Greedy training algorithms for neural networks and applications to PDEs
نویسندگان
چکیده
Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods proven remarkably successful on practical engineering problems, they not shown, theoretically or empirically, to converge the underlying PDE solution with arbitrarily high accuracy. The primary difficulty lies in highly non-convex optimization problems resulting from network discretization, which are difficult treat both and practically. It is our goal this work take a step toward remedying this. For purpose, we develop novel greedy training algorithm shallow networks. Our method applicable variational formulation of also residual minimization pioneered by physics informed (PINNs). We analyze obtain priori error bounds when PDEs function class defined networks, rigorously establishes convergence as size increases. Finally, test several benchmark examples, including dimensional PDEs, confirm theoretical rate. expensive relative traditional approaches finite element methods, view proof concept network-based shows that numerical based upon can be shown converge.
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ژورنال
عنوان ژورنال: Journal of Computational Physics
سال: 2023
ISSN: ['1090-2716', '0021-9991']
DOI: https://doi.org/10.1016/j.jcp.2023.112084