Solving PDEs via learnable quadrature
Keywords: neural network, quadrature rule, PINN
Abstract: Partial differential Equations (PDEs) are an essential tool across science and engineering. Recent work has shown how contemporary developments in machine learning models can directly help in improving methods for solution discovery of PDEs. This line of work falls under the umbrella of Physics-Informed Machine Learning. A key step in solving a PDE is to determine a set of points in the domain where the current iterate of the PDE's solution will be evaluated. The most prevalent strategy here is to use Monte Carlo sampling, but it is widely known to be sub-optimal in lower dimensions. 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 in solving PDEs via neural networks, relative to existing alternatives, on a set of problems commonly 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 a simple use of hyper-networks, a broadly useful capability.
Primary Area: other topics in machine learning (i.e., none of the above)
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Submission Number: 5409
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