Towards solving General Relativity with Physics-Informed Neural Networks
Physics-Informed Neural Networks (PINNs) are a machine-learning framework for approximating solutions to systems of partial differential equations by constraining neural networks to satisfy the underlying physical laws. The resulting continuous representation does not require a predefined computational mesh and can be evaluated at arbitrary points within the training domain. In this work, we investigate the application of PINNs to Einstein's equations of General Relativity. We discuss the mathematical formulation, network architectures, loss functions, and training strategies used to obtain accurate spacetime solutions, and demonstrate the approach through a series of standard benchmarks from Numerical Relativity. We then extend the method to a more astrophysical setting by considering the dynamical evolution of an isolated solitonic boson star. Our simulations show that a PINN operating directly in three-dimensional Cartesian coordinates can reproduce the equilibrium structure of a highly compact boson star and recover its characteristic radial oscillation frequencies. These results establish PINNs as a viable complementary framework for solving coupled matter-Einstein equations and exploring alternative formulations in numerical relativity.