hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition

Abstract

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 of high-order polynomials. The trial space is the space of neural network, which is defined globally over the whole computational domain, while the test space contains the piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with local learning algorithm that can efficiently localize the network parameter optimization. We demonstrate the advantages of hp-VPINNs in accuracy and training cost for several numerical examples of function approximation and solving differential equations.

SciANN: A Keras/Tensorflow wrapper for scientific computations and physics-informed deep learning using artificial neural networks

“…Neural networks have recently been used to solve the variational form of differential equations as well [36,37]. In a recent study [27], the vPINN framework for solving PDEs was introduced and analyzed. Like PINN, it is based on graph-based automatic differentiation.…”

“…Like PINN, it is based on graph-based automatic differentiation. The authors of [27] suggest a Petrov-Galerkin approach, where the test functions are chosen differently from the trial functions. For the test functions, they propose the use of polynomials that vanish on the boundary of the domain.…”

“…We then illustrate SciANN's application to curve fitting, the solution of the Burgers equation, and the identification of the Navier-Stokes equations and the von Mises plasticity model from data. Lastly, we show how to use SciANN in the context of the variational PINN framework [27]. The examples discussed here and several additional applications are freely available at github.com/sciann/sciannapplications.…”

A Physics-Informed Neural Network Framework for PDEs on 3D Surfaces: Time Independent Problems

“…Further, the combination of PINN and adversarial networks can be used for uncertainty quantification (UQ) problems in PDE [13]. The finite element methods have been combined with the PINN to enhance its performance [16,17]. One can even solve the PDEs without the concrete form of it as proposed in [24].…”

Thermodynamically consistent physics-informed neural networks for hyperbolic systems

“…A number of works have considered the use of DNNs in the context of shock problems [26,20,31,32,33], and several works have considered using alternative discretizations in the context of PINNs-like methods, for example Ritz-Galerkin discretizations [1], Petrov-Galerkin methods [34], and mortar methods [35]. To our knowledge, this work marks the first attempt to assimilate traditional finite volume methodology to obtain a thermodynamically consistent treatment of inverse problems in shock physics.…”