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We propose a deep neural network architecture and a training algorithm for computing approximate Lyapunov functions of systems of nonlinear ordinary differential equations. Under the assumption that the system admits a compositional…

This article is concerned with the approximation and expressive powers of deep neural networks. This is an active research area currently producing many interesting papers. The results most commonly found in the literature prove that neural…

First of all we give some reasons that "natural proofs" built not a barrier to prove P $\not=$ NP using Boolean complexity. Then we investigate the approximation method for its extension to prove super-polynomial lower bounds for the…

We address composite optimization problems, which consist in minimizing the sum of a smooth and a merely lower semicontinuous function, without any convexity assumptions. Numerical solutions of these problems can be obtained by proximal…

$p$-adic linear regression is the problem of finding coefficients $\beta$ that minimise $\sum_i |y_i - x_i^\top\beta|_p$. We prove that computing an optimal solution is NP-hard via a polynomial-time reduction from Max Cut using a…

In this paper we study $p$-adic Diophantine approximation on manifolds, specifically multiplicative Diophantine approximation on affine subspaces and a Diophantine dichotomy for analytic $p$-adic manifolds.

Rational and neural network based approximations are efficient tools in modern approximation. These approaches are able to produce accurate approximations to nonsmooth and non-Lipschitz functions, including multivariate domain functions. In…

In this paper, we propose a new and simple approach to the approximation algorithms that are modified and improved from our published results. The computational and graphical examples are presented with the aid of Maple procedures.

We propose a new approach for large-scale high-dynamic range computational imaging. Deep Neural Networks (DNNs) trained end-to-end can solve linear inverse imaging problems almost instantaneously. While unfolded architectures provide…

Universal approximation theorem suggests that a shallow neural network can approximate any function. The input to neurons at each layer is a weighted sum of previous layer neurons and then an activation is applied. These activation…

Many real-world systems can be described by mathematical models that are human-comprehensible, easy to analyze and help explain the system's behavior. Symbolic regression is a method that can automatically generate such models from data.…

Multivariate function approximation is a fundamental problem in machine learning. Classic multivariate function approximations rely on hand-crafted basis functions (e.g., polynomial basis and Fourier basis), which limits their approximation…

We present a procedure to approximate a plane contour by piecewise polynomial functions, depending on various parameters, such as degree, number of local patches, selection of knots. This procedure aims to be adopted to study how…

Polynomial functions have plenty of useful analytical properties, but they are rarely used as learning models because their function class is considered to be restricted. This work shows that when trained properly polynomial functions can…

Polynomial regression is widely used and can help to express nonlinear patterns. However, considering very high polynomial orders may lead to overfitting and poor extrapolation ability for unseen data. The paper presents a method for…

In solving the variational problem, the key is to efficiently find the target function that minimizes or maximizes the specified functional. In this paper, by using the Pade approximant, we suggest a methods for the variational problem. By…

We give a new class of multidimensional $p$-adic continued fraction algorithms. We propose an algorithm in the class for which we can expect that multidimensional $p$-adic version of Lagrange's Theorem holds.

The goal of this note is to show that continuous functions may be approximated using scattered translates of the Poisson kernel.

Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and…

Artificial Neuronal Networks are models widely used for many scientific tasks. One of the well-known field of application is the approximation of high-dimensional problems via Deep Learning. In the present paper we investigate the Deep…