Finite element interpolated neural networks for solving forward and inverse problems

We propose a general framework for solving forward and inverse problems constrained by partial differential equations, where we interpolate neural networks onto finite element spaces to represent the (partial) unknowns. The framework overcomes the challenges related to the imposition of boundary conditions, the choice of collocation points in physics-informed neural networks, and the integration of variational physics-informed neural networks. A numerical experiment set confirms the framework's capability of handling various forward and inverse problems. In particular, the trained neural network generalises well for smooth problems, beating finite element solutions by some orders of magnitude. We finally propose an effective one-loop solver with an initial data fitting step (to obtain a cheap initialisation) to solve inverse problems

AI-augmented stabilized finite element method

An artificial intelligence-augmented Streamline Upwind/Petrov-Galerkin finite element scheme (AiStab-FEM) is proposed for solving singularly perturbed partial differential equations. In particular, an artificial neural network framework is proposed to predict optimal values for the stabilization parameter. The neural network is trained by minimizing a physics-informed cost function, where the equation's mesh and physical parameters are used as input features. Further, the predicted stabilization parameter is normalized with the gradient of the Galerkin solution to treat the boundary/interior layer region adequately. The proposed approach suppresses the undershoots and overshoots in the stabilized finite element solution and outperforms the existing neural network-based partial differential equation solvers such as Physics-Informed Neural Networks and Variational Neural Networks.Comment: 23 pages, 5 figures and 8 table

A Study of the Local Deep Galerkin Method for the Modified Cahn Hilliard Equation

Solving higher order partial differential equations (PDEs) can often prove to be a challenging task due to the involvement of higher-order derivatives of the unknown function, particularly for complex problems. The higher the order of the PDE, the more challenging it becomes to obtain an analytical solution. In such cases, alternative numerical methods are often used, such as finite element method or finite difference method. However, these methods can be computationally expensive and require a significant amount of mathematical expertise to implement. In recent times, there has been significant progress in applying neural networks to various fields, including the solution of physical problems that are governed by PDEs and their corresponding constraints. To achieve this, various techniques have been developed for solving PDEs using neural networks which include the use of deep neural networks, residual neural networks, and autoencoder-based neural networks like Physics-Informed Neural Networks (PINNs) and Physics-Constrained Generative Adversarial Networks (PC-GANs). For my thesis, I will be solving a 2D(Space)+1(Time) 4th order PDE, known as the Cahn Hilliard (CH) Equation, through the utilization of a local deep Galerkin method (LDGM). This method involves introducing intermediate variables and reformulating the fourth order PDE into a system of first-order equations. Therefore, this approach avoids the need for calculating high-order derivatives, making it a more efficient method. Then, a multi-output deep neural network is used to approximate the intermediate variables as well as the solutions to the PDE

Massive finite element computations for deep learning inversion

We focus on the inversion of borehole resistivity measurements in real time. To perform this task, we propose the use of Deep Learning methods. One critical task on this en- deavor is to produce a large database that can be used to train Deep Neural Networks. In this work, we explore different venues to obtain such database conforming the ground truth data via massive finite element computer simulations of the so-called forward prob- lem. This consists of solving multiple Boundary Value Problems governed by a Partial Differential Equation with different material coefficients. After describing the Finite Ele- ment Method, we investigate a venue to achieve high performance for performing a large amount of simulations using a Fourier approximation based Finite Element Method. The idea is to exploit the orthogonality of Fourier basis functions under reasonable assumptions often satisfied in our geophysics applications to reduce the computational cost of building the corresponding systems of linear equations. Solving such systems requires the use of advanced iterative solvers, which will be analyzed during the Ph.D. studies of Carlos Uri- arte

Massive finite element computations for deep learning inversion

We focus on the inversion of borehole resistivity measurements in real time. To perform this task, we propose the use of Deep Learning methods. One critical task on this en- deavor is to produce a large database that can be used to train Deep Neural Networks. In this work, we explore different venues to obtain such database conforming the ground truth data via massive finite element computer simulations of the so-called forward prob- lem. This consists of solving multiple Boundary Value Problems governed by a Partial Differential Equation with different material coefficients. After describing the Finite Ele- ment Method, we investigate a venue to achieve high performance for performing a large amount of simulations using a Fourier approximation based Finite Element Method. The idea is to exploit the orthogonality of Fourier basis functions under reasonable assumptions often satisfied in our geophysics applications to reduce the computational cost of building the corresponding systems of linear equations. Solving such systems requires the use of advanced iterative solvers, which will be analyzed during the Ph.D. studies of Carlos Uri- arte

Artificial Neural Network Methods in Quantum Mechanics

In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for differential and integrodifferential operators, using ANNs. We start by considering the Schr\"odinger equation for the Morse potential that has an analytically known solution, to test the accuracy of the method. We then proceed with the Schr\"odinger and the Dirac equations for a muonic atom, as well as with a non-local Schr\"odinger integrodifferential equation that models the $n+\alpha$ system in the framework of the resonating group method. In two dimensions we consider the well studied Henon-Heiles Hamiltonian and in three dimensions the model problem of three coupled anharmonic oscillators. The method in all of the treated cases proved to be highly accurate, robust and efficient. Hence it is a promising tool for tackling problems of higher complexity and dimensionality.Comment: Latex file, 29pages, 11 psfigs, submitted in CP