相关论文: Deep Learning Schemes For Parabolic Nonlocal Integ…
Over the last few years deep artificial neural networks (DNNs) have very successfully been used in numerical simulations for a wide variety of computational problems including computer vision, image classification, speech recognition,…
Mathematical models for flow and reactive transport in porous media often involve non-linear, degenerate parabolic equations. Their solutions have low regularity, and therefore lower order schemes are used for the numerical approximation.…
At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…
The aim of this work is to show an abstract framework to analyze the numerical approximation for a family of linear degenerate parabolic mixed equations by using a finite element method in space and a Backward-Euler scheme in time. We…
Fast and accurate solution of time-dependent partial differential equations (PDEs) is of key interest in many research fields including physics, engineering, and biology. Generally, implicit schemes are preferred over the explicit ones for…
Motivated by the mathematics literature on the algebraic properties of so-called polynomial vector flows, we propose a technique for approximating nonlinear differential equations by linear differential equations. Although the idea of…
In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied…
We develop a framework for estimating unknown partial differential equations from noisy data, using a deep learning approach. Given noisy samples of a solution to an unknown PDE, our method interpolates the samples using a neural network,…
Evolutionary deep neural networks have emerged as a rapidly growing field of research. This paper studies numerical integrators for such and other classes of nonlinear parametrizations $ u(t) = \Phi(\theta(t)) $, where the evolving…
We propose to use deep learning to estimate parameters in statistical models when standard likelihood estimation methods are computationally infeasible. We show how to estimate parameters from max-stable processes, where inference is…
The numerical approximation of solutions to the compressible Euler and Navier-Stokes equations is a crucial but challenging task with relevance in various fields of science and engineering. Recently, methods from deep learning have been…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
We define some approximation schemes for different kinds of generalized backward stochastic differential systems, considered in the Markovian framework. We propose a mixed approximation scheme for a decoupled system of forward reflected SDE…
We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where…
We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the…
An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ norms, for a numerical scheme consisting of implicit Euler method in time and discontinuous Galerkin method in…
Nonlinear parametric systems have been widely used in modeling nonlinear dynamics in science and engineering. Bifurcation analysis of these nonlinear systems on the parameter space are usually used to study the solution structure such as…
Inverse problems arise anywhere we have indirect measurement. As, in general they are ill-posed, to obtain satisfactory solutions for them needs prior knowledge. Classically, different regularization methods and Bayesian inference based…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
A long-standing problem at the interface of artificial intelligence and applied mathematics is to devise an algorithm capable of achieving human level or even superhuman proficiency in transforming observed data into predictive mathematical…