Related papers: Numerical Integration as an Initial Value Problem
Efficient and robust optimization is essential for neural networks, enabling scientific machine learning models to converge rapidly to very high accuracy -- faithfully capturing complex physical behavior governed by differential equations.…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
The Integral Image algorithm is often applied in tasks that require efficient integration over images, such as object detection. In this paper we discuss theoretical aspects of the algorithm's continuous version. We suggest to define the…
This paper presents high-order numerical methods for solving boundary value problems associated with the Lane-Emden equation, which frequently arises in astrophysics and various nonlinear models. A major challenge in studying this equation…
Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations.…
We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to…
The systems of nonlinear Volterra integral equations of the first kind with jump discontinuous kernels are studied. The iterative numerical method for such nonlinear systems is proposed. Proposed method employs the modified…
A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a…
This chapter presents some numerical methods to solve problems in the fractional calculus of variations and fractional optimal control. Although there are plenty of methods available in the literature, we concentrate mainly on approximating…
This paper introduces an efficient tensor-vector product technique for the rapid and accurate approximation of integral operators within physics-informed deep learning frameworks. Our approach leverages neural network architectures to…
Deep learning often faces the challenge of efficiently processing dynamic inputs, such as sensor data or user inputs. For example, an AI writing assistant is required to update its suggestions in real time as a document is edited.…
We introduce a novel method for encoding integers using smooth real-valued functions whose integral properties implicitly reflect discrete quantities. In contrast to classical representations, where the integer appears as an explicit…
In the number partitioning problem (NPP) one aims to partition a given set of $N$ real numbers into two subsets with approximately equal sum. The NPP is a well-studied optimization problem and is famous for possessing a…
This paper proposes a novel inverse kinematics (IK) solver of articulated robotic systems for path planning. IK is a traditional but essential problem for robot manipulation. Recently, data-driven methods have been proposed to quickly solve…
Ordinary differential equations that model technical systems often contain states, that are considered dangerous for the system. A trajectory that reaches such a state usually indicates a flaw in the design. In this paper, we present and…
The main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The…
In this article we give an algorithm for computing the integral closure of a reduced Noetherian ring R, in case this integral closure is finitely generated over R.
We introduce basic aspects of new operator method, which is very suitable for practical solving differential equations of various types. The main advantage of the method is revealed in opportunity to find compact exact operator solutions of…
We derive several numerical methods for designing optimized first-order algorithms in unconstrained convex optimization settings. Our methods are based on the Performance Estimation Problem (PEP) framework, which casts the worst-case…
We present a new approach to termination analysis of numerical computations in logic programs. Traditional approaches fail to analyse them due to non well-foundedness of the integers. We present a technique that allows to overcome these…