Pseudo-Physics-Informed Neural Operators: Enhancing Operator Learning from Limited Data

TL;DR

This work tackles data scarcity in neural operator learning by introducing Pseudo Physics-Informed Neural Operators (PPI-NO), which learns a data-driven pseudo PDE phi from limited observations and couples it with a neural operator through an alternating optimization scheme. The learned phi approximates local PDE structure using u and its derivatives, facilitated by a neighborhood-aware convolution to mitigate discretization errors, and is used to reconstruct f from the operator's output to regularize training. Across five benchmark tasks and a fatigue modeling application, PPI-NO achieves substantial accuracy gains over standard neural operators, with ablations confirming the value of phi's design choices and the derivative order. The approach broadens physics-informed machine learning to scenarios where detailed physics are unknown, offering a practical path to improved surrogate modeling with limited data while incurring modest memory overhead. In future work, the authors plan to extend the framework to temporal dynamics and initial-condition settings to widen applicability.

Abstract

Neural operators have shown great potential in surrogate modeling. However, training a well-performing neural operator typically requires a substantial amount of data, which can pose a major challenge in complex applications. In such scenarios, detailed physical knowledge can be unavailable or difficult to obtain, and collecting extensive data is often prohibitively expensive. To mitigate this challenge, we propose the Pseudo Physics-Informed Neural Operator (PPI-NO) framework. PPI-NO constructs a surrogate physics system for the target system using partial differential equations (PDEs) derived from simple, rudimentary physics principles, such as basic differential operators. This surrogate system is coupled with a neural operator model, using an alternating update and learning process to iteratively enhance the model's predictive power. While the physics derived via PPI-NO may not mirror the ground-truth underlying physical laws -- hence the term ``pseudo physics'' -- this approach significantly improves the accuracy of standard operator learning models in data-scarce scenarios, which is evidenced by extensive evaluations across five benchmark tasks and a fatigue modeling application.

Figures (9)

• Figure 1: The illustration of "pseudo" physics representation network $\phi$. The input consists of $u$ and its finite difference derivative approximations ($\{\hat{S}_1(u), \ldots, \hat{S}_Q(u)\}$) across different sampling locations. The top row shows a convolution layer that aggregates local neighborhood to compensate the information loss caused by finite difference. The bottom row shows that $\phi$ uses fully connected layers at each sampling location to combine $u$ and its derivatives locally to predict $f$ at the same location.
• Figure 2: PPI-NO learning framework.
• Figure 3: Learning curve of PPI-FNO on Darcy Flow (a), and of PPI-DONet on nonlinear diffusion (b). In (c) and (d) we show how the weight $\lambda$ of "pseudo physics" affects the operator learning performance. The horizontal line in (c) and (d) are the relative $L_2$ errors of standard FNO and DONet.
• Figure 4: Example of semi-elliptic surface crack on a plates merrell2024stress.
• Figure 5: Examples of SIF prediction of FNO and PPI-FNO trained with 600 examples.