Neural Schrödinger Equation: Physical Law as Deep Neural Network

Large-scale photonic computing with nonlinear disordered media

Neural networks find widespread use in scientific and technological applications, yet their implementations in conventional computers have encountered bottlenecks due to ever-expanding computational needs. Photonic computing is a promising neuromorphic platform with potential advantages of massive parallelism, ultralow latency and reduced energy consumption but mostly for computing linear operations. Here we demonstrate a large-scale, high-performance nonlinear photonic neural system based on a disordered polycrystalline slab composed of lithium niobate nanocrystals. Mediated by random quasi-phase-matching and multiple scattering, linear and nonlinear optical speckle features are generated as the interplay between the simultaneous linear random scattering and the second-harmonic generation, defining a complex neural network in which the second-order nonlinearity acts as internal nonlinear activation functions. Benchmarked against linear random projection, such nonlinear mapping embedded with rich physical computational operations shows improved performance across a large collection of machine learning tasks in image classification, regression and graph classification. Demonstrating up to 27,648 input and 3,500 nonlinear output nodes, the combination of optical nonlinearity and random scattering serves as a scalable computing engine for diverse applications.

Higher-order neurodynamical equation for simplex prediction

It is demonstrated that higher-order patterns beyond pairwise relations can significantly enhance the learning capability of existing graph-based models, and simplex is one of the primary form for graphically representing higher-order patterns. Predicting unknown (disappeared) simplices in real-world complex networks can provide us with deeper insights, thereby assisting us in making better decisions. Nevertheless, previous efforts to predict simplices suffer from two issues: (i) they mainly focus on 2- or 3-simplices, and there are few models available for predicting simplices of arbitrary orders, and (ii) they lack the ability to analyze and learn the features of simplices from the perspective of dynamics. In this paper, we present a Higher-order Neurodynamical Equation for Simplex Prediction of arbitrary order (HNESP), which is a framework that combines neural networks and neurodynamics. Specifically, HNESP simulates the dynamical coupling process of nodes in simplicial complexes through different relations (i.e., strong pairwise relation, weak pairwise relation, and simplex) to learn node-level representations, while explaining the learning mechanism of neural networks from neurodynamics. To enrich the higher-order information contained in simplices, we exploit the entropy and normalized multivariate mutual information of different sub-structures of simplices to acquire simplex-level representations. Furthermore, simplex-level representations and multi-layer perceptron are used to quantify the existence probability of simplices. The effectiveness of HNESP is demonstrated by extensive simulations on seven higher-order benchmarks. Experimental results show that HNESP improves the AUC values of the state-of-the-art baselines by an average of 8.32%. Our implementations will be publicly available at: https://github.com/jianruichen/HNESP.

Physical deep learning with biologically inspired training method: gradient-free approach for physical hardware

Ever-growing demand for artificial intelligence has motivated research on unconventional computation based on physical devices. While such computation devices mimic brain-inspired analog information processing, the learning procedures still rely on methods optimized for digital processing such as backpropagation, which is not suitable for physical implementation. Here, we present physical deep learning by extending a biologically inspired training algorithm called direct feedback alignment. Unlike the original algorithm, the proposed method is based on random projection with alternative nonlinear activation. Thus, we can train a physical neural network without knowledge about the physical system and its gradient. In addition, we can emulate the computation for this training on scalable physical hardware. We demonstrate the proof-of-concept using an optoelectronic recurrent neural network called deep reservoir computer. We confirmed the potential for accelerated computation with competitive performance on benchmarks. Our results provide practical solutions for the training and acceleration of neuromorphic computation.