Related papers: Resurgent Extrapolation: Rebuilding a Function fro…
Almost every numerical task can be cast as extrapolation with respect to the fidelity or tolerance parameters of a consistent numerical method. This perspective enables probabilistic uncertainty quantification and optimal experimental…
When approximating a function that depends on a parameter, one encounters many practical examples where linear interpolation or linear approximation with respect to the parameters prove ineffective. This is particularly true for responses…
Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic…
A system of nonlinear differential equations $x^{1+\gamma}\frac{dY}{dx}= F_0(x)+A(x)Y+F(x,Y)$ is considered. We study more precisely the meaning of asymptotic expansion of transformations and solutions than preceding pioneering works, by…
The numerical evaluation of an individual Bessel or Hankel function of large order and large argument is a notoriously problematic issue in physics. Recurrence relations are inefficient when an individual function of high order and argument…
We show how to extrapolate an analytic function (or a smooth signal) by multiplying and dividing its values on geometric sequences that collapse to a point.
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…
The total measurable level of a pathogen is due to many sources, which produce a variety of pulses, overlapping in time, that rise suddenly and then decay. What is measured is the level of the total contribution of the sources at a given…
In this present paper, I propose a derivation of unified interpolation and extrapolation function that predicts new values inside and outside the given range by expanding direct Taylor series on the middle point of given data set.…
We present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations,…
In many applications of machine learning, certain categories of examples may be underrepresented in the training data, causing systems to underperform on such "few-shot" cases at test time. A common remedy is to perform data augmentation,…
The feasibility of extrapolation of completely monotone functions can be quantified by examining the worst case scenario, whereby a pair of completely monotone functions agree on a given interval to a given relative precision, but differ as…
We introduce a novel extrapolation algorithm inspired by quantum mechanics and evaluate its performance against linear prediction. Our method involves mapping function values onto a quantum state and estimating future function values by…
In this paper, we develop and numerically implement a novel approach for solving the inverse source problem of the acoustic wave equation in three dimensions. By injecting a small high-contrast droplet into the medium, we exploit the…
This paper addresses the problem of accurately estimating a function on one domain when only its discrete samples are available on another domain. To answer this challenge, we utilize a neural network, which we train to incorporate prior…
Extreme value theory provides rigorous theory and statistical tools for extrapolation in machine learning, particularly in settings where traditional methods struggle due to data scarcity in the tails. A broad range of tasks benefit from…
This manuscript goes through the fundamental connections between statistical mechanics and estimation theory by focusing on the particular problem of compressive sensing. We first show that the asymptotic analysis of a sparse recovery…
We study the Cowling approximation by analytical means as applied to a system of linear differential equations arising from models of non-radial stellar pulsation. We consider various asymptotic cases, including those of high harmonic…
The notion of interpolation and extrapolation is fundamental in various fields from deep learning to function approximation. Interpolation occurs for a sample $x$ whenever this sample falls inside or on the boundary of the given dataset's…
We discuss some physical consequences of the resurgent structure of Painleve equations and their related conformal block expansions. The resurgent structure of Painleve equations is particularly transparent when expressed in terms of…