Related papers: Newton's method and its hybrid with machine learni…
We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear…
The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of…
In this work, several multilevel decoupled algorithms are proposed for a mixed Navier-Stokes/Darcy model. These algorithms are based on either successively or parallelly solving two linear subdomain problems after solving a coupled…
We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion…
In this work we present an adaptive Newton-type method to solve nonlinear constrained optimization problems in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive…
A discretization method with non-matching grids is proposed for the coupled Stokes-Darcy problem that uses a mortar variable at the interface to couple the marker and cell (MAC) method in the Stokes domain with the Raviart-Thomas mixed…
In this paper, we put forth distributed algorithms for solving loosely coupled unconstrained and constrained optimization problems. Such problems are usually solved using algorithms that are based on a combination of decomposition and first…
In this paper, we develop the numerical theory of decoupled modified characteristic finite element method with different subdomain time steps for the mixed stabilized formulation of nonstationary dual-porosity-Navier-Stokes model. Based on…
This paper studies the performance Newton's iteration applied with Anderson acceleration for solving the incompressible steady Navier-Stokes equations. We manifest that this method converges superlinearly with a good initial guess, and…
This paper is devoted to studying the global and finite convergence of the semi-smooth Newton method for solving a piecewise linear system that arises in cone-constrained quadratic programming problems and absolute value equations. We first…
We introduce a new second-order inertial optimization method for machine learning called INNA. It exploits the geometry of the loss function while only requiring stochastic approximations of the function values and the generalized…
The numerical solution of differential equations using machine learning-based approaches has gained significant popularity. Neural network-based discretization has emerged as a powerful tool for solving differential equations by…
Newton's method is a fundamental technique in optimization with quadratic convergence within a neighborhood around the optimum. However reaching this neighborhood is often slow and dominates the computational costs. We exploit two…
This paper is concerned with stochastic incompressible Navier-Stokes equations with multiplicative noise in two dimensions with respect to periodic boundary conditions. Based on the Helmholtz decomposition of the multiplicative noise,…
Machine Learning models incorporating multiple layered learning networks have been seen to provide effective models for various classification problems. The resulting optimization problem to solve for the optimal vector minimizing the…
This paper focuses on proposing a deep learning initialized iterative method (Int-Deep) for low-dimensional nonlinear partial differential equations (PDEs). The corresponding framework consists of two phases. In the first phase, an…
Deep learning involves a difficult non-convex optimization problem with a large number of weights between any two adjacent layers of a deep structure. To handle large data sets or complicated networks, distributed training is needed, but…
Deep learning involves a difficult non-convex optimization problem, which is often solved by stochastic gradient (SG) methods. While SG is usually effective, it may not be robust in some situations. Recently, Newton methods have been…
In this work we introduce and analyse a new low-order method for the variable-density incompressible Navier-Stokes equations. The main novelty of the proposed method lies in the support of general meshes, possibly including polygonal or…
A recent paper [J. A. Evans, D. Kamensky, Y. Bazilevs, "Variational multiscale modeling with discretely divergence-free subscales", Computers & Mathematics with Applications, 80 (2020) 2517-2537] introduced a novel stabilized finite element…