A Tutorial on the Use of Physics-Informed Neural Networks to Compute the Spectrum of Quantum Systems

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 Artificial Intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states.