Solving Singular Liouville Equations Using Deep Learning

Deep learning has been applied to solving high-dimensional PDEs and successfully breaks the curse of dimensionality. However, it has barely been applied to finding singular solutions to certain PDEs, whose boundary conditions are absent and singular behavior is not a priori known. In this paper, we treat one example of such equations, the singular Liouville equations, which naturally arise when studying the celebrated Einstein equation in general relativity by using deep learning. We introduce a method of jointly training multiple deep neural networks to dynamically learn the singular behaviors of the solution and successfully capture both the smooth and singular parts of such equations.