PIP$^2$ Net: Physics-informed Partition Penalty Deep Operator Network

TL;DR

This paper introduces PIP$^2$ Net, a physics-informed deep operator network that uses partition penalties to improve stability, accuracy, and robustness in learning operators for nonlinear PDEs.

Contribution

It develops a novel partition penalty framework for DeepONet, enhancing stability and expressiveness while reducing data requirements for operator learning.

Findings

PIP$^2$ Net outperforms existing DeepONet variants in accuracy.

It demonstrates robustness across multiple nonlinear PDEs.

The method improves stability and expressiveness of operator learning.

Abstract

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 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 \emph{P}hysics-\emph{i}nformed \emph{P}artition \emph{P}enalty Deep Operator Network (PIP$^{2}$ Net) introduces a simplified and more principled partition penalty that improved the coordinated trunk outputs that leads to more expressiveness without sacrificing the flexibility of DeepONet. We evaluate PIP$^{2}$ 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.