What is it about?
Many problems in science and engineering require solving the same partial differential equations (PDEs) under different conditions. Operator learning has become a promising way to speed up this process by learning the mapping from input functions to their corresponding solutions. Existing operator learning methods have achieved encouraging results, but they can still become unstable when learning complex nonlinear problems. In this work, we introduce PIP² Net, a physics-informed operator learning framework that improves the stability of DeepONet by adding a simple partition penalty to the trunk network. This penalty encourages the model to learn more balanced internal representations while keeping the original architecture unchanged. We test PIP² Net on three representative nonlinear PDEs and compare it with several existing operator learning methods. The results show that PIP² Net provides consistently more accurate and more robust predictions across all benchmark problems.
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Why is it important?
Fast and reliable solutions of PDEs are needed in many scientific and engineering applications, where the same equations often need to be solved repeatedly. Improving the stability of operator learning models can make these computations more dependable while reducing the need for repeated numerical simulations. Our work shows that a simple structural constraint, together with physics-informed learning, can improve the stability and predictive performance of operator learning without changing the overall DeepONet framework. We hope PIP² Net provides a practical approach for building more reliable operator learning models for scientific computing.
Perspectives
One motivation behind this work was to better understand why operator learning models sometimes become unstable, even when physical information is incorporated during training. Rather than designing a completely new architecture, we wanted to explore whether a simple structural idea could improve how these models learn and represent complex physical systems. One thing we found particularly interesting was that a partition penalty inspired by classical numerical analysis consistently improved performance across different nonlinear PDEs while leaving the overall DeepONet framework unchanged. This reinforced our view that carefully designed structural regularization can play an important role in operator learning. For us, one of the most rewarding aspects of this project was seeing that a simple idea could strengthen an existing operator learning framework without making it more complicated.
huiqiang lun
Shanghai University of Finance and Economics
Read the Original
This page is a summary of: PIP<sup>2</sup> Net: Physics-informed partition penalty deep operator network, Electronic Research Archive, January 2026, Tsinghua University Press,
DOI: 10.3934/era.2026090.
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