Approximation of One-Dimensional Darcy–Brinkman–Forchheimer Model by Physics Informed Deep Learning Feedforward Artificial Neural Network and Finite Element Methods: A Comparative Study - Martinez, Mara; Rao, B. Veena S. N.; Mallikarjunaiah, S. M. (International journal of applied and computational mathematics, 2024)
In the last few years, a new research program such as deep learning neural networks (or simulated neural networks )—a class of machine learning algorithms—has gained a lot of attention due to its applicability in various science and engineering fields. From the numerical methods point of view, in recent years new approximation methods for the solution of differential equations based on the supervised/unsupervised machine learning algorithms have been widely implemented. Since such machine learning algorithms are superior for any optimization problem compared to any traditional mesh-based approximation methods. For a specific choice of the total loss function —an l 2 -type mean square function—we propose an unsupervised feedforward deep learning artificial neural network framework to approximate the solution to a one-dimensional Darcy–Brinkman–Forchheimer (DBF) porous medium model. In the present work, we address the issue of choosing hyperparameters based on the learning and efficacy of approximating the physics-informed solution. An issue which was largely ignored in the literature. We also implement continuous Galerkin-type finite elements for the solution of a sequence of linear problems which were obtained from the Newton’s method by linearizing the nonlinear boundary value problem (BVP). A comparative investigation is performed between the neural networks and Galerkin finite element methods. A detailed sensitivity analysis is done on the accuracy of the numerical solution concerning the size of the training data set, the number of hidden layers, the number of neurons in each layer, the activation functions, and the learning rate. We observe that the optimal network architecture can approximate the solution to the nonlinear model with a small data set while to achieve the same accuracy the traditional numerical methods demands a bigger set of discrete points in the domain.