PIP<sup>2</sup> Net: Physics-informed partition penalty deep operator network

Hongjin Mi; Huiqiang Lun; Changhong Mou; Yeyu Zhang; Hongjin Mi; Huiqiang Lun; Changhong Mou; Yeyu Zhang

doi:10.3934/era.2026090

Electronic Research Archive

2026, Volume 34, Issue 3: 2009-2037. doi: 10.3934/era.2026090

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PIP² Net: Physics-informed partition penalty deep operator network

1.
School of Mathematics, Shanghai University of Finance and Economics, No. 777 Guoding Road, Shanghai 200433, China
2.
Faculty of Liberal Arts and Professional Studies, York University, 4700 Keele St, North York, ON M3J1P3, Canada
3.
Department of Mathematics and Statistics, Utah State University, 900 Old Main Hill, Logan, UT 84322, USA

Received: 16 December 2025 Revised: 01 February 2026 Accepted: 12 February 2026 Published: 05 March 2026

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.
- deep operator networks,
- physics-informed learning,
- partition-of-unity,
- nonlinear PDEs,
- spatiotemporal dynamics
Citation: Hongjin Mi, Huiqiang Lun, Changhong Mou, Yeyu Zhang. PIP2 Net: Physics-informed partition penalty deep operator network[J]. Electronic Research Archive, 2026, 34(3): 2009-2037. doi: 10.3934/era.2026090

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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 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.

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