PINNs with Learnable Quadrature
Sourav Pal, Kamyar Azizzadenesheli, Vikas Singh
The growing body of work on Physics-Informed Neural Networks (PINNs) seeks to use machine learning strategies to improve methods for solution discovery of Partial Differential Equations (PDEs). While classical solvers may remain the preferred tool of choice in the short-term, PINNs can be viewed as complementary. The expectation is that in some specific use cases, they can even be effective, standalone. A key step in training PINNs is selecting domain points for loss evaluation, where Monte Carlo sampling remains the dominant but often suboptimal in low dimension settings, common in physics. We leverage recent advances in asymptotic expansions of quadrature nodes and weights (for weight functions belonging to the modified Gauss-Jacobi family) together with suitable adjustments for parameterization towards a data-driven framework for learnable quadrature rules. A direct benefit is a performance improvement of PINNs, relative to existing alternatives, on a wide range of problems  studied in the literature. Beyond finding a standard solution for an instance of a single PDE, our construction enables learning rules to predict solutions for a given family of PDEs via hyper-networks, a useful capability for PINNs.