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Solving systems of linear equations is a problem occuring frequently in waterengineering applications. Usually the size of the problem is too large to besolved via direct factorization. One can resort to iterative approaches, inparticular the conjugate gradients method if the matrix is symmetric positivedefinite. Preconditioners further enhance the rate of convergence but hithertoonly handcrafted ones requiring expert knowledge have been used. We propose aninnovative approach employing Machine Learning, in particular a ConvolutionalNeural Network, to unassistedly design preconditioning matrices specificallyfor the problem at hand. Based on an in-depth case study in fluid simulation weare able to show that our learned preconditioner is able to improve theconvergence rate even beyond well established methods like incomplete Choleskyfactorization or Algebraic MultiGrid.

Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems