Related papers: Constructing general rough differential equations …
Projection-based model reduction has become a popular approach to reduce the cost associated with integrating large-scale dynamical systems so they can be used in many-query settings such as optimization and uncertainty quantification. For…
Systems biology relies on mathematical models that often involve complex and intractable likelihood functions, posing challenges for efficient inference and model selection. Generative models, such as normalizing flows, have shown…
We reconstruct the velocity field of incompressible flows given a finite set of measurements. For the spatial approximation, we introduce the Sparse Fourier divergence-free (SFdf) approximation based on a discrete $L^2$ projection. Within…
We develop the theory of linear evolution equations associated with the adjacency matrix of a graph, focusing in particular on infinite graphs of two kinds: uniformly locally finite graphs as well as locally finite line graphs. We discuss…
The theta process is a stochastic process of number theoretical origin arising as a scaling limit of quadratic Weyl sums. It can be described in terms of the geodesic flow and an automorphic function on a homogeneous space. This process has…
Flow-based generative modeling in continuous spaces exploit Tweedie's formula to express the denoiser (learned in training) as a score function (used in sampling). In contrast, this relation has been largely missing in the discrete setting…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
Standard diffuse approximations of the Willmore flow often lead to intersecting phase boundaries that in many cases do not correspond to the intended sharp interface evolution. Here we introduce a new two-variable diffuse approximation that…
In this paper, approximate solutions for a class of fractional Lane - Emden type equations based on the series expansion method are presented. Various examples are introduced and discussed. The recurrence relation for the components of the…
Learning causal relationships between variables is a well-studied problem in statistics, with many important applications in science. However, modeling real-world systems remain challenging, as most existing algorithms assume that the…
We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial…
We give a brief account on a basic result (Lemma \ref{lem2}) which is a very useful tool in proving various convergence theorems in the framework of the iterative approximation of fixed points of demicontractive mappings in Hilbert spaces.…
High dimensional data reduction techniques are provided by using partial least squares within deep learning. Our framework provides a nonlinear extension of PLS together with a disciplined approach to feature selection and architecture…
In this paper, we consider the problem of piecewise affine abstraction of nonlinear systems, i.e., the overapproximation of its nonlinear dynamics by a pair of piecewise affine functions that "includes" the dynamical characteristics of the…
The derivative expansion approach to the calculation of the interaction between two surfaces, is a generalization of the proximity force approximation, a technique of widespread use in different areas of physics. The derivative expansion…
This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the…
We prove existence and uniqueness results for (mild) solutions to some non-linear parabolic evolution equations with a rough forcing term. Our method of proof relies on a careful exploitation of the interplay between the spatial and time…
We will develop a nonlinear upscaling method for nonlinear transport equation. The proposed scheme gives a coarse scale equation for the cell average of the solution. In order to compute the parameters in the coarse scale equation, a local…
In this paper, approximate analytical solutions of nonlinear Emden-Fowler type equations are obtained by the differential transform method (DTM). The DTM is a numerical as well as analytical method for solving integral equations, ordinary…
In two recent papers we introduced some new techniques for constructing an extension of a probability-preserving system $T:\mathbb{Z}^d\curvearrowright (X,\mu)$ that enjoys certain desirable properties in connexion with the asymptotic…