Related papers: Deep learning for gradient flows using the Brezis-…
The scarcity of labeled data is a long-standing challenge for many machine learning tasks. We propose our gradient flow method to leverage the existing dataset (i.e., source) to generate new samples that are close to the dataset of interest…
A deep learning approach to numerically approximate the solution to the Eikonal equation is introduced. The proposed method is built on the fast marching scheme which comprises of two components: a local numerical solver and an update…
In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high…
We develop novel neural network-based implicit particle methods to compute high-dimensional Wasserstein-type gradient flows with linear and nonlinear mobility functions. The main idea is to use the Lagrangian formulation in the…
Numerical simulation of fluids plays an essential role in modeling many physical phenomena, such as weather, climate, aerodynamics and plasma physics. Fluids are well described by the Navier-Stokes equations, but solving these equations at…
This paper aims at studying the difference between Ritz-Galerkin (R-G) method and deep neural network (DNN) method in solving partial differential equations (PDEs) to better understand deep learning. To this end, we consider solving a…
We introduce a generative learning framework to model high-dimensional parametric systems using gradient guidance and virtual observations. We consider systems described by Partial Differential Equations (PDEs) discretized with structured…
In recent years, data-driven methods have been developed to learn dynamical systems and partial differential equations (PDE). The goal of such work is discovering unknown physics and the corresponding equations. However, prior to achieving…
Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for…
In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…
We introduce a deep neural network based method for solving a class of elliptic partial differential equations. We approximate the solution of the PDE with a deep neural network which is trained under the guidance of a probabilistic…
Solutions of certain partial differential equations (PDEs) are often represented by the steepest descent curves of corresponding functionals. Minimizing movement scheme was developed in order to study such curves in metric spaces.…
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…
We present a subspace method based on neural networks for solving the partial differential equation in weak form with high accuracy. The basic idea of our method is to use some functions based on neural networks as base functions to span a…
Achievement of solutions in Navier-Stokes equation is one of challenging quests, especially for its closure problem. For achievement of particular solutions, there are variety of numerical simulations including Direct Numerical Simulation…
Recent works in deep learning have shown that integrating differentiable physics simulators into the training process can greatly improve the quality of results. Although this combination represents a more complex optimization task than…
In this work we propose a deep adaptive sampling (DAS) method for solving partial differential equations (PDEs), where deep neural networks are utilized to approximate the solutions of PDEs and deep generative models are employed to…
High-dimensional partial differential equations (PDEs) pose significant challenges for numerical computation due to the curse of dimensionality, which limits the applicability of traditional mesh-based methods. Since 2017, the Deep BSDE…
Deep neural networks have been shown to provide accurate function approximations in high dimensions. However, fitting network parameters requires informative training data that are often challenging to collect in science and engineering…
In this review, we survey the latest approaches and techniques developed to overcome the spectral bias towards low frequency of deep neural network learning methods in learning multiple-frequency solutions of partial differential equations.…