Physics-informed neural networks (PINNs) are a widely used tool in today's Machine Learning (ML) landscape. They consist of Neural Networks (NNs) that, during the training phase, learn to solve the differential equations governing the physical laws of a system in a model-independent way. When these differential equations correspond to Hamilton equations of motion, we refer to them as Hamiltonian Neural Networks (HNNs). The HNN paradigm was introduced in [1] and in the present work we closely follow the strategy proposed in [2]. The key advantages of HNNs over standard numerical integrators can be summarized as follows: the predicted solution is analytical in time and not limited to a discrete set of time steps; conservation laws, symmetries, constraints and prior knowledge of the system can be easily incorporated at the level of the architecture and of the loss function to improve the predictability of the HNN; the minimization process of the loss function occurs under the constraint that the solution satisfies the system of equations at all times simultaneously and independently, thus avoiding any iterative mechanism.

Reprinted with permission from Quanta Magazine's Abstractions blog. Despite its ability to bend both minds and space, an Einsteinian black hole looks so simple a child could draw it. The point is the singularity, an infinitely dense, unimaginably small dot contorting space so radically that anything nearby falls straight in, leaving behind a vacuum. The spherical boundary marks the event horizon, the point of no return between the vacuum and the rest of the universe. But according to Einstein's theory of gravity, the event horizon isn't anything that an unlucky astronaut would immediately notice if she were to cross it.