相关论文: Desingularization of Implicit Analytic Differentia…

The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet for fluid problems it only achieves first-order spatial accuracy near embedded boundaries for the velocity field and…

In this paper we the formulation of inverse problems as constrained minimization problems and their iterative solution by gradient or Newton type. We carry out a convergence analysis in the sense of regularization methods and discuss…

Advances in differentiable numerical integrators have enabled the use of gradient descent techniques to learn ordinary differential equations (ODEs). In the context of machine learning, differentiable solvers are central for Neural ODEs…

This paper deals with the index reduction problem for the class of quasi-regular DAE systems. It is shown that any of these systems can be transformed to a generically equivalent first order DAE system consisting of a single purely…

Convergence guarantees for optimization over bounded-rank matrices are delicate to obtain because the feasible set is a non-smooth and non-convex algebraic variety. Existing techniques include projected gradient descent, fixed-rank…

An implicit purification scheme is proposed for calculation of the temperature-dependent, grand canonical single-particle density matrix, given as a Fermi operator expansion in terms of the Hamiltonian. The computational complexity is shown…

Many engineering problems involve learning hidden dynamics from indirect observations, where the physical processes are described by systems of partial differential equations (PDE). Gradient-based optimization methods are considered…

A method for model reduction in nonlinear ODE systems is demonstrated through computational examples. The method does not require an implicit separation of time-scales in the fine dynamics to be effective. From the computational standpoint,…

Explicit step-truncation tensor methods have recently proven successful in integrating initial value problems for high-dimensional partial differential equations (PDEs). However, the combination of non-linearity and stiffness may introduce…

In this paper we establish a connection between non-convex optimization methods for training deep neural networks and nonlinear partial differential equations (PDEs). Relaxation techniques arising in statistical physics which have already…

The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE…

We study conditions of reduction of the multidimensional wave equation - a system of the d'Alembert and Hamilton equations. We prove necessary conditions for compatibility of such system of the reduction conditions. Possible types of the…

The quest for analytical solutions to differential equations has traditionally been constrained by the need for extensive mathematical expertise. Machine learning methods like genetic algorithms have shown promise in this domain, but are…

In this paper, the elliptic PDE-constrained optimization problem with box constraints on the control is studied. To numerically solve the problem, we apply the 'optimize-discretize-optimize' strategy. Specifically, the alternating direction…

Solutions to fractional models inherently exhibit non-smooth behavior, which significantly deteriorates the accuracy and therefore efficiency of existing numerical methods. We develop a two-stage data-infused computational framework for…

In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…

Solving partial differential equations (PDEs) using an annealing-based approach involves solving generalized eigenvalue problems. Discretizing a PDE yields a system of linear equations (SLE). Solving an SLE can be formulated as a general…

We propose a reduced-order modeling approach for nonlinear, parameter-dependent ordinary differential equations (ODE). Dimensionality reduction is achieved using nonlinear maps represented by autoencoders. The resulting low-dimensional ODE…

We construct stable periodic solutions for a simple form nonlinear delay differential equation (DDE) with a periodic coefficient. The equation involves one underlying nonlinearity with the multiplicative periodic coefficient. The well-known…

Structure-preserving linearly implicit exponential integrators are constructed for Hamiltonian partial differential equations with linear constant damping. Linearly implicit integrators are derived by polarizing the polynomial terms of the…