Solving Hamiltonian constraint equation with physics-informed neural networks
Phys. Rev. D 114, 023032 – Published 17 July, 2026
DOI: https://doi.org/10.1103/619s-p6md
Abstract
Numerical relativity (NR), solving Einstein equations numerically, plays an important role in source modeling for gravitational wave astronomy. Traditional methods for NR including the finite difference method, spectral method, and finite element method have been well developed. But newly developed neural network methods for partial differential equations (PDE) have not been well studied yet for NR. We present a physics-informed neural network (PINN) method to solve the Hamiltonian constraint equation for binary black hole (BBH) initial data in NR. This equation is a highly nonlinear elliptic PDE, posing significant challenges for conventional PINN approaches. To overcome these difficulties, we introduce a set of new techniques. We show that our PINN together with these techniques can successfully solve the Hamiltonian constraint equation for generic BBH systems. Validation against the traditional results demonstrates the high accuracy and robustness of our method, revealing the immense potential of constructing a PINN-based initial data solution to all BBH systems for NR.
