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This Editorial paper provides an overview of a selection of contributions from researchers working in the fields of Subdivision, Applied Geometric and Algebraic Methods, Isogeometric Analysis, Refinability and Geometric Deep Learning, with special emphasis on numerical computation, advanced algorithms, their analysis, and their applications. [ABSTRACT FROM AUTHOR]

• Two new computational algorithms are proposed for estimating the Lyapunov exponents from time series data in order to test the null hypothesis of a chaotic behavior. • The results obtained provide robust evidence that the Jacobian indirect methods provide better estimates than traditional direct methods in all the experiments we have conducted. • We have shown empirically that the algorithms proposed are robust to the presence of (small) measurement errors because the results obtained are comparable to those which are noise free. • The empirical size of the algorithms proposed decreased and the empirical power increased as the sample size increased which means that our hypothesis tests are consistent and reliable. • This paper opens a new research line where new contributions may appear considering new machine learning methods and deep learning algorithms for estimating the Lyapunov exponents from time series data. Most of the existing methods and techniques for the detection of chaotic behaviour from empirical time series try to quantify the well-known sensitivity to initial conditions through the estimation of the so-called Lyapunov exponents corresponding to the data generating system, even if this system is unknown. Some of these methods are designed to operate in noise-free environments, such as those methods that directly quantify the separation rate of two initially close trajectories. As an alternative, this paper provides two nonlinear indirect regression methods for estimating the Lyapunov exponents on a noisy environment. We extend the global Jacobian method, by using local polynomial kernel regressions and local neural net kernel models. We apply such methods to several noise-contaminated time series coming from different data generating processes. The results show that in general, the Jacobian indirect methods provide better results than the traditional direct methods for both clean and noisy time series. Moreover, the local Jacobian indirect methods provide more robust and accurate fit than the global ones, with the methods using local networks obtaining more accurate results than those using local polynomials. [ABSTRACT FROM AUTHOR]

• We propose a community detection algorithm based on a deep sparse autoencoder. • We combine the path weight matrix with the weighted adjacent paths of the node to obtain the similarity matrix. • The feature matrix has stronger ability to express the features of the network. • The proposed algorithm can more accurately identify community structures. At present, community detection methods are mostly focused on the investigation at unweighted networks. However, real-world networks are always complex, and unweighted networks are not sufficient to reflect the connections among real-world objects. Hence, this paper proposes a community detection algorithm based on a deep sparse autoencoder. First, the second-order neighbors of the nodes are identified, and we can obtain the path weight matrix for the second-order neighbors of the node. We combine the path weight matrix with the weighted adjacent paths of the node to obtain the similarity matrix, which can not only reflect the similarity relationships among connected nodes in the network topology but also the similarity relationships among nodes and second-order neighbors. Then, based on the unsupervised deep learning method, the feature matrix which has a stronger ability to express the features of the network can be obtained by constructing a deep sparse autoencoder. Finally, the K-means algorithm is adopted to cluster the low-dimensional feature matrix and obtain the community structure. The experimental results indicate that compared with 4 typical community detection algorithms, the algorithm proposed here can more accurately identify community structures. Additionally, the results of parameter experiments show that compared with the community structure found by the K-means algorithm, which directly uses the high-dimensional adjacency matrix, the community structure detected by the WCD algorithm in this paper is more accurate. [ABSTRACT FROM AUTHOR]

In this paper, the optimal consensus control problem with second-order dynamics consisting of leader and follower agents is discussed. For optimal consensus problem, the optimal control policies rely on algebraic Riccati equations (AREs) equation, which are difficult to solve. Furthermore, both the follower agents' and the leader agent's dynamics are assumed to be completely unknown. As the consensus problem based on feedback control, the second-order discrete-time multi-agent systems (DT-MASs) model with directed topology is formulated to the optimal tracking control problem via online deep reinforcement learning method. Based on graph theory, matrix analysis, Lyapunov stability, deep learning and optimal control, the optimality of value function and the stability of the consensus error systems for the unknown second-order systems are guaranteed for each agent. The results show that the designed policy iteration algorithm not only stabilizes the distributed dynamic systems, but also makes all agents' position and velocity states reach consensus, respectively. Finally, the correctness of our theoretical results is illustrated under two numerical simulations based on the designing model-free actor-critic networks. [ABSTRACT FROM AUTHOR]

In this paper, we propose a neural network-based method for approximating expected exposures and potential future exposures of Bermudan options. In a first phase, the method relies on the Deep Optimal Stopping algorithm (DOS) proposed by Becker, Cheridito, and Jentzen (2019), which learns the optimal stopping rule from Monte-Carlo samples of the underlying risk factors. Cashflow paths are then created by applying the learned stopping strategy on a new set of realizations of the risk factors. Furthermore, in a second phase the cashflow paths are projected onto the risk factors to obtain approximations of pathwise option values. The regression step is carried out by ordinary least squares as well as neural networks, and it is shown that the latter results in more accurate approximations. The expected exposure is formulated, both in terms of the cashflow paths and in terms of the pathwise option values and it is shown that a simple Monte-Carlo average yields accurate approximations in both cases. The potential future exposure is estimated by the empirical α -percentile. Finally, it is shown that the expected exposures, as well as the potential future exposures can be computed under either, the risk neutral measure, or the real world measure, without having to re-train the neural networks. [ABSTRACT FROM AUTHOR]