuss the challenges and perspectives in neuromorphic physics, which include developing the algorithms and the hardware hand in hand, making significant advances with small toy systems, as well as building large scale networks.

The correlation of phase fluctuations in any type of oscillator fundamentally defines its spectral shape. However, in nonlinear oscillators, such as spin torque nano oscillators, the frequency spectrum can become particularly complex. This is specifically true when not only considering thermal but also colored $1/f$ flicker noise processes, which are crucial in the context of the oscillator's long term stability. In this study, we address the frequency spectrum of spin torque oscillators in the regime of large-amplitude steady oscillations experimentally and as well theoretically. We particularly take both thermal and flicker noise into account. We perform a series of measurements of the phase noise and the spectrum on spin torque vortex oscillators, notably varying the measurement time duration. Furthermore, we develop the modelling of thermal and flicker noise in Thiele equation based simulations. We also derive the complete phase variance in the framework of the nonlinear auto-oscillator theory and deduce the actual frequency spectrum. We investigate its dependence on the measurement time duration and compare with the experimental results. Long term stability is important in several of the recent applicative developments of spin torque oscillators. This study brings some insights on how to better address this issue.

The brain naturally binds events from different sources in unique concepts. It is hypothesized that this process occurs through the transient mutual synchronization of neurons located in different regions of the brain when the stimulus is presented. This mechanism of binding through synchronization can be directly implemented in neural networks composed of coupled oscillators. To do so, the oscillators must be able to mutually synchronize for the range of inputs corresponding to a single class, and otherwise remain desynchronized. Here we show that the outstanding ability of spintronic nano-oscillators to mutually synchronize and the possibility to precisely control the occurrence of mutual synchronization by tuning the oscillator frequencies over wide ranges allows pattern recognition. We demonstrate experimentally on a simple task that three spintronic nano-oscillators can bind consecutive events and thus recognize and distinguish temporal sequences. This work is a step forward in the construction of neural networks that exploit the non-linear dynamic properties of their components to perform brain-inspired computations.

Magnetic tunnel junctions are nanoscale spintronic devices with microwave generation and detection capabilities. Here we use the rectification effect called "spin-diode" in a magnetic tunnel junction to wirelessly detect the microwave emission of another junction in the auto-oscillatory regime. We show that the rectified spin-diode voltage measured at the receiving junction end can be reconstructed from the independently measured auto-oscillation and spin diode spectra in each junction. Finally we adapt the auto-oscillator model to the case of spin-torque oscillator and spin-torque diode and we show that accurately reproduces the experimentally observed features. These results will be useful to design circuits and chips based on spintronic nanodevices communicating through microwaves.

Arrays of spin-torque nano-oscillators are promising for broadband microwave signal detection and processing, as well as for neuromorphic computing. In many of these applications, the oscillators should be engineered to have equally-spaced frequencies and equal sensitivity to microwave inputs. Here we design spin-torque nano-oscillator arrays with these rules and estimate their optimum size for a given sensitivity, as well as the frequency range that they cover. For this purpose, we explore analytically and numerically conditions to obtain vortex spin-torque nano-oscillators with equally-spaced gyrotropic oscillation frequencies and having all similar synchronization bandwidths to input microwave signals. We show that arrays of hundreds of oscillators covering ranges of several hundred MHz can be built taking into account nanofabrication constraints.

Equilibrium Propagation (EP) is a biologically inspired learning algorithm for convergent recurrent neural networks, i.e. RNNs that are fed by a static input x and settle to a steady state. Training convergent RNNs consists in adjusting the weights until the steady state of output neurons coincides with a target y. Convergent RNNs can also be trained with the more conventional Backpropagation Through Time (BPTT) algorithm. In its original formulation EP was described in the case of real-time neuronal dynamics, which is computationally costly. In this work, we introduce a discrete-time version of EP with simplified equations and with reduced simulation time, bringing EP closer to practical machine learning tasks. We first prove theoretically, as well as numerically that the neural and weight updates of EP, computed by forward-time dynamics, are step-by-step equal to the ones obtained by BPTT, with gradients computed backward in time. The equality is strict when the transition function of the dynamics derives from a primitive function and the steady state is maintained long enough. We then show for more standard discrete-time neural network dynamics that the same property is approximately respected and we subsequently demonstrate training with EP with equivalent performance to BPTT. In particular, we define the first convolutional architecture trained with EP achieving ~ 1% test error on MNIST, which is the lowest error reported with EP. These results can guide the development of deep neural networks trained with EP.

The reservoir computing neural network architecture is widely used to test hardware systems for neuromorphic computing. One of the preferred tasks for bench-marking such devices is automatic speech recognition. However, this task requires acoustic transformations from sound waveforms with varying amplitudes to frequency domain maps that can be seen as feature extraction techniques. Depending on the conversion method, these may obscure the contribution of the neuromorphic hardware to the overall speech recognition performance. Here, we quantify and separate the contributions of the acoustic transformations and the neuromorphic hardware to the speech recognition success rate. We show that the non-linearity in the acoustic transformation plays a critical role in feature extraction. We compute the gain in word success rate provided by a reservoir computing device compared to the acoustic transformation only, and show that it is an appropriate benchmark for comparing different hardware. Finally, we experimentally and numerically quantify the impact of the different acoustic transformations for neuromorphic hardware based on magnetic nano-oscillators., Comment: 13 pages, 5 figures

Spin-torque nano-oscillators can emulate neurons at the nanoscale. Recent works show that the non-linearity of their oscillation amplitude can be leveraged to achieve waveform classification for an input signal encoded in the amplitude of the input voltage. Here we show that the frequency and the phase of the oscillator can also be used to recognize waveforms. For this purpose, we phase-lock the oscillator to the input waveform, which carries information in its modulated frequency. In this way we considerably decrease amplitude, phase and frequency noise. We show that this method allows classifying sine and square waveforms with an accuracy above 99% when decoding the output from the oscillator amplitude, phase or frequency. We find that recognition rates are directly related to the noise and non-linearity of each variable. These results prove that spin-torque nano-oscillators offer an interesting platform to implement different computing schemes leveraging their rich dynamical features.

Spintronic neurons which emit sharp voltage spikes are required for the realization of hardware neural networks enabling fast data processing with low-power consumption. In many neuroscience and computer science models, neurons are abstracted as non-linear oscillators. Magnetic nano-oscillators called spin-torque nano-oscillators are interesting candidates for imitating neurons at nanoscale. These oscillators, however, emit sinusoidal waveforms without spiking while biological neurons are relaxation oscillators that emit sharp voltage spikes. Here we propose a simple way to imitate neuron spiking in high-magnetoresistance nanoscale spin valves where both magnetic layers are free and thin enough to be switched by spin torque. Our numerical-simulation results show that the windmill motion induced by spin torque in the proposed spintronic neurons gives rise to spikes whose shape and frequency, set by the charging and discharging times, can be tuned through the amplitude of injected dc current. We also found that these devices can exhibit chaotic oscillations. Chaotic-like neuron dynamics has been observed in the brain, and it is desirable in some neuromorphic computing applications whereas it should be avoided in others. We demonstrate that the degree of chaos can be tuned in a wide range by engineering the magnetic stack and anisotropies and by changing the dc current. The proposed spintronic neuron is a promising building block for hardware neuromorphic chips leveraging non-linear dynamics for computing.

Pattern classification architectures leveraging the physics of coupled nano-oscillators have been demonstrated as promising alternative computing approaches, but lack effective learning algorithms. In this work, we propose a nano-oscillator based classification architecture where the natural frequencies of the oscillators are learned linear combinations of the inputs, and define an offline learning algorithm based on gradient back-propagation. Our results show significant classification improvements over a related approach with online learning. We also compare our architecture with a standard neural network on a simple machine learning case, which suggests that our approach is economical in terms of numbers of adjustable parameters. The introduced architecture is also compatible with existing nano-technologies: the architecture does not require changes in the coupling between nano-oscillators, and it is tolerant to oscillator phase noise.

Many emerging alternative models of computation require massive numbers of random bits, but their generation at low energy is currently a challenge. The superparamagnetic tunnel junction, a spintronic device based on the same technology as spin torque magnetoresistive random access memory has recently been proposed as a solution, as this device naturally switches between two easy to measure resistance states, due only to thermal noise. Reading the state of the junction naturally provides random bits, without the need of write operations. In this work, we evaluate a circuit solution for reading the state of superparamagnetic tunnel junction. We see that the circuit may induce a small read disturb effect for scaled superparamagnetic tunnel junctions, but this effect is naturally corrected in the whitening process needed to ensure the quality of the generated random bits. These results suggest that superparamagnetic tunnel junctions could generate truly random bits at 20 fJ/bit, including overheads, orders of magnitudes below CMOS-based solutions.

The brain, which uses redundancy and continuous learning to overcome the unreliability of its components, provides a promising path to building computing systems that are robust to the unreliability of their constituent nanodevices. In this work, we illustrate this path by a computing system based on population coding with magnetic tunnel junctions that implement both neurons and synaptic weights. We show that equipping such a system with continuous learning enables it to recover from the loss of neurons and makes it possible to use unreliable synaptic weights (i.e. low energy barrier magnetic memories). There is a tradeoff between power consumption and precision because low energy barrier memories consume less energy than high barrier ones. For a given precision, there is an optimal number of neurons and an optimal energy barrier for the weights that leads to minimum power consumption.

Substantial evidence indicates that the brain uses principles of non-linear dynamics in neural processes, providing inspiration for computing with nanoelectronic devices. However, training neural networks composed of dynamical nanodevices requires finely controlling and tuning their coupled oscillations. In this work, we show that the outstanding tunability of spintronic nano-oscillators can solve this challenge. We successfully train a hardware network of four spin-torque nano-oscillators to recognize spoken vowels by tuning their frequencies according to an automatic real-time learning rule. We show that the high experimental recognition rates stem from the high frequency tunability of the oscillators and their mutual coupling. Our results demonstrate that non-trivial pattern classification tasks can be achieved with small hardware neural networks by endowing them with non-linear dynamical features: here, oscillations and synchronization. This demonstration is a milestone for spintronics-based neuromorphic computing.