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Several neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance one may include…

Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…

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…

We introduce a new representation learning approach for domain adaptation, in which data at training and test time come from similar but different distributions. Our approach is directly inspired by the theory on domain adaptation…

Deep Neural Networks (DNNs) have recently been achieving state-of-the-art performance on a variety of computer vision related tasks. However, their computational cost limits their ability to be implemented in embedded systems with…

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…

Approximating the solutions of boundary value problems governed by partial differential equations with neural networks is challenging, largely due to the difficult training process. This difficulty can be partly explained by the spectral…

Neural networks (NNs) have gained significant attention across various engineering disciplines, particularly in design optimization, where they are used to build surrogate models for high-dimensional regression problems. Despite their power…

Recent research efforts have shown that neural architectures can be effective in conventional information extraction tasks such as named entity recognition, yielding state-of-the-art results on standard newswire datasets. However, despite…

As one of the fundamental tasks in computer vision, semantic segmentation plays an important role in real world applications. Although numerous deep learning models have made notable progress on several mainstream datasets with the rapid…

We develop a randomized Newton's method for solving differential equations, based on a fully connected neural network discretization. In particular, the randomized Newton's method randomly chooses equations from the overdetermined nonlinear…

The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here,…

To enhance solution accuracy and training efficiency in neural network approximation to partial differential equations, partitioned neural networks can be used as a solution surrogate instead of a single large and deep neural network…

The neural network-based approach to solving partial differential equations has attracted considerable attention due to its simplicity and flexibility in representing the solution of the partial differential equation. In training a neural…

Over-parameterization is one of the inherent characteristics of modern deep neural networks, which can often be overcome by leveraging regularization methods, such as Dropout. Usually, these methods are applied globally and all the input…

In this paper we propose a method to generate suitably refined finite element meshes using neural networks. As a model problem we consider a linear elasticity problem on a planar domain (possibly with holes) having a polygonal boundary. We…

Motivated by the gap between theoretical optimal approximation rates of deep neural networks (DNNs) and the accuracy realized in practice, we seek to improve the training of DNNs. The adoption of an adaptive basis viewpoint of DNNs leads to…

Convolutional neural network-based approaches for semantic segmentation rely on supervision with pixel-level ground truth, but may not generalize well to unseen image domains. As the labeling process is tedious and labor intensive,…

This paper proposes a domain decomposition subspace neural network method for efficiently solving linear and nonlinear partial differential equations. By combining the principles of domain decomposition and subspace neural networks, the…

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…