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We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter epsilon tends to zero, their solutions converge…
The cutoff method, which cuts off the values of a function less than a given number, is studied for the numerical computation of nonnegative solutions of parabolic partial differential equations. A convergence analysis is given for a broad…
The theory of plasma physics offers a number of nontrivial examples of partial differential equations, which can be successfully treated with symmetry methods. We propose three different examples which may illustrate the reciprocal…
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…
A symmetry group method is used to obtain exact solutions for a semilinear radial heat equation in $n>1$ dimensions with a general power nonlinearity. The method involves an ansatz technique to solve an equivalent first-order PDE system of…
Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries,…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
We study a class of evolutionary partial differential systems with two components related to second order (in time) non-evolutionary equations of odd order in spatial variable. We develop the formal diagonalisation method in symbolic…
There was proposed the method of a factorization of PDE. The method is based on reduction of complicated systems to more easy ones (for example, due to dimension decrease). This concept is proposed in general case for the arbitrary PDE…
In this paper, we introduce a novel concept for learning of the parameters in a neural network. Our idea is grounded on modeling a learning problem that addresses a trade-off between (i) satisfying local objectives at each node and (ii)…
We propose the symmetry reduction method of partial differential equations to the system of differential equations with fewer number of independent variables. We also obtain generalized sufficient conditions for the solution found by…
Employing a limiting case of a conjecture for constructing piecewise separable-variables functions, the elements of the Pseudoanalytic Function Theory are used for numerically approaching solutions of the forward Dirichlet boundary value…
In the present paper we continue our reconsideration about the foundations for a thermostatistical description of the called Hamiltonian nonextensive systems (see in cond-mat/0604290). After reviewing the selfsimilarity concept and the…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
In this paper we address the problem of establishing correspondences between different instances of the same object. The problem is posed as finding the geometric transformation that aligns a given image pair. We use a convolutional neural…
This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
In this thesis, we study the one parameter point transformations which leave invariant the differential equations. In particular we study the Lie and the Noether point symmetries of second order differential equations. We establish a new…
We advance a variational method to prove qualitative properties such as symmetries, monotonicity, upper and lower bounds, sign properties, and comparison principles for a large class of doubly-nonlinear evolutionary problems including…
Machine learning (ML) and deep learning models are extensively used for parameter optimization and regression problems. However, not all inverse problems in ML are ``identifiable,'' indicating that model parameters may not be uniquely…