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Despite their great success, neural networks still remain as black-boxes due to the lack of interpretability. Here we propose a new analyzing method, namely the weight pathway analysis (WPA), to make them transparent. We consider weights in pathways that link neurons longitudinally from input neurons to output neurons, or simply weight pathways, as the basic units for understanding a neural network, and decompose a neural network into a series of subnetworks of such weight pathways. A visualization scheme of the subnetworks is presented that gives longitudinal perspectives of the network like radiographs, making the internal structures of the network visible. Impacts of parameter adjustments or structural changes to the network can be visualized via such radiographs. Characteristic maps are established for subnetworks to characterize the enhancement or suppression of the influence of input samples on each output neuron. Using WPA, we discover that neural network store and utilize information in a holographic way, that is, subnetworks encode all training samples in a coherent structure and thus only by investigating the weight pathways can one explore samples stored in the network. Furthermore, with WPA, we reveal fundamental learning modes of a neural network: the linear learning mode and the nonlinear learning mode. The former extracts linearly separable features while the latter extracts linearly inseparable features. The hidden-layer neurons self-organize into different classes for establishing learning modes and for reaching the training goal. The finding of learning modes provides us the theoretical ground for understanding some of the fundamental problems of machine learning, such as the dynamics of learning process, the role of linear and nonlinear neurons, as well as the role of network width and depth., Comment: 16 pages, 10 figures

We study the thermalization dynamics of one-dimensional diatomic lattices (which represents the simplest system possessing multi-branch phonons), exemplified by the famous Fermi-Pasta-Ulam-Tsingou (FPUT)-$\beta$ and the Toda models. Here we focus on how the system relaxes to the equilibrium state when part of highest-frequency optical modes are initially excited, which is called the anti-FPUT problem comparing with the original FPUT problem (low frequency excitations of the monatomic lattice). It is shown numerically that the final thermalization time $T_{\rm eq}$ of the diatomic FPUT-$\beta$ chain depends on whether its acoustic modes are thermalized, whereas the $T_{\rm eq}$ of the diatomic Toda chain depends on the optical ones; in addition, the metastable state of both models have different energy distributions and lifetimes. Despite these differences, in the near-integrable region, the $T_{\rm eq}$ of both models still follows the same scaling law, i.e., $T_{\rm eq}$ is inversely proportional to the square of the perturbation strength. Finally, comparisons of the thermalization behavior between different models under various initial conditions are briefly summarized., Comment: 13 pages, 6 figures

The symmetry of interparticle interaction potential (IIP) has a crucial influence on the thermodynamic and transport properties of solids. Here we focus on the effect of the asymmetry of IIP on thermalization properties. In general, asymmetry and nonlinearity interweave with each other. To explore the effects of asymmetry and nonlinearity on thermalization separately, we here introduce an asymmetric harmonic (AH) model, whose IIP has the only asymmetry but no nonlinearity in the sense that the frequency of a system is independent of input energy. Through extensive numerical simulations, a power-law relationship between the thermalization time $T_{\rm eq}$ and the perturbation strength (here the degree of asymmetry) is still confirmed, yet a larger exponent is observed instead of the previously found inverse square law in the thermodynamic limit. Then the quartic (symmetric) nonlinearity is added into the AH model, and the thermalization behavior under the coexistence of asymmetry and nonlinearity is systematically studied. It is found that Matthiessen's rule gives a better estimation of the $T_{\rm eq}$ of the system. This work further suggests that the asymmetry plays a distinctive role in the properties of relaxation of a system., Comment: 7 pages, 5 figs