Parsimonious Universal Function Approximator for Elastic and Elastoplastic Cavity Expansion Problems

Physics-informed neural networks (PINNs) have prevailed as an effective universal function approximator to solve a wide range of governing partial differential equations (PDEs) and ordinary differential equations (ODEs), including the ones in geotechnical engineering. However, PINNs’ ability to solve elastoplastic problems remains obscure, especially for the cavity expansion problem, which is a canonical problem in geotechnics. The challenge is that the PDEs for elastoplasticity are highly nonlinear, and the corresponding loss functions contain multiple terms that are hard to balance, which may result in gradient pathologies of PINNs and even failure of convergence. Thus, we enhance PINNs with a novel parsimonious loss function to handle the expanded cavity problem considering elastic and elastoplastic constitutive models. This novel parsimonious loss function is first proposed to balance the simplicity and accuracy of PINNs. The proposed method is applied to diverse material behaviors in the cavity expansion problem including isotropic, anisotropic elastic media, and elastic-perfectly plastic media with Tresca and Mohr-Coulomb yield criteria. The results indicate that the use of a parsimonious prior information-based loss function is highly beneficial to deriving the approximate solutions of complex PDEs with high accuracy. The present method allows for accurate derivation of solutions for both elastic and plastic mechanical responses of an expanded cavity. It also provides insights into how PINNs can be further advanced to solve more complex problems in geotechnical practice.