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Various traditional numerical methods for solving initial value problems of differential equations often produce local solutions near the initial value point, despite the problems having larger interval solutions. Even current popular…
Ensembles of neural networks typically outperform individual networks but incur large computational costs, whereas weight aggregation produces less costly, yet also less accurate, aggregate models. We introduce partial fusion of networks,…
In this paper, we deal with a Cauchy problem for a nonlinear fractional differential equation with the Caputo derivative of order $\alpha \in (0, 1)$. As initial data, we consider a pair consisting of an initial point, which does not…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
Deep learning has excelled in image recognition tasks through neural networks inspired by the human brain. However, the necessity for large models to improve prediction accuracy introduces significant computational demands and extended…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
Fractional differential equations (FDEs) describe subdiffusion behavior of dynamical systems. Its non-local structure requires taking into account the whole evolution history during the time integration, which then possibly causes…
In this paper we extend the notion of an $\alpha$-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling…
We propose a simple recursive algorithm that allows the computation of the first- and second-order derivatives with respect to the inputs of an arbitrary deep feed forward neural network (DFNN). The algorithm naturally incorporates the…
We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose derivatives are…
In this paper we study linear and nonlinear fractional differential equations involving the Caputo fractional derivative with Mittag-Leffler non-singular kernel of order $0<\alpha<1.$ We first obtain a new estimate of the fractional…
In this paper, neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. Neural network work approximation methods have advantages over classical approaches in that they…
In the present paper, we address a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from…
The problem of modeling forest tree growth curves with an artificial neural network (NN) is examined. The NN parametric form is shown to be a suitable model if each forest tree plot is assumed to consist of several differently growing…
In this paper the Green formula for the operator of fractional differentiation in Caputo sense is proved. By using this formula the integral representation of all regular in a rectangular domains solutions is obtained in the form of the…
A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent…
In this paper, the fractional differential matrices based on the Jacobi-Gauss points are derived with respect to the Caputo and Riemann-Liouville fractional derivative operators. The spectral radii of the fractional differential matrices…
In this paper invariant subspace method has been employed for solving linear and non-linear fractional partial differential equations involving Caputo derivative. A variety of illustrative examples are solved to demonstrate the…
This paper presents a comparative study three numerical schemes such as Linear, Quadratic and Quadratic-Linear scheme for the fractional integro-differential equations defined in terms of the Caputo fractional derivatives. The error…
We propose a novel Caputo fractional derivative-based optimization algorithm. Upon defining the Caputo fractional gradient with respect to the Cartesian coordinate, we present a generic Caputo fractional gradient descent (CFGD) method. We…