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