Related papers: N-Adaptive Ritz Method: A Neural Network Enriched …

Mesh generation is essential for accurate and efficient computational fluid dynamics simulations. To resolve critical features in the flow, adaptive mesh refinement (AMR) is routinely employed in certain regions of the computational domain,…

Recent years have witnessed growing interests in solving partial differential equations by deep neural networks, especially in the high-dimensional case. Unlike classical numerical methods, such as finite difference method and finite…

We propose an adaptive refinement algorithm to solve total variation regularized measure optimization problems. The method iteratively constructs dyadic partitions of the unit cube based on i) the resolution of discretized dual problems and…

We present a new physics informed neural network (PINN) algorithm for solving brittle fracture problems. While most of the PINN algorithms available in the literature minimize the residual of the governing partial differential equation, the…

We proposed the boundary-integral type neural networks (BINN) for the boundary value problems in computational mechanics. The boundary integral equations are employed to transfer all the unknowns to the boundary, then the unknowns are…

We explore the probabilistic partition of unity network (PPOU-Net) model in the context of high-dimensional regression problems and propose a general framework focusing on adaptive dimensionality reduction. With the proposed framework, the…

Physics-informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering…

Using unitary (instead of general) matrices in artificial neural networks (ANNs) is a promising way to solve the gradient explosion/vanishing problem, as well as to enable ANNs to learn long-term correlations in the data. This approach…

This work develops an epsilon-uniform finite element method for singularly perturbed boundary value problems. A surprising and remarkable observation is illustrated: By moving one node arbitrarily in between its adjacent nodes, the new…

Monotone finite difference methods provide stable convergent discretizations of a class of degenerate elliptic and parabolic Partial Differential Equations (PDEs). These methods are best suited to regular rectangular grids, which leads to…

Due to their susceptibility to adversarial perturbations, neural networks (NNs) are hardly used in safety-critical applications. One measure of robustness to such perturbations in the input is the Lipschitz constant of the input-output map…

We propose a novel method for fast and accurate training of physics-informed neural networks (PINNs) to find solutions to boundary value problems (BVPs) and initial boundary value problems (IBVPs). By combining the methods of training deep…

A type of adaptive finite element method for the eigenvalue problems is proposed based on the multilevel correction scheme. In this method, adaptive finite element method to solve eigenvalue problems involves solving associated boundary…

In this work, we bridge standard adaptive mesh refinement and coarsening on scalable octree background meshes and robust unfitted finite element formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics…

We propose a new semi-analytic physics informed neural network (PINN) to solve singularly perturbed boundary value problems. The PINN is a scientific machine learning framework that offers a promising perspective for finding numerical…

We extend the finite element interpolated neural network (FEINN) framework from partial differential equations (PDEs) with weak solutions in $H^1$ to PDEs with weak solutions in $H(\textbf{curl})$ or $H(\textbf{div})$. To this end, we…

In this paper, a shallow Ritz-type neural network for solving elliptic equations with delta function singular sources on an interface is developed. There are three novel features in the present work; namely, (i) the delta function…

To address the challenge of segmenting noisy images with blurred or fragmented boundaries, this paper presents a robust version of Variational Model Based Tailored UNet (VM_TUNet), a hybrid framework that integrates variational methods with…

This work considers stochastic Galerkin approximations of linear elliptic partial differential equations (PDEs) with stochastic forcing terms and stochastic diffusion coefficients, that cannot be bounded uniformly away from zero and…

Solving Partial Differential Equations (PDEs) using neural networks presents different challenges, including integration errors and spectral bias, often leading to poor approximations. In addition, standard neural network-based methods,…