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Simulations of complex physical systems are typically realized by discretizing partial differential equations (PDEs) on unstructured meshes. While neural networks have recently been explored for surrogate and reduced order modeling of PDE…

Physics-informed deep learning has emerged as a promising alternative for solving partial differential equations. However, for complex problems, training these networks can still be challenging, often resulting in unsatisfactory accuracy…

In this paper, we introduce the Deep Finite Volume Method (DFVM), an innovative deep learning framework tailored for solving high-order (order \(\geq 2\)) partial differential equations (PDEs). Our approach centers on a novel loss function…

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…

A state-of-the-art deep domain decomposition method (D3M) based on the variational principle is proposed for partial differential equations (PDEs). The solution of PDEs can be formulated as the solution of a constrained optimization…

In this work, we propose a learning method for solving the linear transport equation under the diffusive scaling. Due to the multiscale nature of our model equation, the model is challenging to solve by using conventional methods. We employ…

In this paper, we develop a new optimization framework for the least squares learning problem via fully connected neural networks or physics-informed neural networks. The gradient descent sometimes behaves inefficiently in deep learning…

Many types of physics-informed neural network models have been proposed in recent years as approaches for learning solutions to differential equations. When a particular task requires solving a differential equation at multiple…

Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…

Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation…

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…

In recent years a large literature on deep learning based methods for the numerical solution partial differential equations has emerged; results for integro-differential equations on the other hand are scarce. In this paper we study deep…

Solving time-dependent partial differential equations (PDEs) that exhibit sharp gradients or local singularities is computationally demanding, as traditional physics-informed neural networks (PINNs) often suffer from inefficient point…

Partial differential equations (PDEs) play a crucial role in studying a vast number of problems in science and engineering. Numerically solving nonlinear and/or high-dimensional PDEs is often a challenging task. Inspired by the traditional…

In this paper, a physics-informed multiresolution wavelet neural network (PIMWNN) method is proposed for solving partial differential equations (PDEs). This method uses the multiresolution wavelet neural network (MWNN) to approximate…

Partial-differential-equation (PDE)-constrained optimization is a well-worn technique for acquiring optimal parameters of systems governed by PDEs. However, this approach is limited to providing a single set of optimal parameters per…

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.…

Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…

Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…

Enhancing neural networks with knowledge of physical equations has become an efficient way of solving various physics problems, from fluid flow to electromagnetism. Graph neural networks show promise in accurately representing irregularly…