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The inability of conventional electronic architectures to efficiently solve large combinatorial problems motivates the development of novel computational hardware. There has been much effort toward developing application-specific hardware across many different fields of engineering, such as integrated circuits, memristors, and photonics. However, unleashing the potential of such architectures requires the development of algorithms which optimally exploit their fundamental properties. Here, we present the Photonic Recurrent Ising Sampler (PRIS), a heuristic method tailored for parallel architectures allowing fast and efficient sampling from distributions of arbitrary Ising problems. Since the PRIS relies on vector-to-fixed matrix multiplications, we suggest the implementation of the PRIS in photonic parallel networks, which realize these operations at an unprecedented speed. The PRIS provides sample solutions to the ground state of Ising models, by converging in probability to their associated Gibbs distribution. The PRIS also relies on intrinsic dynamic noise and eigenvalue dropout to find ground states more efficiently. Our work suggests speedups in heuristic methods via photonic implementations of the PRIS.

Symbolic regression is a powerful technique to discover analytic equations that describe data, which can lead to explainable models and the ability to predict unseen data. In contrast, neural networks have achieved amazing levels of accuracy on image recognition and natural language processing tasks, but they are often seen as black-box models that are difficult to interpret and typically extrapolate poorly. In this article, we use a neural network-based architecture for symbolic regression called the equation learner (EQL) network and integrate it with other deep learning architectures such that the whole system can be trained end-to-end through backpropagation. To demonstrate the power of such systems, we study their performance on several substantially different tasks. First, we show that the neural network can perform symbolic regression and learn the form of several functions. Next, we present an MNIST arithmetic task where a convolutional network extracts the digits. Finally, we demonstrate the prediction of dynamical systems where an unknown parameter is extracted through an encoder. We find that the EQL-based architecture can extrapolate quite well outside of the training data set compared with a standard neural network-based architecture, paving the way for deep learning to be applied in scientific exploration and discovery.

denotes equal contribution.Conventional computing architectures have no known efficient algorithms for combinatorial optimization tasks, which are encountered in fundamental areas and real-world practical problems including logistics, social networks, and cryptography. Physical machines have recently been proposed and implemented as an alternative to conventional exact and heuristic solvers for the Ising problem, one such optimization task that requires finding the ground state spin configuration of an arbitrary Ising graph. However, these physical approaches usually suffer from decreased ground state convergence probability or universality for high edge-density graphs or arbitrary graph weights, respectively. We experimentally demonstrate a proof-of-principle integrated nanophotonic recurrent Ising sampler (INPRIS) capable of converging to the ground state of various 4-spin graphs with high probability. The INPRIS exploits experimental physical noise as a resource to speed up the ground state search. By injecting additional extrinsic noise during the algorithm iterations, the INPRIS explores larger regions of the phase space, thus allowing one to probe noise-dependent physical observables. Since the recurrent photonic transformation that our machine imparts is a fixed function of the graph problem, and could thus be implemented with optoelectronic architectures that enable GHz clock rates (such as passive or non-volatile photonic circuits that do not require reprogramming at each iteration), our work paves a way for orders-of-magnitude speedups in exploring the solution space of combinatorially hard problems.Combinatorial optimization is critical for a broad array of tasks, including artificial intelligence, bioinformatics, cryptography, scheduling, trajectory planning, and circuit design [1-3]. However, combinatorial problems typically fall into the nondeterministic-polynomial hard (NP-hard) problem class, becoming computationally intractable at large scale for traditional algorithms. This challenge motivates the search for alternatives to conventional (von Neumann) computing architectures that can efficiently solve such problems. The Ising problem, which consists of finding the ground state spin configuration of a quadratic Hamiltonian defined by a symmetric matrix K and spins of unit amplitude σ 1≤i≤N ∈ {−1, 1} N ,has garnered significant attention as many other combinatorial problems can be polynomially reduced to an Ising problem [4, 5]. Therefore, any technique for finding the ground state of arbitrary Ising problems, which is an NP-hard computational task, may extend to a wide range of other computationally intensive optimization problems. There is currently no known efficient classical algorithm to find the exact ground state of an arbitrary Ising graph, so heuristic and meta-heuristic algorithms are often implemented as a means of quickly obtaining approximate solutions [6]. Various physical systems have been proposed as Ising machines, as the evolution of many natural systems (ferromagnets [7], lipid m...

The prediction and design of photonic features have traditionally been guided by theory-driven computational methods, spanning a wide range of direct solvers and optimization techniques. Motivated by enormous advances in the field of machine learning, there has recently been a growing interest in developing complementary data-driven methods for photonics. Here, we demonstrate several predictive and generative data-driven approaches for the characterization and inverse design of photonic crystals. Concretely, we built a data set of 20,000 two-dimensional photonic crystal unit cells and their associated band structures, enabling the training of supervised learning models. Using these data set, we demonstrate a high-accuracy convolutional neural network for band structure prediction, with orders-of-magnitude speedup compared to conventional theory-driven solvers. Separately, we demonstrate an approach to high-throughput inverse design of photonic crystals via generative adversarial networks, with the design goal of substantial transverse-magnetic band gaps. Our work highlights photonic crystals as a natural application domain and test bed for the development of data-driven tools in photonics and the natural sciences.