Related papers: Stability and Bifurcation Analysis of Nonlinear PD…
We present a unified theoretical framework for analyzing the stability and consistency of Physics-Informed Neural Networks (PINNs), grounded in operator coercivity, variational formulations, and non-asymptotic perturbation theory. PINNs…
We present a linear stability analysis of physics-informed random projection neural networks (PI-RPNNs), for the numerical solution of {the initial value problem (IVP)} of (stiff) ODEs. We begin by proving that PI-RPNNs are uniform…
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a…
We present a method for solving linear and nonlinear PDEs based on the variable projection (VarPro) framework and artificial neural networks (ANN). For linear PDEs, enforcing the boundary/initial value problem on the collocation points…
Background: Deep learning techniques, particularly neural networks, have revolutionized computational physics, offering powerful tools for solving complex partial differential equations (PDEs). However, ensuring stability and efficiency…
Physics-informed neural networks based on automatic differentiation (AD-PINNs) and their finite-difference counterparts (FD-PINNs) are widely used for solving partial differential equations (PDEs), yet their analytical properties remain…
Physics-Informed Neural Networks (PINNs) are machine learning tools that approximate the solution of general partial differential equations (PDEs) by adding them in some form as terms of the loss/cost function of a Neural Network. Most…
This research introduces an extended application of neural networks for solving nonlinear partial differential equations (PDEs). A neural network, combined with a pseudo-arclength continuation, is proposed to construct bifurcation diagrams…
Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs). PINNs are based on simple architectures, and learn the behavior of complex…
Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks…
Physics-informed neural networks (PINNs) often exhibit weight matrices that appear statistically random after training, yet their implications for signal propagation and stability remain unsatisfactorily understood, let alone the…
The recently developed physics-informed machine learning has made great progress for solving nonlinear partial differential equations (PDEs), however, it may fail to provide reasonable approximations to the PDEs with discontinuous…
Motivated by recent research on Physics-Informed Neural Networks (PINNs), we make the first attempt to introduce the PINNs for numerical simulation of the elliptic Partial Differential Equations (PDEs) on 3D manifolds. PINNs are one of the…
The present study investigates the dynamics of nonlocal beams by establishing a consistent stress-driven integral elastic using the Physics-Informed Neural Network (PINN) approach. Specifically, a PINN is developed to compute the first…
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…
The Lane-Emden equation, a nonlinear second-order ordinary differential equation, plays a fundamental role in theoretical physics and astrophysics, particularly in modeling the structure of stellar interiors. Also referred to as the…
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…
Physics-Informed Neural Networks (PINNs) are a novel computational approach for solving partial differential equations (PDEs) with noisy and sparse initial and boundary data. Although, efficient quantification of epistemic and aleatoric…
Partial Differential Equations (PDEs) are notoriously difficult to solve. In general, closed-form solutions are not available and numerical approximation schemes are computationally expensive. In this paper, we propose to approach the…
Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive…