dNNsolve: an efficient NN-based PDE solver
  Neural Networks (NNs) can be used to solve Ordinary and Partial Differential
Equations (ODEs and PDEs) by redefining the question as an optimization
problem. The objective function to be optimized is the sum of the squares of
the PDE to be solved and of the initial/boundary conditions. A feed forward NN
is trained to minimise this loss function evaluated on a set of collocation
points sampled from the domain where the problem is defined. A compact and
smooth solution, that only depends on the weights of the trained NN, is then
obtained. This approach is often referred to as PINN, from Physics Informed
Neural Network~\cite{raissi2017physics_1, raissi2017physics_2}. Despite the
success of the PINN approach in solving various classes of PDEs, an
implementation of this idea that is capable of solving a large class of ODEs
and PDEs with good accuracy and without the need to finely tune the
hyperparameters of the network, is not available yet. In this paper, we
introduce a new implementation of this concept - called dNNsolve - that makes
use of dual Neural Networks to solve ODEs/PDEs. These include: i) sine and
sigmoidal activation functions, that provide a more efficient basis to capture
both secular and periodic patterns in the solutions; ii) a newly designed
architecture, that makes it easy for the the NN to approximate the solution
using the basis functions mentioned above. We show that dNNsolve is capable of
solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions, without
the need of hyperparameter fine-tuning.
