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Deep neural networks (DNN) has received increasing attention in machine learning applications in the last several years. Recently, a non-asymptotic error bound has been developed to measure the performance of the fully connected DNN…
The physics-informed neural network (PINN) is effective in solving the partial differential equation (PDE) by capturing the physics constraints as a part of the training loss function through the Automatic Differentiation (AD). This study…
In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied…
We present a method to solve initial and boundary value problems using artificial neural networks. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the boundary (or initial) conditions…
Deep neural networks (DNNs) have shown great success in many machine learning tasks. Their training is challenging since the loss surface of the network architecture is generally non-convex, or even non-smooth. How and under what…
The study of parametric differential equations plays a crucial role in weather forecasting and epidemiological modeling. These phenomena are better represented using fractional derivatives due to their inherent memory or hereditary effects.…
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…
Deep neural networks (DNNs) have been widely applied to solve real-world regression problems. However, selecting optimal network structures remains a significant challenge. This study addresses this issue by linking neuron selection in DNNs…
In this paper we develop a non-diffusive neural network (NDNN) algorithm for accurately solving weak solutions to hyperbolic conservation laws. The principle is to construct these weak solutions by computing smooth local solutions in…
Large-scale deep neural networks (DNN) have been successfully used in a number of tasks from image recognition to natural language processing. They are trained using large training sets on large models, making them computationally and…
More competent learning models are demanded for data processing due to increasingly greater amounts of data available in applications. Data that we encounter often have certain embedded sparsity structures. That is, if they are represented…
Utilizing physics-informed neural networks (PINN) to solve partial differential equations (PDEs) becomes a hot issue and also shows its great powers, but still suffers from the dilemmas of limited predicted accuracy in the sampling domain…
Nonlinear differential equations are challenging to solve numerically and are important to understanding the dynamics of many physical systems. Deep neural networks have been applied to help alleviate the computational cost that is…
Neural network-based solvers for partial differential equations (PDEs) have attracted considerable attention, yet they often face challenges in accuracy and computational efficiency. In this work, we focus on time-dependent PDEs and observe…
In this article, we propose a novel Stabilized Physics Informed Neural Networks method (SPINNs) for solving wave equations. In general, this method not only demonstrates theoretical convergence but also exhibits higher efficiency compared…
With the advantages of fast calculating speed and high precision, the physics-informed neural network method opens up a new approach for numerically solving nonlinear partial differential equations. Based on conserved quantities, we devise…
A new method to solve computationally challenging (random) parametric obstacle problems is developed and analyzed, where the parameters can influence the related partial differential equation (PDE) and determine the position and surface…
In this paper we use deep feedforward artificial neural networks to approximate solutions to partial differential equations in complex geometries. We show how to modify the backpropagation algorithm to compute the partial derivatives of the…
A feedforward neural network, including hidden layers, motivated by nonlinear functions (such as Tanh, ReLU, and Sigmoid functions), exhibits uniform approximation properties in Sobolev space, and discontinuous neural networks can reduce…
The state-of-the-art deep neural networks (DNNs) have significant computational and data management requirements. The size of both training data and models continue to increase. Sparsification and pruning methods are shown to be effective…