Vol. 9, No. 1 (2025): Regular Issue

In the present work, we develop and assess a compact neural-network architecture called hereafter the Combinatorial Operation Neural Network (CombOpNet) which is designed to mitigate the curse of dimensionality of Sparse Identification of Nonlinear Dynamics (SINDy). Within CombOpNet, nonlinear terms in dynamical equations are represented by a chain of multiplication of univariate functions. These functions correspond to neurons of a multi-layer neural network which allows the construction of nonlinear terms in dynamical equations by employing forward propagation. This way, the CombOpNet can form and perform summations of all possible combinations of univariate functions now using far fewer weights, which themselves can reconstruct the elements in the coefficient matrix of SINDy. If $n$ and $p$ denote the number of dimensions and order of nonlinearity, respectively, the present method reduces the number of SINDy coefficients from $n\binom{n+p}{n}$ to $\mathcal{O}(n^3p)$. This reduction facilitates the scalability of SINDy, making SINDy applicable to multidimensional systems such as power grids and climate models. We discuss the implementation of CombOpNet in Tensorflow, and demonstrate its applicability to several complex nonlinear dynamical systems with an eye towards solving high-dimensional problems.

One of the major challenges in today's science and engineering fields is how to handle complex systems. Reductionist principles have helped scientists and engineers to develop uncanny an understanding of the universe ranging from mechanical to electro-magnetic properties and phenomena. Ever since that famous quote from Albert Einstein, ``God does not play dice," people often wonder what happened to obvious uncertainties in life and in nature. What would have happened if the Asteroid missed the Yucatan Peninsula in Mexico and the Earth 65 million years ago? The research in chaotic nonlinear systems seems to point us in a direction that even the deterministic nonlinear system can produce unpredictable results. Is it possible that when a large number of factors or entities interact nonlinearly with each other, they eventually yield a completely different system which is quantifiable with an entirely different set of phenomenological rules? Of course, the intermediate stage is characterized by a so-called positive Lyapunov exponent, fractals, and bifurcations. It is not difficult to imagine that a large or infinitely large positive Lyapunov exponent tends to produce a stochastic or random system. Overall, the accurate description of physical, chemical, and biological phenomena over a wide range of spatial and temporal scales is extremely difficult if not feasible. Nevertheless, although the intricate nature of turbulence poses seemingly insurmountable challenges to scientists and engineers with its spatial and temporal chaotic behaviors, many useful strategies have been discovered and implemented in various engineering applications. In the same spirit, in this study, various hierarchical modeling techniques have been explored in multi-scale and multi-physics modeling of proteins and cells. In addition, singular value decomposition, the key concept of importance to many reduced order modeling strategies, is reiterated with respect to four fundamental subspaces, namely, left null space, column space, null space, and row space for a rectangular matrix representative of any linear space tangent to a curved space or differentiable manifold. Using the singular value decomposition, it is possible to identify the hidden spatial and temporal correlations and patterns between variables and material properties. Hopefully, a computational protocol with phenomenological rules derived from experimental and computational approaches can be established for the modeling of complex systems.

The present study aims to investigate a new modeling method of single degree of freedom nonlinear vibration isolation system. Grey-box theory is used to model the system. Grey-box model is a combination of white-box model and black-box model. The white-box model is a simple component of vibration isolation system expressed by the differential models. In contrast to the white-box model, the black-box model is a NARX (Nonlinear AutoregRessive with eXogenous input) model, which is used to represent the complex structural components in the vibration isolation system that cannot be built with differential model. Based on the interface analysis, the grey-box model of vibration isolation system is obtained by integrating the white box model and the black box model. Compared with traditional physical modeling and finite element simulation, the method proposed in this paper is more rapid and cost saving. In addition, the grey-box modeling method can also be applied to the design of vibration isolators. Finally, a single degree of freedom nonlinear vibration isolation system is taken as an example to verify the correctness of the proposed method. The results of the study indicate that the proposed method is effective.

The $\psi$-Hilfer implicit fractional boundary value problem with impulses is investigated in this paper. The qualitative analysis is carried out by means of Krasnosel'skii's fixed point theorem and the Banach contraction principle. An example is provided to help interpret our results, which are also analysed graphically.

Recent progress in industries have led to nanofluids with superior thermal attributes to clear fluids. These improvements act as motivation for the present work, which emphasizes the heat and mass transfer properties of magnetohydrodynamic incompressible nanofluid flow towards a stretching sheet with the addition of thermal radiation. The mathematical model is framed in the knowledge of fundamental conversation laws and followed by transmuted into a dimensionless form employing similarity variables. The set of these converted equations are numerically treated through Wolfram language. An influence of germane parameters on boundary layer are scrutinized and graphically visualized to deliver the applicability of the present model. Furthermore, the machine learning technique for predicting the physical nature of flow is innovated as a novelty. In place of the classical simulation method, this work uncover the new technique to anticipate the physical quantities more accurately. The findings demonstrate that the thermal profile is an escalating term of the nonlinear radiation parameter. The outcomes are authenticated by comparing the current exploration with some earlier studies in certain instances and the reliability is highlighted with a aid of table format. The multiple linear regression accurately predicted the measurement of engineering concerns with the minimal error $10^{-3}$. The present optimization technique delivers a robust and intellectual perspective on industrial processes, such as solar technologies, polymer extrusion, metal processing, rubber sheet production, etc.

We present the wing flutter model as a dynamic structure with odd elasticity. The wing flutter model is an elastically supported airfoil in the flow. The lift force contributes to the non-Hermitian stiffness to the existing elastic support. The stability analysis of the flutter model demonstrates the onset of oscillations due to the odd elasticity. Surprisingly, we found linear coordinate transformations such that the equations of motion in the new coordinates are Hermitian: with symmetric mass matrix and symmetric stiffness matrix.

The main purpose of this study is to introduce a generalization of the shift two-sided map, that is, the generalized two-sided shift map. In this paper, we discuss periodic points that are dense in $\Sigma_{m}$ regarding the generalized shift map $\sigma_{n}$: ${\Sigma}_m\to{\Sigma}_m$. We consider a particular property of the dynamical system namely, the topologically transitivity. Here, we prove that the generalized shift map is topologically mixing and hence is topologically transitive on ${\Sigma}_m$. Few strong chaotic properties of the generalized shift map have been discussed. We know that the two-sided shift maps are automorphisms and the one-sided shift maps are endomorphisms. These maps can be conjugate or semi-conjugate to some automorphisms or endomorphisms which admit appropriate Markov partitions. We also discuss other maps such as $\alpha $-map and $\beta $-map are chaotic in bi-infinite symbol space.