A physics-energy dual-informed neural network with one-shot learning

The strong-form collocation point method (CPM) can effectively minimize the residuals within the computational domain, but it is sensitive to boundary conditions. The deep Ritz method (DRM), based on the minimum potential energy principle, reduces dependence on boundary conditions but suffers from limited convergence stability. To balance convergence efficiency and boundary adaptability, this study proposes a physics-energy dual-informed neural network (PEDINN) with one-shot learning. This approach constructs and minimizes a mixed loss function that integrates governing-equation residuals with the system energy functional, thereby effectively combining the stable convergence capability of CPM with the boundary adaptability of DRM. Furthermore, the load is taken as an input parameter of the network. An energy-guided method is introduced by establishing regression relationships between the load and both the strain energy and external work, all extracted from one finite element method (FEM) solution sample. These relationships are embedded as soft constraints in the loss function. Case studies show that PEDINN enables rapid prediction for both in-sample and out-of-sample loading conditions. While maintaining high prediction accuracy, it significantly improves convergence efficiency, boundary adaptability, and engineering applicability.