用物理启发的核函数提升神经算子的可解释性与泛化能力
PIKFNO: An Interpretable Neural Operator Based on Physics Informed Kernel Function
- 将物理方程导出的核函数显式嵌入神经算子结构
- 数据量少时仍保持高精度与强泛化能力
- 适合需物理一致性与可解释性的科学计算场景
本文提出一种新型可解释神经算子框架——物理启发核函数神经算子（PIKFNO），将由控制方程导出的物理启发核函数显式融入神经算子架构。与依赖深度网络隐式学习基函数的传统神经算子（如DeepONet）不同，PIKFNO通过物理启发核函数约束主干网络，使其算子结构与无网格配置方法中的核展开一致。提出了两种构建策略：一是直接从数据学习核函数，所学核可视为非奇异基本解；二是通过对解析基本解进行变换构建。数值实验表明，PIKFNO在训练数据有限条件下仍实现高预测精度，显著提升可解释性并具备优越泛化能力。该框架为开发高效、物理一致且可解释的神经算子提供了新路径。
原文摘要 · Abstract (English)
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.
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