Related papers: Solving high-dimensional partial differential equa…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…
We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural…
Partial differential equations frequently appear in the natural sciences and related disciplines. Solving them is often challenging, particularly in high dimensions, due to the "curse of dimensionality". In this work, we explore the…
Relying on the classical connection between Backward Stochastic Differential Equations (BSDEs) and non-linear parabolic partial differential equations (PDEs), we propose a new probabilistic learning scheme for solving high-dimensional…
We propose a new deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) and forward-backward stochastic differential equations with jumps (FBSDEJs). This novel algorithm can be viewed as an…
Partial differential equations (PDEs) are a fundamental tool in the modeling of many real world phenomena. In a number of such real world phenomena the PDEs under consideration contain gradient-dependent nonlinearities and are…
Deep neural networks and other deep learning methods have very successfully been applied to the numerical approximation of high-dimensional nonlinear parabolic partial differential equations (PDEs), which are widely used in finance,…
In this paper, we propose forward and backward stochastic differential equations (FBSDEs) based deep neural network (DNN) learning algorithms for the solution of high dimensional quasilinear parabolic partial differential equations (PDEs),…
In this work we apply the Deep Galerkin Method (DGM) described in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations that arise in quantitative finance applications including option pricing, optimal…
The paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a…
Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov…
We propose a finite-dimensional control-based method to approximate solution operators for evolutional partial differential equations (PDEs), particularly in high-dimensions. By employing a general reduced-order model, such as a deep neural…
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high…
Partial differential equations (PDEs) are widely used across the physical and computational sciences. Decades of research and engineering went into designing fast iterative solution methods. Existing solvers are general purpose, but may be…
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…
We propose a new method, called a deep-genetic algorithm (deep-GA), to accelerate the performance of the so-called deep-BSDE method, which is a deep learning algorithm to solve high dimensional partial differential equations through their…
In this work, we present a novel forward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations (BSDEs). Motivated by the fact that differential deep learning can…
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…
Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using…