PIKFNO：一种基于物理信息核函数的可解释神经算子
PIKFNO: An Interpretable Neural Operator Based on Physics Informed Kernel Function
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中文总结 AI 辅助
本研究提出PIKFNO框架，通过物理信息核函数约束主干网络，采用两种核函数构建策略，在有限训练数据下提升了神经算子的预测精度、可解释性与泛化性能，为开发高性能可解释神经算子提供新路径。
中文摘要 AI 辅助
本研究提出了一种新的可解释神经算子框架，称为物理信息核函数神经算子（PIKFNO），它将从控制方程推导而来的物理信息核函数明确整合到神经算子架构中。与DeepONet等传统神经算子依赖深度网络隐式学习基函数不同，PIKFNO通过物理信息核函数约束主干网络，使其算子结构与无网格配点法中使用的核展开式对齐。本文引入了两种构建策略：一种直接从数据中学习核函数，所学习的核可视为非奇异基本解；另一种通过解析基本解的变换来构建核函数。数值实验表明，PIKFNO在有限训练数据下可实现高预测精度，同时显著提升了可解释性和泛化性能。该框架为开发高效、物理一致性强且可解释的神经算子提供了新途径。
英文摘要
This work proposes a new interpretable neural operator framework, termed the Physics Informed Kernel Function Neural Operator (PIKFNO), which explicitly incorporates physics informed kernel functions derived from governing equations into the neural operator architecture. Unlike traditional neural operators such as DeepONet, which rely on deep networks to implicitly learn basis functions, PIKFNO constrains the trunk network through physics informed kernel functions, thereby aligning its operator structure with the kernel expansions used in meshless collocation methods. Two construction strategies are introduced: one learns kernel functions directly from data, where the learned kernel can be regarded as a nonsingular fundamental solution, while the other builds them through transformations of analytical fundamental solutions. Numerical experiments demonstrate that PIKFNO achieves high predictive accuracy with substantially improved interpretability and superior generalization under limited training data. The proposed framework offers a new pathway for developing efficient, physically consistent, and interpretable neural operators.
