Operator learning augmented physics-informed neural networks for partial differential equations exhibiting sharp features
Phys. Rev. E 113, 035306 – Published 9 March, 2026
DOI: https://doi.org/10.1103/xyjy-gx6f
Abstract
In this work, we develop an operator learning augmented physics-informed neural network (OL-PINN) framework for solving partial differential equations with sharp solution features and singular behaviors. The method combines the standard PINN architecture with a pretrained deep operator network (DeepONet), which is trained on a related class of smooth problems and serves as an operator prior to provide reliable extrapolations. This integration acts as a regularization mechanism that enhances stability, improves efficiency, and reduces the reliance on large numbers of residual points. The proposed OL-PINN is systematically validated on several benchmark problems, including the nonlinear diffusion–reaction equation, Burgers equation, lid-driven cavity flow, and the two-dimensional Navier-Stokes equations in vorticity form. Numerical experiments demonstrate that OL-PINN achieves higher predictive accuracy and more stable convergence than vanilla PINNs and DeepONet extrapolations, even when trained with only a small number of residual points. Importantly, OL-PINN proves effective in addressing ill-posed problems with incomplete boundary conditions, which are intractable for classical numerical methods. Furthermore, OL-PINN is able to predict new small-scale flow structures, such as emergent vortices at higher Reynolds numbers, highlighting its potential as a promising tool for simulating turbulent or near-turbulent flows.
