Related papers: Numerical Integration as an Initial Value Problem
While quantum computing provides an exponential advantage in solving linear differential equations, there are relatively few quantum algorithms for solving nonlinear differential equations. In our work, based on the homotopy perturbation…
The finite element method is used to approximately solve boundary value problems for differential equations. The method discretises the parameter space and finds an approximate solution by solving a large system of linear equations. Here we…
In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product…
Solving constrained nonlinear optimization problems (CNLPs) is a longstanding problem that arises in various fields, e.g., economics, computer science, and engineering. We propose optimization-informed neural networks (OINN), a deep…
Most ordinary differential equation (ODE) models used to describe biological or physical systems must be solved approximately using numerical methods. Perniciously, even those solvers which seem sufficiently accurate for the forward…
This paper addresses the training of Neural Ordinary Differential Equations (neural ODEs), and in particular explores the interplay between numerical integration techniques, stability regions, step size, and initialization techniques. It is…
Partial differential equation (PDE)-constrained optimization, where an optimization problem is subject to PDE constraints, arises in various applications such as design, control, and inference. Solving such problems is computationally…
A general framework of Numerical Singular Integrals (NSI) method based on the Integration By Parts (IBP) has been developed for integrals involving singular and nearly singular integrands, or NSI-IBP. Through a general integration by parts…
We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretisation errors…
The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of…
In this paper, we present a new numerical method to solve fractional differential equations. Given a fractional derivative of arbitrary real order, we present an approximation formula for the fractional operator that involves integer-order…
Algorithms for the computation of the real zeros of hypergeometric functions which are solutions of second order ODEs are described. The algorithms are based on global fixed point iterations which apply to families of functions satisfying…
We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional…
We present an efficient and elementary algorithm for computing the number of primes up to $N$ in $\tilde{O}(\sqrt N)$ time, improving upon the existing combinatorial methods that require $\tilde{O}(N ^ {2/3})$ time. Our method has a similar…
While integer arithmetic has been widely adopted for improved performance in deep quantized neural network inference, training remains a task primarily executed using floating point arithmetic. This is because both high dynamic range and…
We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local…
Integer programming (IP), as the name suggests is an integer-variable-based approach commonly used to formulate real-world optimization problems with constraints. Currently, quantum algorithms reformulate the IP into an unconstrained form…
In this article we propose a new adaptive numerical quadrature procedure which includes both local subdivision of the integration domain, as well as local variation of the number of quadrature points employed on each subinterval. In this…
Many real-world dynamical systems are associated with first integrals (a.k.a. invariant quantities), which are quantities that remain unchanged over time. The discovery and understanding of first integrals are fundamental and important…
In this paper we consider a fully third order nonlinear boundary value problem which is of great interest of many researchers. First we establish the existence, uniqueness of solution. Next, we propose simple iterative methods on both…