Solving higher-order Lane-Emden-Fowler type equations using physics-informed neural networks: benchmark tests comparing soft and hard constraints

In this paper, numerical methods using Physics-Informed Neural Networks (PINNs) are presented with the aim to solve higher-order ordinary differential equations (ODEs). Indeed, this deep-learning technique is successfully applied for solving different classes of singular ODEs, namely the well known second-order Lane-Emden equations, third order-order Emden-Fowler equations, and fourth-order Lane-Emden-Fowler equations. Two variants of PINNs technique are considered and compared. First, a minimization procedure is used to constrain the total loss function of the neural network, in which the equation residual is considered with some weight to form a physics-based loss and added to the training data loss that contains the initial/boundary conditions. Second, a specific choice of trial solutions ensuring these conditions as hard constraints is done in order to satisfy the differential equation, contrary to the first variant based on training data where the constraints appear as soft ones. Advantages and drawbacks of PINNs variants are highlighted.

Solving differential equations using physics informed deep learning: a hand-on tutorial with benchmark tests

We revisit the original approach of using deep learning and neural networks to solve differential equations by incorporating the knowledge of the equation. This is done by adding a dedicated term to the loss function during the optimization procedure in the training process. The so-called physics-informed neural networks (PINNs) are tested on a variety of academic ordinary differential equations in order to highlight the benefits and drawbacks of this approach with respect to standard integration methods. We focus on the possibility to use the least possible amount of data into the training process. The principles of PINNs for solving differential equations by enforcing physical laws via penalizing terms are reviewed. A tutorial on a simple equation model illustrates how to put into practice the method for ordinary differential equations. Benchmark tests show that a very small amount of training data is sufficient to predict the solution when the non linearity of the problem is weak. However, this is not the case in strongly non linear problems where a priori knowledge of training data over some partial or the whole time integration interval is necessary.