让神经算子突破训练数据限制，稳定处理未知函数。
Extending Neural Operators: Robust Handling of Functions Beyond the Training Set
- 用核逼近理论构建扩展框架，分析输入输出空间关系
- 理论证明可准确预测外推时的近似误差，支持导数捕捉
- 适合需泛化到新几何结构的PDE求解场景
我们建立了一个严格的框架，使神经算子能可靠处理分布外输入函数。通过核逼近技术，将输入输出函数空间刻画为再生核希尔伯特空间（RKHS），并给出理论保证：确保扩展的可靠性及预测近似精度。我们还建立了特定核选择与对应Sobolev原生空间之间的形式关系，使得扩展后的神经算子不仅能准确捕捉函数值，还能可靠计算其导数。方法在具有点云表示的流形上椭圆型偏微分方程求解中进行了实证验证，涵盖几何贡献建模。报告了影响扩展方法精度与计算性能的关键因素。
原文摘要 · Abstract (English)
We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of Reproducing Kernel Hilbert Spaces (RKHSs). We provide theorems on the requirements for reliable extensions and their predicted approximation accuracy. We also establish formal relationships between specific kernel choices and their corresponding Sobolev Native Spaces. This connection further allows the extended neural operators to reliably capture not only function values but also their derivatives. Our methods are empirically validated through the solution of elliptic partial differential equations (PDEs) involving operators on manifolds having point-cloud representations and handling geometric contributions. We report results on key factors impacting the accuracy and computational performance of the extension approaches.
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