  Quantum many-body systems are of great interest for many research areas,
including physics, biology and chemistry. However, their simulation is
extremely challenging, due to the exponential growth of the Hilbert space with
the system size, making it exceedingly difficult to parameterize the wave
functions of large systems by using exact methods. Neural networks and machine
learning in general are a way to face this challenge. For instance, methods
like Tensor networks and Neural Quantum States are being investigated as
promising tools to obtain the wave function of a quantum mechanical system. In
this tutorial, we focus on a particularly promising class of deep learning
algorithms. We explain how to construct a Physics-Informed Neural Network
(PINN) able to solve the Schr\"odinger equation for a given potential, by
finding its eigenvalues and eigenfunctions. This technique is unsupervised, and
utilizes a novel computational method in a manner that is barely explored.
PINNs are a deep learning method that exploits Automatic Differentiation to
solve Integro-Differential Equations in a mesh-free way. We show how to find
both the ground and the excited states. The method discovers the states
progressively by starting from the ground state. We explain how to introduce
inductive biases in the loss to exploit further knowledge of the physical
system. Such additional constraints allow for a faster and more accurate
convergence. This technique can then be enhanced by a smart choice of
collocation points in order to take advantage of the mesh-free nature of the
PINN. The methods are made explicit by applying them to the infinite potential
well and the particle in a ring, a challenging problem to be learned by an AI
agent due to the presence of complex-valued eigenfunctions and degenerate
states.
