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# PIFNO-LAW: Learned Adaptive Weighting for Physics-Informed Fourier Neural Operators
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Official dataset and documentation for **PIFNO-LAW**, a novel dual-operator framework designed to solve partial differential equations (PDEs) with discontinuous solutions (shocks) under limited-data scenarios. This repository focuses on the open-source **Burgers' Equation** benchmarks.
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## 📌 Overview
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Physics-informed Fourier neural operators (PIFNO) show great promise for scientific machine learning but frequently struggle with sharp discontinuities. High localized residuals at shock fronts tend to destabilize training. **PIFNO-LAW** overcomes this limitation by introducing an auxiliary Fourier Neural Operator that learns an optimal, dynamic weight field for the physics loss.
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### Key Innovations
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- **Dual-Operator Design**: An auxiliary FNO, conditioned on the local solution and its derivatives, identifies shocks dynamically.
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- **Dual-Penalty Regularization**: A novel objective that maintains a robust penalty in smooth regions while proactively suppressing chaotic residuals at shocks, replacing unstable minimax/adversarial training paradigms.
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- **Data Efficiency**: Specifically designed for robust "few-shot" learning where high-fidelity simulation data is sparse.
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## 📊 Dataset Description
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- `x`: Spatial grid
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- `t`: Temporal grid
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- `u`: Scalar solution field
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## 📖 Citation
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If you use this dataset or the PIFNO-LAW architecture in your research, please cite our study:
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```bibtex
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@article{Hu2026PIFNOLAW,
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title={Learned Adaptive Weighting for Physics-Informed Fourier Neural Operators: Solving Discontinuous PDEs based on Limited Data},
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author={Hu, Xin and An, Bo and Guan, Yongke and Xu, Liang and Yu, Min and Li, Dong},
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journal={Unpublished},
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year={2026}
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}
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```
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## 📊 Dataset Description
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- `x`: Spatial grid
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- `t`: Temporal grid
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- `u`: Scalar solution field
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