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While neural networks can be approximated by linear models as their width increases, certain properties of wide neural networks cannot be captured by linear models. In this work we show that recently proposed Neural Quadratic Models can…
A polynomial of degree $\ge 2$ with coefficients in the ring of $p$-adic numbers $\mathbb{Z}_p$ is studied as a dynamical system on $\mathbb{Z}_p$. It is proved that the dynamical behavior of such a system is totally described by its…
When a dynamical system contains several different modes of oscillations it may behave in a variety of ways: If the modes oscillate at their own individual frequencies, it exhibits quasiperiodic behavior; when the modes lock to one another…
Nonlinear state-space modelling is a very powerful black-box modelling approach. However powerful, the resulting models tend to be complex, described by a large number of parameters. In many cases interpretability is preferred over…
Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real-world systems remains challenging due to issues in mathematical…
A nonlinear algebraic equation system of 5 variables is numerically solved, which is derived from the application of the Fourier transform to a differential equation system that allows modeling the behavior of the temperatures and the…
A quadratic dynamical system with practical applications is taken into considered. This system is transformed into a new bilinear system with Hadamard products by means of the implicit matrix structure. The corresponding quadratic bilinear…
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…
We consider linear dynamical systems with quadratic output. We first define the two transfer functions, a single-variable and a multivariate one, that fully describe the dynamics of these special nonlinear systems. Then, using the samples…
Many high-dimensional uncertainty quantification problems are solved by polynomial dimensional decomposition (PDD), which represents Fourier-like series expansion in terms of random orthonormal polynomials with increasing dimensions. This…
In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a…
Two effective methods for writing the dynamical equations for non-holonomic systems are illustrated. They are based on the two types of representation of the constraints: by parametric equations or by implicit equations. They can be applied…
Identification of nonlinear dynamical systems is crucial across various fields, facilitating tasks such as control, prediction, optimization, and fault detection. Many applications require methods capable of handling complex systems while…
Univariate polynomial root-finding is both classical and important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the polynomial…
In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the…
Recently, various systems of nonlinear difference equations, of different forms, were studied. In this existing work, two earlier published papers, due respectively to Bayram and Das. [Appl. Math. Sci. (Ruse), 4(7) (2010) pp. 817-821] and…
Dynamic probabilistic networks are a compact representation of complex stochastic processes. In this paper we examine how to learn the structure of a DPN from data. We extend structure scoring rules for standard probabilistic networks to…
Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this work, we combine deep learning techniques and classic numerical methods for differential equations to address two…
Recently, a system identification method based on center manifold is proposed to identify polynomial nonlinear systems with uncontrollable linearization. This note presents a numerical example to show the effectiveness of this method.
The authors proposed a general way to find particular solutions for overdetermined systems of PDEs previously, where the number of equations is greater than the number of unknown functions. In this paper, we propose an algorithm for finding…