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Many scientific phenomena are modeled by Partial Differential Equations (PDEs). The development of data gathering tools along with the advances in machine learning (ML) techniques have raised opportunities for data-driven identification of…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of…
Many conservative partial differential equations correspond to geodesic equations on groups of diffeomorphisms. Stability of their solutions can be studied by examining sectional curvature of these groups: negative curvature in all sections…
The present article considers stability of the solutions to nonlinear and nonautonomous compartmental systems governed by ordinary differential equations (ODEs). In particular, compartmental systems with a right-hand side that can be…
We extend the definition of $n$-dimensional difference equations to complex order $\alpha\in \mathbb{C} $. We investigate the stability of linear systems defined by an $n$-dimensional matrix $A$ and derive conditions for the stability of…
This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result…
In this work, we study continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study existence…
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as…
In this paper, we study a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises. Our method consists in studying first the nonlocal SPDEs and showing then the convergence of the family of these…
We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…
The existence, uniqueness, and asymptotic stability of modulo periodic Poisson stable solutions of dynamic equations on a periodic time scale are investigated. The model under investigation involves a term which is constructed via a Poisson…
We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity…
Nonlinear partial differential equations are central to physics, engineering, and finance. Except in a limited number of integrable cases, their solution generally requires numerical methods whose cost becomes prohibitive in…
Rapidly developing machine learning methods has stimulated research interest in computationally reconstructing differential equations (DEs) from observational data which may provide additional insight into underlying causative mechanisms.…
A central theme in Iachello's quest for understanding simple ordered patterns in complex quantum systems, is the concept of dynamical symmetry. Relying on his seminal contributions, we present further generalization of this notion to that…
Every linear system of partial differential equations (PDEs) admits a scaling symmetry in its dependent variables. In conjunction with other admitted symmetries of linear type, the associated invariant solution condition poses a linear…
Irregular sampling intervals and missing values in real-world time series data present challenges for conventional methods that assume consistent intervals and complete data. Neural Ordinary Differential Equations (Neural ODEs) offer an…
Systems of non-autonomous parabolic partial differential equations over a bounded domain with nonlinear term of Carath\'eodory type are considered. Appropriate topologies on sets of Lipschitz Carath\'eodory maps are defined in order to have…
These notes present an alternative approach to the asymptotic stability of stochastic partial differential equations driven by multiplicative noise, applicable to a wide range of dissipative systems. The method builds on general criteria…