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We propose energy natural gradient descent, a natural gradient method with respect to a Hessian-induced Riemannian metric as an optimization algorithm for physics-informed neural networks (PINNs) and the deep Ritz method. As a main…
Deep learning methods have gained considerable interest in the numerical solution of various partial differential equations (PDEs). One particular focus is physics-informed neural networks (PINN), which integrate physical principles into…
We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-refinement via domain decomposition and projection onto space…
We propose a new physics-informed neural network framework, IDPINN, based on the enhancement of initialization and domain decomposition to improve prediction accuracy. We train a PINN using a small dataset to obtain an initial network…
Physics-informed neural network (PINN) is a data-driven approach to solve equations. It is successful in many applications; however, the accuracy of the PINN is not satisfactory when it is used to solve multiscale equations. Homogenization…
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 deep learning has emerged as a promising framework for solving partial differential equations (PDEs). Nevertheless, training these models on complex problems remains challenging, often leading to limited accuracy and…
We develop innovative algorithms for solving the strong-constraint formulation of four-dimensional variational data assimilation in large-scale applications. We present a space-time decomposition approach that employs domain decomposition…
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…
While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains $\Omega \subset \mathbb{R} ^d, $ $d=1,2,3$ in association with…
A method is presented that allows to reduce a problem described by differential equations with initial and boundary conditions to the problem described only by differential equations. The advantage of using the modified problem for…
Physics-informed neural networks (PINNs) provide a means of obtaining approximate solutions of partial differential equations and systems through the minimisation of an objective function which includes the evaluation of a residual function…
In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for…
We introduce a new approach for solving forward systems of differential equations using a combination of splitting methods and physics-informed neural networks (PINNs). The proposed method, splitting PINN, effectively addresses the…
In recent years, physics-informed neural networks (PINNs) have gained significant attention for solving differential equations, although they suffer from two fundamental limitations, namely, spectral bias inherent in neural networks and…
As a promising framework for resolving partial differential equations (PDEs), Physics-Informed Neural Networks (PINNs) have received widespread attention from industrial and scientific fields. However, lack of expressive ability and…
In this paper, we propose a method for computing eigenvalues of elliptic problems using Deep Learning techniques. A key feature of our approach is that it is independent of the space dimension and can compute arbitrary eigenvalues without…
Physics informed neural networks (PINNs) have emerged as a powerful tool to provide robust and accurate approximations of solutions to partial differential equations (PDEs). However, PINNs face serious difficulties and challenges when…
Physics-Informed Neural Networks (PINNs) have been successfully applied to solve Partial Differential Equations (PDEs). Their loss function is founded on a strong residual minimization scheme. Variational Physics-Informed Neural Networks…
Solving time-dependent Partial Differential Equations (PDEs) is one of the most critical problems in computational science. While Physics-Informed Neural Networks (PINNs) offer a promising framework for approximating PDE solutions, their…