Related papers: Deep Feynman-Kac Methods for High-dimensional Semi…
We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for…
Recent advances in deep learning makes solving parabolic partial differential equations (PDEs) in high dimensional spaces possible via forward-backward stochastic differential equation (FBSDE) formulations. The implementation of most…
The Feynman-Kac formulae (FKF) express local solutions of partial differential equations (PDEs) as expectations with respect to some complementary stochastic differential equation (SDE). Repeatedly sampling paths from the complementary SDE…
In recent years, Scientific Machine Learning (SciML) methods for solving partial differential equations (PDEs) have gained increasing popularity. Within such a paradigm, Physics-Informed Neural Networks (PINNs) are novel deep learning…
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…
Inferring parameters of high-dimensional partial differential equations (PDEs) poses significant computational and inferential challenges, primarily due to the curse of dimensionality and the inherent limitations of traditional numerical…
Due to the curse of dimensionality, solving high dimensional parabolic partial differential equations (PDEs) has been a challenging problem for decades. Recently, a weak adversarial network (WAN) proposed in (Y.Zang et al., 2020) offered a…
Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computation domains, etc.…
Recently, the deep learning method has been used for solving forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). It has good accuracy and performance for high-dimensional…
We propose an algorithm based on variational quantum imaginary time evolution for solving the Feynman-Kac partial differential equation resulting from a multidimensional system of stochastic differential equations. We utilize the…
Feynman's path integral approach is to sum over all possible spatio-temporal paths to reproduce the quantum wave function and the corresponding time evolution, which has enormous potential to reveal quantum processes in classical view.…
We introduce forward-backward stochastic differential equations, highlighting the connection between solutions of these and solutions of partial differential equations, related by the Feynman-Kac theorem. We review the technique of…
Physics-informed neural networks (PINNs) have recently emerged as a promising way to compute the solutions of partial differential equations (PDEs) using deep neural networks. However, despite their significant success in various fields, it…
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),…
We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the…
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…
This study analyzes the derivative-free loss method to solve a certain class of elliptic PDEs and fluid problems using neural networks. The approach leverages the Feynman-Kac formulation, incorporating stochastic walkers and their averaged…
We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear…
Deep neural networks (DNNs) are increasingly used to solve partial differential equations (PDEs) that naturally arise while modeling a wide range of systems and physical phenomena. However, the accuracy of such DNNs decreases as the PDE…
The approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively…