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Fast and accurate simulation of dynamical systems is a fundamental challenge across scientific and engineering domains. Traditional numerical integrators often face a trade-off between accuracy and computational efficiency, while existing…
Simple Quantum Integro-Differential Solver (SQuIDS) is a C++ code designed to solve semi-analytically the evolution of a set of density matrices and scalar functions. This is done efficiently by expressing all operators in an SU(N) basis.…
There has been a long history of using ordinary differential equations (ODEs) to understand the dynamics of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to…
The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of \'Elie Cartan. This solver returns a target equation equivalent to the equation to be…
We propose a physical analogy between finding the solution of an ordinary differential equation (ODE) and a $N$ particle problem in statistical mechanics. It uses the fact that the solution of an ODE is equivalent to obtain the minimum of a…
There exists a huge number of numerical methods that iteratively construct approximations to the solution $y(x)$ of an ordinary differential equation (ODE) $y'(x)=f(x,y)$ starting from an initial value $y_0=y(x_0)$ and using a finite…
The research in Artificial Intelligence methods with potential applications in science has become an essential task in the scientific community last years. Physics Informed Neural Networks (PINNs) is one of this methods and represent a…
Differential equations arise in mathematics, physics,medicine, pharmacology, communications, image processing and animation, etc. An Ordinary Differential Equation (ODE) is a differential equation if it involves derivatives with respect to…
The C++ library deal.II provides classes and functions to solve stationary problems with finite elements on one- to threedimensional domains. It also supports the typical way to solve time-dependent problems using time-stepping schemes,…
MFEM is an open-source, lightweight, flexible and scalable C++ library for modular finite element methods that features arbitrary high-order finite element meshes and spaces, support for a wide variety of discretization approaches and…
As quantum computing hardware systems continue to advance, the research and development of performant, scalable, and extensible software architectures, languages, models, and compilers is equally as important in order to bring this novel…
Inverse problem or parameter estimation of ordinary differential equations (ODEs), the iterative process of minimizing the mismatch between model-predicted and experimental states by tuning the parameter values within an optimization…
Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using…
In the article we discuss the notion of the generalized invariant manifold introduced in our previous study. In the literature the method of the differential constraints is well known as a tool for constructing particular solutions for the…
Ordinary and partial differential equations (DE) are used extensively in scientific and mathematical domains to model physical systems. Current literature has focused primarily on deep neural network (DNN) based methods for solving a…
We present Matrix Distributed Processing, a C++ library for fast development of efficient parallel algorithms. MDP is based on MPI and consists of a collection of C++ classes and functions such as lattice, site and field. Once an algorithm…
We describe here a library aimed at automating the solution of partial differential equations using the finite element method. By employing novel techniques for automated code generation, the library combines a high level of expressiveness…
We propose a novel second-order optimization framework for training the emerging deep continuous-time models, specifically the Neural Ordinary Differential Equations (Neural ODEs). Since their training already involves expensive gradient…
In a recent article, the class of functions from the integers to the integers computable in polynomial time has been characterized using discrete ordinary differential equations (ODE), also known as finite differences. Doing so, we pointed…
Quantum computers have the potential to efficiently solve a system of nonlinear ordinary differential equations (ODEs), which play a crucial role in various industries and scientific fields. However, it remains unclear which system of…