Related papers: Multi-Grade Deep Learning for Partial Differential…

Wave propagation problems are typically formulated as partial differential equations (PDEs) on unbounded domains to be solved. The classical approach to solving such problems involves truncating them to problems on bounded domains by…

Despite the notable success of deep neural networks (DNNs) in solving complex tasks, the training process still remains considerable challenges. A primary obstacle is the substantial time required for training, particularly as high…

Partial differential equations (PDEs) are indispensable for modeling many physical phenomena and also commonly used for solving image processing tasks. In the latter area, PDE-based approaches interpret image data as discretizations of…

Solving partial differential equations (PDEs) on manifolds defined by randomly sampled point clouds is a challenging problem in scientific computing and has broad applications in various fields. In this paper, we develop a two-step…

Surface partial differential equations arise in numerous scientific and engineering applications. Their numerical solution on static and evolving surfaces remains challenging due to geometric complexity and, for evolving geometries, the…

The finite element method, finite difference method, finite volume method and spectral method have achieved great success in solving partial differential equations. However, the high accuracy of traditional numerical methods is at the cost…

Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that…

Dynamic Bayesian Networks (DBNs), renowned for their interpretability, have become increasingly vital in representing complex stochastic processes in various domains such as gene expression analysis, healthcare, and traffic prediction.…

The main aim of this study is to introduce a 2-layered Artificial Neural Network (ANN) for solving the Black-Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed…

Recent developments in mechanical, aerospace, and structural engineering have driven a growing need for efficient ways to model and analyse structures at much larger and more complex scales than before. While established numerical methods…

We propose TopDis (Topological Disentanglement), a method for learning disentangled representations via adding a multi-scale topological loss term. Disentanglement is a crucial property of data representations substantial for the…

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…

Learning the evolutionary dynamics of Partial Differential Equations (PDEs) is critical in understanding dynamic systems, yet current methods insufficiently learn their representations. This is largely due to the multi-scale nature of the…

Semilinear parabolic partial differential equations (PDEs) are fundamental to modeling complex dynamical systems across scientific domains. The Deep Backward Stochastic Differential Equation (BSDE) method is a promising approach for…

Many scientific and industrial applications require solving Partial Differential Equations (PDEs) to describe the physical phenomena of interest. Some examples can be found in the fields of aerodynamics, astrodynamics, combustion and many…

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 computational domains, etc.…

First-order methods like stochastic gradient descent(SGD) are recently the popular optimization method to train deep neural networks (DNNs), but second-order methods are scarcely used because of the overpriced computing cost in getting the…

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.…

The discovery of Partial Differential Equations (PDEs) is an essential task for applied science and engineering. However, data-driven discovery of PDEs is generally challenging, primarily stemming from the sensitivity of the discovered…

This work presents a two-stage adaptive framework for progressively developing deep neural network (DNN) architectures that generalize well for a given training data set. In the first stage, a layerwise training approach is adopted where a…