发表机构
Hohai University; Bauhaus-University Weimar（河海大学; 包豪斯大学魏玛）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
KernelOnet通过显式核函数构建可解释神经算子，提供数据驱动、物理信息和混合三种核，在基准与波导问题上以更少参数实现高精度，并支持无监督训练与高效推理。
AI 中文摘要
本文提出了一种可解释的神经算子框架——核算子网络（KernelOnet），该框架将核函数显式地融入神经算子架构，使得算子结构与边界型核展开方法中使用的核展开形式相匹配。与DeepONet等传统神经算子不同，后者通过深度网络隐式学习基函数，而KernelOnet用显式核替代了主干网络，并提供三种互补的核：一种数据驱动的可学习核，其中神经网络参数化从数据中学习的径向基函数，对于常系数线性问题，该核可视为非奇异基本解；一种物理信息核，将解析基本解等物理信息嵌入网络结构，使得展开自动满足控制方程，并且可以仅基于边界条件进行无监督训练，无需内部解数据；以及一种混合核，根据控制方程的线性主部将解分解为由解析基本解张成的齐次部分和由低秩学习修正核承载的源项部分，从而在缺乏解析基本解的非线性问题上平衡物理先验与数据拟合。在三个基准测试和一个浅水波导工程问题中，KernelOnet取得了高精度；在与DeepONet可比较的情况下，它以更少的可学习参数实现了更高的精度。其无监督配置不需要内部解标签，且其每次查询的推理成本远低于逐实例求解器，为通用神经算子难以处理的无界外部域声传播提供了一条有效途径。
英文摘要
This paper proposes an interpretable neural operator framework, the Kernel Operator Network (KernelOnet), which incorporates kernel functions explicitly into the neural operator architecture, so that the operator structure matches the kernel-expansion form used in boundary-type kernel-expansion methods. Unlike traditional neural operators such as DeepONet, which learn basis functions implicitly through deep networks, KernelOnet replaces the trunk network with explicit kernels and offers three complementary kernels: a data-driven learnable kernel, in which a neural network parameterizes a radial basis function learned from data, and which for constant-coefficient linear problems can be regarded as a non-singular fundamental solution; a physics-informed kernel, which embeds physical information such as analytic fundamental solutions into the network structure, so that the expansion satisfies the governing equation automatically and can be trained without supervision on boundary conditions alone, with no interior solution data; and a hybrid kernel, which splits the solution, according to the linear principal part of the governing equation, into a homogeneous part spanned by analytic fundamental solutions and a source part carried by low-rank learned correction kernels, thereby balancing physical priors against data fitting on nonlinear problems lacking an analytic fundamental solution. On three benchmarks and one engineering problem in a shallow-water waveguide, KernelOnet attains high accuracy; where comparable with DeepONet, it is more accurate with fewer learnable parameters. Its unsupervised configuration needs no interior solution labels, and its per-query inference cost is far below that of per-instance solvers, offering an effective route to acoustic propagation in unbounded exterior domains that general-purpose neural operators struggle to handle.
