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The Fractional Diffusion Equation (FDE) is a mathematical model that describes anomalous transport phenomena characterized by non-local and long-range dependencies which deviate from the traditional behavior of diffusion. Solving this…
Probabilistic solvers for ordinary differential equations (ODEs) have emerged as an efficient framework for uncertainty quantification and inference on dynamical systems. In this work, we explain the mathematical assumptions and detailed…
Moment estimation for stochastic differential equations (SDEs) is fundamental to the formal reasoning and verification of stochastic dynamical systems, yet remains challenging and is rarely available in closed form. In this paper, we study…
The development of data-driven approaches for solving differential equations has been followed by a plethora of applications in science and engineering across a multitude of disciplines and remains a central focus of active scientific…
Uncertain fractional differential equation (UFDE) is a kind of differential equation about uncertain process. As an significant mathematical tool to describe the evolution process of dynamic system, UFDE is better than the ordinary…
Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology,…
The conjoining of dynamical systems and deep learning has become a topic of great interest. In particular, neural differential equations (NDEs) demonstrate that neural networks and differential equation are two sides of the same coin.…
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…
Implicit computational complexity is a lively area of theoretical computer science, which aims to provide machine-independent characterizations of relevant complexity classes. % for uniformity with subsequent uses >> 1960s (but feel free to…
A novel approach to approximate solutions of Stochastic Differential Equations (SDEs) by Deep Neural Networks is derived and analysed. The architecture is inspired by the notion of Deep Operator Networks (DeepONets), which is based on…
This paper proposes a novel non-iterative method to solve power system differential algebraic equations (DAEs) using the differential transformation, a mathematical tool that can obtain power series coefficients by transformation rules…
Boolean Delay Equations (BDEs) are semi-discrete dynamical models with Boolean-valued variables that evolve in continuous time. Systems of BDEs can be classified into conservative or dissipative, in a manner that parallels the…
Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with some representative datasets. Recently, an augmented framework has been…
Within recent years, considerable progress has been made regarding high-performance solvers for Partial Differential Equations (PDEs), yielding potential gains in efficiency compared to industry standard tools. However, the latter largely…
Conventional finite-difference schemes for solving partial differential equations are based on approximating derivatives by finite-differences. In this work, an alternative theory is proposed which view finite-difference schemes as…
Delay Differential Equations (DDEs) are a class of differential equations that can model diverse scientific phenomena. However, identifying the parameters, especially the time delay, that make a DDE's predictions match experimental results…
A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs),…
Differential equations where the graph of some derivative of a function is composed of a finite number of similarity transformations of the graph of the function itself are defined. We call these self-similar differential equations (SSDEs)…
This article introduces a framework for measuring the uncertain behaviour of a changing system in terms of the solution of a class of fractional stochastic differential equations (fsDEs). This is accomplished via operational matrices based…