Related papers: A deep backward regression-based scheme for high-d…
We propose a probabilistic numerical algorithm to solve Backward Stochastic Differential Equations (BSDEs) with nonnegative jumps, a class of BSDEs introduced in [9] for representing fully nonlinear HJB equations. In particular, this allows…
In this paper we extend the hierarchical model reduction framework based on reduced basis techniques for the application to nonlinear partial differential equations. The major new ingredient to accomplish this goal is the introduction of…
Local temporal patterns in real-world time series continuously shift, rendering globally shared transformations suboptimal. Current deep forecasting models, despite their scale and complexity, rely on fixed weight matrices applied uniformly…
In this paper, we present a numerical scheme to solve the initial-boundary value problem for backward stochastic partial differential equations of parabolic type. Based on the Galerkin method, we approximate the original equation by a…
In this work, in order to obtain higher-order schemes for solving forward backward stochastic differential equations, we adopt the high-order multi-step method in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36(4) (2014),…
We present a new deep primal-dual backward stochastic differential equation framework based on stopping time iteration to solve optimal stopping problems. A novel loss function is proposed to learn the conditional expectation, which…
In many important machine learning applications, the training distribution used to learn a probabilistic classifier differs from the testing distribution on which the classifier will be used to make predictions. Traditional methods correct…
Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering. In recent years, a great number of computational approaches have been developed, most of them relying on a combination…
A novel approach for selecting appropriate reconstructions is implemented to the hyperbolic conservation laws in the high-order local polynomial-based framework, e.g., the discontinuous Galerkin (DG) and flux reconstruction (FR) schemes.…
In this work, we derive two-sided a posteriori error estimates for the dual-weighted residual (DWR) method. We consider both single and multiple goal functionals. Using a saturation assumption, we derive lower bounds yielding the efficiency…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems…
In recent years we have witnessed a growth in mathematics for deep learning, which has been used to solve inverse problems of partial differential equations (PDEs). However, most deep learning-based inversion methods either require paired…
In this paper we study different algorithms for backward stochastic differential equations (BSDE in short) basing on random walk framework for 1-dimensional Brownian motion. Implicit and explicit schemes for both BSDE and reflected BSDE are…
Fractional Laplace equations are becoming important tools for mathematical modeling and prediction. Recent years have shown much progress in developing accurate and robust algorithms to numerically solve such problems, yet most solvers for…
We present the Deep Picard Iteration (DPI) method, a new deep learning approach for solving high-dimensional partial differential equations (PDEs). The core innovation of DPI lies in its use of Picard iteration to reformulate the typically…
This paper introduces a new approximation scheme for solving high-dimensional semilinear partial differential equations (PDEs) and backward stochastic differential equations (BSDEs). First, we decompose a target semilinear PDE (BSDE) into…
The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination…
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,…
Machine learning based partial differential equations (PDEs) solvers have received great attention in recent years. Most progress in this area has been driven by deep neural networks such as physics-informed neural networks (PINNs) and…