Improving the accuracy of the deep energy method

Abstract

The deep energy method (DEM) employs the principle of minimum potential energy to train neural network models to predict displacement at a state of equilibrium under given boundary conditions. The accuracy of the model is contingent upon choosing appropriate hyperparameters. The hyperparameters have traditionally been chosen based on literature or through manual iterations. The displacements predicted using hyperparameters suggested in the literature do not ensure the minimum potential energy of the system. Additionally, they do not necessarily generalize to different load cases. Selecting hyperparameters through manual trial and error and grid search algorithms can be highly time-consuming. We propose a systematic approach using the Bayesian optimization algorithms and random search to identify optimal values for these parameters. Seven hyperparameters are optimized to obtain the minimum potential energy of the system under compression, tension, and bending loads cases. In addition to Bayesian optimization, Fourier feature mapping is also introduced to improve accuracy. The models trained using optimal hyperparameters and Fourier feature mapping could accurately predict deflections compared to finite element analysis for linear elastic materials. The deflections obtained for tension and compression load cases are found to be more sensitive to values of hyperparameters compared to bending. The approach can be easily extended to 3D and other material models.