相关论文: Deep learning for gradient flows using the Brezis-…

I provide an introduction to the application of deep learning and neural networks for solving partial differential equations (PDEs). The approach, known as physics-informed neural networks (PINNs), involves minimizing the residual of the…

Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately…

We develop a Bregman proximal gradient method for structure learning on linear structural causal models. While the problem is non-convex, has high curvature and is in fact NP-hard, Bregman gradient methods allow us to neutralize at least…

This paper studies deep neural networks for solving extremely large linear systems arising from highdimensional problems. Because of the curse of dimensionality, it is expensive to store both the solution and right-hand side vector in such…

This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the…

Partial differential equations (PDEs) are often computationally challenging to solve, and in many settings many related PDEs must be be solved either at every timestep or for a variety of candidate boundary conditions, parameters, or…

The reaction-diffusion equation is one of the cornerstones equations in applied science and engineering. In the present study, a deep neural network has been trained in order to predict the solution of the equation with different…

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…

In this paper, we consider the problem of learning prediction models for spatiotemporal physical processes driven by unknown partial differential equations (PDEs). We propose a deep learning framework that learns the underlying dynamics and…

The deep-learning-based least squares method has shown successful results in solving high-dimensional non-linear partial differential equations (PDEs). However, this method usually converges slowly. To speed up the convergence of this…

We present a new hybrid physics-based machine-learning approach to reservoir modeling. The methodology relies on a series of deep adversarial neural network architecture with physics-based regularization. The network is used to simulate the…

This paper presents a novel deep learning framework for solving multiple optimal stopping problems in high dimensions. While deep learning has recently shown promise for single stopping problems, the multiple exercise case involves complex…

In this paper, we propose a new adaptation of the D-iteration algorithm to numerically solve the differential equations. This problem can be reinterpreted in 2D or 3D (or higher dimensions) as a limit of a diffusion process where the…

Owing to the remarkable development of deep learning technology, there have been a series of efforts to build deep learning-based climate models. Whereas most of them utilize recurrent neural networks and/or graph neural networks, we design…

We develop a new coarse-scale approximation strategy for the nonlinear single-continuum Richards equation as an unsaturated flow over heterogeneous non-periodic media, using the online generalized multiscale finite element method (online…

Deep learning is a form of machine learning for nonlinear high dimensional pattern matching and prediction. By taking a Bayesian probabilistic perspective, we provide a number of insights into more efficient algorithms for optimisation and…

Understanding and interpreting the decisions made by deep learning models is valuable in many domains. In computer vision, computing heatmaps from a deep network is a popular approach for visualizing and understanding deep networks.…

Diffusion-based models on continuous spaces have seen substantial recent progress through the mathematical framework of gradient flows, leveraging the Wasserstein-2 (${W}_2$) metric via the Jordan-Kinderlehrer-Otto (JKO) scheme. Despite the…

We present an efficient deep learning technique for the model reduction of the Navier-Stokes equations for unsteady flow problems. The proposed technique relies on the Convolutional Neural Network (CNN) and the stochastic gradient descent…

This paper solves the discretised multiphase flow equations using tools and methods from machine-learning libraries. The idea comes from the observation that convolutional layers can be used to express a discretisation as a neural network…