PIP² Net: Physics-informed partition penalty deep operator network

Operator learning has become a powerful tool for accelerating the solution of parameterized partial differential equations (PDEs), enabling rapid prediction of full spatiotemporal fields for new initial conditions or forcing functions. Existing architectures such as the deep operator network (DeepONet) and the Fourier neural operator (FNO) show strong empirical performance, but often require large training datasets, lack explicit physical structure, and may suffer from instability in their trunk-network features, where mode imbalance or collapse can hinder accurate operator approximation. Motivated by the stability and locality of classical partition-of-unity (PoU) methods, we investigate PoU-based regularization techniques for operator learning and develop a revised formulation of the existing POU–PI–DeepONet framework. The resulting physics-informed partition penalty deep operator network (PIP² Net) introduces a simplified and more principled partition penalty that improves the coordinated trunk outputs, which leads to more expressiveness without sacrificing the flexibility of DeepONet. We evaluate PIP² Net on three nonlinear PDEs: the viscous Burgers equation, the Allen–Cahn equation, and a diffusion–reaction system. The results show that it consistently outperforms DeepONet, PI-DeepONet, and POU-DeepONet in prediction accuracy and robustness.