Investigating Guiding Information for Adaptive Collocation Point Sampling in PINNs
arxiv(2024)
摘要
Physics-informed neural networks (PINNs) provide a means of obtaining
approximate solutions of partial differential equations and systems through the
minimisation of an objective function which includes the evaluation of a
residual function at a set of collocation points within the domain. The quality
of a PINNs solution depends upon numerous parameters, including the number and
distribution of these collocation points. In this paper we consider a number of
strategies for selecting these points and investigate their impact on the
overall accuracy of the method. In particular, we suggest that no single
approach is likely to be “optimal” but we show how a number of important
metrics can have an impact in improving the quality of the results obtained
when using a fixed number of residual evaluations. We illustrate these
approaches through the use of two benchmark test problems: Burgers' equation
and the Allen-Cahn equation.
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