An Experimental Analysis of effects of different Training Methods for the Accuracy and the Computational Cost of PINNs
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The recent developments in numerical methods regard the employment of mesh-free approaches for solving Partial Differential Equations and find their best application for irregular domains that usually describe real applications. Thinking about the cultural heritage field, the possibility to apply mash-free approaches allows the exploitation of mathematical models for cultural assets digital twins for achieving predictive maintenance. In this field, the application of mesh-free approaches permits the employment of points acquired from the cultural assets digitalization, according to their irregular form, for solving Partial Differential Equations related to the considered physical phenomena. In particular, this work aims to analyse the effects of different training algorithms on the solution of PDE through Physics-Informed Neural Networks (PINNs). In fact, for real applications, employed training methods can influence the accuracy and the time for identifying the solution. Therefore, during this work, results obtained through Gradient Descent, Stochastic Gradient Descent, Adam, and L-BFGS methods will be evaluated on academic problems, studying the accuracy and the computation cost. For this purpose, the study requires, on one side, the variation of the number of layers and neurons of the network and, on the other side, the variation of the cardinality of the training set.
