SEM: a shallow energy method for finite deformation hyperelasticity problems

In existing works, the deep energy method (DEM) is developed based on deep neural networks to handle partial differential equation problems of finite deformation hyperelasticity. Inspired by the DEM, we propose a shallow energy method (SEM) which utilizes simple and efficient radial basis function neural networks (RBFNNs) with only one hidden layer as the approximator of the continuous displacement fields of the investigated structures. In our method, the energy items, including the internal energy and the external energy, are calculated through numerical integration techniques and form the overall potential energy functional with respect to the parameters of the RBFNNs. Then, the minimization problem of the potential energy is carried out by the L-BFGS algorithm embedded in Python optimizer packages. Finally, the numerical approximation of the true displacement field is yielded and compared to those obtained by the DEM and the FEM, to illustrate the validity, the accuracy and other good performance of the proposed SEM in dealing with problems of finite deformation hyperelasticity.