用物理约束的分块有理柯尔莫戈洛夫网络，高效求解时变偏微分方程。
PO-CKAN:Physics Informed Deep Operator Kolmogorov Arnold Networks with Chunk Rational Structure
- 分块有理柯尔莫戈洛夫网络提升函数逼近能力
- 在布辛涅斯克方程上误差降低48%以上
- 适合需快速预测参数化方程的科研与工程场景
我们提出PO-CKAN，一种基于分块有理柯尔莫戈洛夫-阿诺德网络（CKAN）的物理信息深度算子网络框架，用于逼近偏微分方程（PDE）的解算子。该框架采用DeepONet结构，其分支与主干子网络由有理KAN模块构成，并融合物理信息神经网络（PINN）机制，通过残差损失强制满足物理一致性。该设计可高效学习具有时空一致性的解算子，训练后能快速预测不同参数、初值或边界条件下时变参数化PDE的解。在典型基准测试中，PO-CKAN表现优异：在黏度ν=0.01的布辛涅斯克方程上，相比PI-DeepONet，均方相对L²误差降低约48%；在Eikonal和扩散-反应问题上也达到竞争性精度。
原文摘要 · Abstract (English)
We propose PO-CKAN, a physics-informed deep operator framework based on Chunkwise Rational Kolmogorov--Arnold Networks (KANs), for approximating the solution operators of partial differential equations. This framework leverages a Deep Operator Network (DeepONet) architecture that incorporates Chunkwise Rational Kolmogorov-Arnold Network (CKAN) sub-networks for enhanced function approximation. The principles of Physics-Informed Neural Networks (PINNs) are integrated into the operator learning framework to enforce physical consistency. This design enables the efficient learning of physically consistent spatio-temporal solution operators and allows for rapid prediction for parametric time-dependent PDEs with varying inputs (e.g., parameters, initial/boundary conditions) after training. Validated on challenging benchmark problems, PO-CKAN demonstrates accurate operator learning with results closely matching high-fidelity solutions. PO-CKAN adopts a DeepONet-style branch--trunk architecture with its sub-networks instantiated as rational KAN modules, and enforces physical consistency via a PDE residual (PINN-style) loss. On Burgers' equation with $ν=0.01$, PO-CKAN reduces the mean relative $L^2$ error by approximately 48\% compared to PI-DeepONet, and achieves competitive accuracy on the Eikonal and diffusion--reaction benchmarks.
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