Extension of the Finite-Element Physics-Informed Neural Network (FE-PINN)

We present an extension to finite element–based physics-informed neural networks (FE-PINNs) that improves both computational efficiency and adaptability across problem classes. In prior work, we addressed one of two key limitations of the architecture by introducing a CUDA accelerated stencil-transformation algorithm that parallelizes the inverse isoparametric mapping, substantially reducing training time and memory overhead. In this work, we address the second limitation: universal adaptability of the residual-loss formulation. While the original FE-PINN residual evaluation was implemented for linear-elastic problem sets, we now present a Firedrake-based software package that defines residual losses directly from variational forms, enabling users to impose arbitrary constitutive laws and boundary conditions to generate new families of nonlinear problems. This establishes a general-purpose FE-PINN training pipeline in which user-imposed physics enters through changes to the weak form rather than redesigning the architecture, supporting ongoing work in nonlinear deformation and providing a path toward broader continuum mechanics extensions.