Deep neural networks for solving extremely large linear systems
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In this paper, we study deep neural networks for solving extremely large linear systems arising from physically relevant problems. Because of the curse of dimensionality, it is expensive to store both solution and right hand side vectors in such extremely large linear systems. Our idea is to employ a neural network to characterize the solution with parameters being much fewer than the size of the solution. We present an error analysis of the proposed method provided that the solution vector can be approximated by the continuous quantity, which is in the Barron space. Several numerical examples arising from partial differential equations, queueing problems and probabilistic Boolean networks are presented to demonstrate that solutions of linear systems with sizes ranging from septillion (10^24) to nonillion (10^30) can be learned quite accurately.
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