Jan Trynda , Paweł Maczuga , Albert Oliver-Serra , Luis Emilio García-Castillo , Robert Schaefer , Maciej Woźniak
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An h-adaptive collocation method for Physics-Informed Neural Networks
Despite their flexibility and success in solving partial differential equations, Physics-Informed Neural Networks (PINNs) often suffer from convergence issues, even failing to converge, particularly in problems with steep gradients or localized features. Several remedies have been suggested to solve this problem, but one of the most promising is the dynamical adaptation of the collocation points. This paper explores a novel adaptive sampling method, of a stochastic nature, based on the Adaptive Mesh Refinement used in the Finite Element Method. The error estimates in our refinement algorithm are based on the value of the residual loss function. We tested our method against a variety of 1D and 2D benchmark problems that exhibit steep gradients near certain boundaries, with promising results.
期刊介绍:
Computational Science is a rapidly growing multi- and interdisciplinary field that uses advanced computing and data analysis to understand and solve complex problems. It has reached a level of predictive capability that now firmly complements the traditional pillars of experimentation and theory.
The recent advances in experimental techniques such as detectors, on-line sensor networks and high-resolution imaging techniques, have opened up new windows into physical and biological processes at many levels of detail. The resulting data explosion allows for detailed data driven modeling and simulation.
This new discipline in science combines computational thinking, modern computational methods, devices and collateral technologies to address problems far beyond the scope of traditional numerical methods.
Computational science typically unifies three distinct elements:
• Modeling, Algorithms and Simulations (e.g. numerical and non-numerical, discrete and continuous);
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• Computer and information science that develops and optimizes the advanced system hardware, software, networking, and data management components (e.g. problem solving environments).