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Physics informed neural network (PINN) based solution methods for differential equations have recently shown success in a variety of scientific computing applications. Several authors have reported difficulties, however, when using PINNs to…
Physics-informed neural networks (PINNs) have recently received much attention due to their capabilities in solving both forward and inverse problems. For training a deep neural network associated with a PINN, one typically constructs a…
In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods…
This paper develops convolutional neural network (CNN) methods for simultaneous approximation and elliptic boundary value problems on compact Riemannian manifolds. We establish simultaneous Sobolev approximation results for single- and…
Physics-Informed Neural Networks (PINNs) have been recognized as a mesh-free alternative to solve partial differential equations where physics information is incorporated. However, in dealing with problems characterized by high stiffness or…
Physics-informed neural networks (PINNs) have emerged as promising methods for solving partial differential equations (PDEs) by embedding physical laws within neural architectures. However, these classical approaches often require a large…
Physics-informed neural networks (PINNs) are a promising approach for solving partial differential equations (PDEs). Their training, however, is often difficult because multiple loss terms induced by PDE residuals and boundary or initial…
We study the posterior contraction rate of Bayesian Physics-Informed Neural Networks (PINNs) for solving a general class of elliptic partial differential equations (PDEs). We focus on learning of the elliptic equation with a non-homogeneous…
Solving high-dimensional partial differential equations is a recurrent challenge in economics, science and engineering. In recent years, a great number of computational approaches have been developed, most of them relying on a combination…
Whilst the Universal Approximation Theorem guarantees the existence of approximations to Sobolev functions -- the natural function spaces for PDEs -- by Neural Networks (NNs) of sufficient size, low-regularity solutions may lead to poor…
We consider the discretization of elliptic boundary-value problems by variational physics-informed neural networks (VPINNs), in which test functions are continuous, piecewise linear functions on a triangulation of the domain. We define an a…
The dual norm characterisation of weak solutions of second-order linear elliptic partial differential equations is mathematically natural but computationally intractable: evaluating the $H^{-1}$ norm of a residual requires a supremum over…
We establish the convergence of physics-informed neural networks (PINNs) for time-dependent fractional diffusion equations posed on bounded domains. The presence of fractional Laplacian operators introduces nonlocal behavior and regularity…
Physics-Informed Neural Networks (PINNs) solve physical systems by incorporating governing partial differential equations directly into neural network training. In electromagnetism, where well-established methodologies such as FDTD and FEM…
Physics-informed neural networks (PINNs) have recently emerged as effective methods for solving partial differential equations (PDEs) in various problems. Substantial research focuses on the failure modes of PINNs due to their frequent…
Motivated by many applications in complex domains with boundaries exposed to large topological changes or deformations, fictitious domain methods regard the actual domain of interest as being embedded in a fixed Cartesian background. This…
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…
We consider a least-squares variational kernel-based method for numerical solution of second order elliptic partial differential equations on a multi-dimensional domain. In this setting it is not assumed that the differential operator is…
We consider the approximation of a class of dynamic partial differential equations (PDE) of second order in time by the physics-informed neural network (PINN) approach, and provide an error analysis of PINN for the wave equation, the…
This paper explores the difficulties in solving partial differential equations (PDEs) using physics-informed neural networks (PINNs). PINNs use physics as a regularization term in the objective function. However, a drawback of this approach…