Related papers: Deep learning based numerical approximation algori…
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,…
Neural network-based solvers for partial differential equations (PDEs) have attracted considerable attention, yet they often face challenges in accuracy and computational efficiency. In this work, we focus on time-dependent PDEs and observe…
We perform a comprehensive numerical study of the effect of approximation-theoretical results for neural networks on practical learning problems in the context of numerical analysis. As the underlying model, we study the…
This paper proposes two efficient approximation methods to solve high-dimensional fully nonlinear partial differential equations (NPDEs) and second-order backward stochastic differential equations (2BSDEs), where such high-dimensional fully…
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…
We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called direction and norm decomposition method, proposes to approximate the required solution $X_t$ by…
In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational…
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection…
We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central…
In this paper, we introduce PDE-LEARN, a novel deep learning algorithm that can identify governing partial differential equations (PDEs) directly from noisy, limited measurements of a physical system of interest. PDE-LEARN uses a Rational…
We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs…
Data-driven methods have recently made great progress in the discovery of partial differential equations (PDEs) from spatial-temporal data. However, several challenges remain to be solved, including sparse noisy data, incomplete candidate…
Although stochastic approximation learning methods have been widely used in the machine learning literature for over 50 years, formal theoretical analyses of specific machine learning algorithms are less common because stochastic…
Stochastic neighbor embedding (SNE) and related nonlinear manifold learning algorithms achieve high-quality low-dimensional representations of similarity data, but are notoriously slow to train. We propose a generic formulation of embedding…
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,…
We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous…
We present a novel solution method for It\^o stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation between two successive intervals. The main…
We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any…
Starting with some fundamental concepts, in this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations (PDEs). We start by using Lagrange and…
The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard…