Jun Ohkubo

Naoki Sugishita, Jun Ohkubo

We propose a new approach to constructing a neural network for predicting expectations of stochastic differential equations. The proposed method does not need data sets of inputs and outputs; instead, the information obtained from the time-evolution equations, i.e., the corresponding dual process, is directly compared with the weights in the neural network. As a demonstration, we construct neural networks for the Ornstein-Uhlenbeck process and the noisy van der Pol system. The remarkable feature of learned networks with the proposed method is the accuracy of inputs near the origin. Hence, it would be possible to avoid the overfitting problem because the learned network does not depend on training data sets.

Yuki Furue, Makiko Konoshima, Hirotaka Tamura et al.

Annealing machines specialized for combinatorial optimization problems have been developed, and some companies offer services to use those machines. Such specialized machines can only handle binary variables, and their input format is the quadratic unconstrained binary optimization (QUBO) formulation. Therefore, discretization is necessary to solve problems with continuous variables. However, there is a severe constraint on the number of binary variables with such machines. Although the simple binary expansion in the previous research requires many binary variables, we need to reduce the number of such variables in the QUBO formulation due to the constraint. We propose a discretization method that involves using correlations of continuous variables. We numerically show that the proposed method reduces the number of necessary binary variables in the QUBO formulation without a significant loss in prediction accuracy.

Fumito Kimura, Jun Ohkubo

Recent developments in hardware, such as photonic integrated circuits and optical devices, are driving demand for research on constructing machine learning architectures tailored for linear operations. Hence, it is valuable to explore methods for constructing learning machines with only linear operations after simple nonlinear preprocessing. In this study, we propose a framework to extract a linearized model from a pre-trained neural network for classification tasks by integrating Koopman operator theory with knowledge distillation. Numerical demonstrations on the MNIST and the Fashion-MNIST datasets reveal that the proposed model consistently outperforms the conventional least-squares-based Koopman approximation in both classification accuracy and numerical stability.

Tatsuya Naoi, Jun Ohkubo

Nonlinear coupled systems are ubiquitous in science and engineering. The analysis and modeling of such systems is challenging due to their high dimensionality and complex interactions among subsystems. In recent years, operator-theoretic methods based on the Koopman operator have attracted attention as a powerful tool for analyzing and modeling nonlinear dynamical systems. Extended dynamic mode decomposition (EDMD) is one of the most popular methods to approximate the Koopman operator. However, EDMD is a purely data-driven method, and it could be unstable and inaccurate for coupled systems under limited data availability. In this paper, we propose a method to learn the Koopman operator for coupled systems using the differential equations governing each subsystem. We also demonstrate its effectiveness through numerical experiments on coupled oscillator systems.

Tatsuya Kishimoto, Jun Ohkubo

Identifying governing equations of nonlinear dynamical systems from data is challenging. While sparse identification of nonlinear dynamics (SINDy) and its extensions are widely used for system identification, operator-logarithm approaches use the logarithm to avoid time differentiation, enabling larger sampling intervals. However, they still suffer from the curse of dimensionality. Then, we propose a data-driven method to compute the Koopman generator in a low-rank tensor train (TT) format by taking logarithms of Koopman eigenvalues while preserving the TT format. Experiments on 4-dimensional Lotka-Volterra and 10-dimensional Lorenz-96 systems show accurate recovery of vector field coefficients and scalability to higher-dimensional systems.

Hikaru Homma, Jun Ohkubo

The choice of parameters in neural networks is crucial in the performance, and an oracle distribution derived from the ridgelet transform enables us to obtain suitable initial parameters. In other words, the distribution of parameters is connected to the integral representation of target functions. The oracle distribution allows us to avoid the conventional backpropagation learning process; only a linear regression is enough to construct the neural network in simple cases. This study provides a new look at the oracle distributions and ridgelet transforms, i.e., an aspect of importance sampling. In addition, we propose extensions of the parameter sampling methods. We demonstrate the aspect of importance sampling and the proposed sampling algorithms via one-dimensional and high-dimensional examples; the results imply that the magnitude of weight parameters could be more crucial than the intercept parameters.

Felix Köster, Kazutaka Kanno, Jun Ohkubo et al.

Photonic reservoir computing has been successfully utilized in time-series prediction as the need for hardware implementations has increased. Prediction of chaotic time series remains a significant challenge, an area where the conventional reservoir computing framework encounters limitations of prediction accuracy. We introduce an attention mechanism to the reservoir computing model in the output stage. This attention layer is designed to prioritize distinct features and temporal sequences, thereby substantially enhancing the prediction accuracy. Our results show that a photonic reservoir computer enhanced with the attention mechanism exhibits improved prediction capabilities for smaller reservoirs. These advancements highlight the transformative possibilities of reservoir computing for practical applications where accurate prediction of chaotic time series is crucial.

Naoki Sugishita, Kayo Kinjo, Jun Ohkubo

Nonlinearity plays a crucial role in deep neural networks. In this paper, we investigate the degree to which the nonlinearity of the neural network is essential. For this purpose, we employ the Koopman operator, extended dynamic mode decomposition, and the tensor-train format. The Koopman operator approach has been recently developed in physics and nonlinear sciences; the Koopman operator deals with the time evolution in the observable space instead of the state space. Since we can replace the nonlinearity in the state space with the linearity in the observable space, it is a hopeful candidate for understanding complex behavior in nonlinear systems. Here, we analyze learned neural networks for the classification problems. As a result, the replacement of the nonlinear middle layers with the Koopman matrix yields enough accuracy in numerical experiments. In addition, we confirm that the pruning of the Koopman matrix gives sufficient accuracy even at high compression ratios. These results indicate the possibility of extracting some features in the neural networks with the Koopman operator approach.

Hikaru Homma, Jun Ohkubo

Initialization of neural network parameters, such as weights and biases, has a crucial impact on learning performance; if chosen well, we can even avoid the need for additional training with backpropagation. For example, algorithms based on the ridgelet transform or the SWIM (sampling where it matters) concept have been proposed for initialization. On the other hand, it is well-known that neural networks tend to learn coarse information in the earlier layers. The feature is called spectral bias. In this work, we investigate the effects of utilizing information on the spectral bias in the initialization of neural networks. Hence, we propose a framework that adjusts the scale factors in the SWIM algorithm to capture low-frequency components in the early-stage hidden layers and to represent high-frequency components in the late-stage hidden layers. Numerical experiments on a one-dimensional regression task and the MNIST classification task demonstrate that the proposed method outperforms the conventional initialization algorithms. This work clarifies the importance of intrinsic spectral properties in learning neural networks, and the finding yields an effective parameter initialization strategy that enhances their training performance.

Jun Ohkubo

It is sometimes difficult to achieve a complete observation for a full set of observables, and partial observations are necessary. For deterministic systems, the Mori-Zwanzig formalism provides a theoretical framework for handling partial observations. Recently, data-driven algorithms based on the Koopman operator theory have made significant progress, and there is a discussion to connect the Mori-Zwanzig formalism with the Koopman operator theory. In this work, we discuss the effects of partial observation in stochastic systems using the Koopman operator theory. The discussion clarifies the importance of distinguishing the state space and the function space in stochastic systems. Even in stochastic systems, the delay embedding technique is beneficial for partial observation, and several numerical experiments showed a power-law behavior of the accuracy for the amplitude of the additive noise. We also discuss the relation between the exponent of the power-law behavior and the effects of partial observation.

Ichiro Ohta, Shota Koyanagi, Kayo Kinjo et al.

In recent years, there has been a growing interest in data-driven approaches in physics, such as extended dynamic mode decomposition (EDMD). The EDMD algorithm focuses on nonlinear time-evolution systems, and the constructed Koopman matrix yields the next-time prediction with only linear matrix-product operations. Note that data-driven approaches generally require a large dataset. However, assume that one has some prior knowledge, even if it may be ambiguous. Then, one could achieve sufficient learning from only a small dataset by taking advantage of the prior knowledge. This paper yields methods for incorporating ambiguous prior knowledge into the EDMD algorithm. The ambiguous prior knowledge in this paper corresponds to the underlying time-evolution equations with unknown parameters. First, we apply the proposed method to forward problems, i.e., prediction tasks. Second, we propose a scheme to apply the proposed method to inverse problems, i.e., parameter estimation tasks. We demonstrate the learning with only a small dataset using guiding examples, i.e., the Duffing and the van der Pol systems.

Tomoya Nishikata, Jun Ohkubo

Machine learning methods allow the prediction of nonlinear dynamical systems from data alone. The Koopman operator is one of them, which enables us to employ linear analysis for nonlinear dynamical systems. The linear characteristics of the Koopman operator are hopeful to understand the nonlinear dynamics and perform rapid predictions. The extended dynamic mode decomposition (EDMD) is one of the methods to approximate the Koopman operator as a finite-dimensional matrix. In this work, we propose a method to compress the Koopman matrix using hierarchical clustering. Numerical demonstrations for the cart-pole model and comparisons with the conventional singular value decomposition (SVD) are shown; the results indicate that the hierarchical clustering performs better than the naive SVD compressions.

Yoshiki Sato, Makiko Konoshima, Hirotaka Tamura et al.

Ising formulations are widely utilized to solve combinatorial optimization problems, and a variety of quantum or semiconductor-based hardware has recently been made available. In combinatorial optimization problems, the existence of local minima in energy landscapes is problematic to use to seek the global minimum. We note that the aim of the optimization is not to obtain exact samplings from the Boltzmann distribution, and there is thus no need to satisfy detailed balance conditions. In light of this fact, we develop an algorithm to get out of the local minima efficiently while it does not yield the exact samplings. For this purpose, we utilize a feature that characterizes locality in the current state, which is easy to obtain with a type of specialized hardware. Furthermore, as the proposed algorithm is based on a rejection-free algorithm, the computational cost is low. In this work, after presenting the details of the proposed algorithm, we report the results of numerical experiments that demonstrate the effectiveness of the proposed feature and algorithm.

Kotaro Furuya, Jun Ohkubo

A semi-supervised learning method for spiking neural networks is proposed. The proposed method consists of supervised learning by backpropagation and subsequent unsupervised learning by spike-timing-dependent plasticity (STDP), which is a biologically plausible learning rule. Numerical experiments show that the proposed method improves the accuracy without additional labeling when a small amount of labeled data is used. This feature has not been achieved by existing semi-supervised learning methods of discriminative models. It is possible to implement the proposed learning method for event-driven systems. Hence, it would be highly efficient in real-time problems if it were implemented on neuromorphic hardware. The results suggest that STDP plays an important role other than self-organization when applied after supervised learning, which differs from the previous method of using STDP as pre-training interpreted as self-organization.

Tomohiro Yokota, Makiko Konoshima, Hirotaka Tamura et al.

We propose a quadratic unconstrained binary optimization (QUBO) formulation of the l1-norm, which enables us to perform sparse estimation of Ising-type annealing methods such as quantum annealing. The QUBO formulation is derived using the Legendre transformation and the Wolfe theorem, which have recently been employed to derive the QUBO formulations of ReLU-type functions. It is shown that a simple application of the derivation method to the l1-norm case results in a redundant variable. Finally a simplified QUBO formulation is obtained by removing the redundant variable.