CT-PIKAN：用于求解曲线域偏微分方程的坐标变换物理信息柯尔莫哥洛夫-阿诺尔德网络（结合基于自动微分的度量评估）
CT-PIKAN: Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Network with Autograd-Based Metric Evaluation for Solving PDEs in Curvilinear Domains
浏览论文内容
中文总结 AI 辅助
本研究提出CT-PIKAN框架，通过坐标变换结合自动微分，解决曲线域偏微分方程求解的几何适配问题，在多类基准问题上实现高效准确的求解。
中文摘要 AI 辅助
物理信息柯尔莫哥洛夫-阿诺尔德网络（Physics-Informed Kolmogorov-Arnold Networks，PIKAN）是近期兴起的一类高效神经偏微分方程求解器，它结合了基于样条的柯尔莫哥洛夫-阿诺尔德表示的表达能力与物理信息学习。然而，现有PIKAN公式主要针对笛卡尔域开发，无法自然适配曲线域带来的几何复杂性。本研究提出坐标变换物理信息柯尔莫哥洛夫-阿诺尔德网络（Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Networks，CT-PIKAN），这是一种几何感知框架，可通过坐标变换求解任意形状域上的偏微分方程：用平滑映射将物理域转换为规则计算域，转换后的控制方程直接在物理信息损失中施加。与需要手动推导度量系数的传统变换型PINN方法不同，CT-PIKAN利用自动微分（autograd）直接从坐标映射评估雅可比矩阵、度量张量和变换后的微分算子，消除了解析推导，提升了实现灵活性。为建立该框架，首先构建了基于无数据B样条的PIKAN，并在二维平流方程上进行验证；随后在代表性的椭圆型、抛物型和双曲型基准问题（包括极坐标和波浪曲线坐标下的泊松方程、热传导方程和平流方程）上评估CT-PIKAN方法。该框架为微分几何与物理信息柯尔莫哥洛夫-阿诺尔德网络的结合提供了通用且可扩展的方法，可在复杂域上实现高效、准确的偏微分方程求解。
英文摘要
Physics-Informed Kolmogorov-Arnold Networks have recently emerged as an effective class of neural solvers for partial differential equations, combining the expressive power of spline-based Kolmogorov-Arnold representations with physics-informed learning. However, existing PIKAN formulations are primarily developed for Cartesian domains and cannot naturally accommodate the geometric complexity introduced by curvilinear domains. In this work, we propose Coordinate-Transformed Physics-Informed Kolmogorov-Arnold Networks (CT-PIKAN), a geometry-aware framework for solving PDEs on arbitrarily shaped domains through coordinate transformation. A smooth mapping transforms the physical domain into a regular computational domain, while the transformed governing equations are enforced directly within the physics-informed loss. Unlike conventional transformed PINN approaches that require manually derived metric coefficients, CT-PIKAN employs automatic differentiation to evaluate Jacobians, metric tensors, and transformed differential operators directly from the coordinate mapping, eliminating analytical derivations and improving implementation flexibility. To establish the proposed framework, a data-free B-spline-based PIKAN is first constructed and validated on the two-dimensional advection equation. The CT-PIKAN methodology is subsequently assessed on representative elliptic, parabolic, and hyperbolic benchmark problems, including the Poisson, heat, and advection equations formulated in polar and wavy curvilinear coordinates. The proposed framework provides a general and extensible methodology for integrating differential geometry with physics-informed Kolmogorov-Arnold networks, enabling efficient and accurate PDE solutions on complex domains.
