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Surrogate neural network-based partial differential equation (PDE) solvers have the potential to solve PDEs in an accelerated manner, but they are largely limited to systems featuring fixed domain sizes, geometric layouts, and boundary…
Domain decomposition provides an effective way to tackle the dilemma of physics-informed neural networks (PINN) which struggle to accurately and efficiently solve partial differential equations (PDEs) in the whole domain, but the lack of…
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…
By integrating physics-informed neural network (PINN) techniques with domain decomposition method, a deep domain decomposition method is presented for solving elliptic variational inequality problems. Based on the Ritz variation method, the…
In the present work, a multi-scale framework for neural network enhanced methods is proposed for approximation of function and solution of partial differential equations (PDEs). By introducing the multi-scale concept, the total solution of…
Many problems in science and engineering can be represented by a set of partial differential equations (PDEs) through mathematical modeling. Mechanism-based computation following PDEs has long been an essential paradigm for studying topics…
The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…
Deep Neural Networks (DNNs) have already become a crucial computational approach to revealing the spatial patterns in the human brain; however, there are three major shortcomings in utilizing DNNs to detect the spatial patterns in…
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…
Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for…
Extreme learning machine (ELM) is a methodology for solving partial differential equations (PDEs) using a single hidden layer feed-forward neural network. It presets the weight/bias coefficients in the hidden layer with random values, which…
The usual approach to model reduction for parametric partial differential equations (PDEs) is to construct a linear space $V_n$ which approximates well the solution manifold $\mathcal{M}$ consisting of all solutions $u(y)$ with $y$ the…
Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the…
The use of deep learning methods for solving PDEs is a field in full expansion. In particular, Physical Informed Neural Networks, that implement a sampling of the physical domain and use a loss function that penalizes the violation of the…
Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the optimal construction of these methods requires tedious analysis and is often available only in…
At present, deep learning based methods are being employed to resolve the computational challenges of high-dimensional partial differential equations (PDEs). But the computation of the high order derivatives of neural networks is costly,…
We propose a two-scale neural network method for solving partial differential equations (PDEs) with small parameters using physics-informed neural networks (PINNs). We directly incorporate the small parameters into the architecture of…
The Dirichlet-Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are…
A multiscale numerical method is proposed for the solution of semi-linear elliptic stochastic partial differential equations with localized uncertainties and non-linearities, the uncertainties being modeled by a set of random parameters. It…
Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the…