基于Trefftz方法的PINN与标准PINN的比较：聚焦于结构保持
Comparison of Trefftz-Based PINNs and Standard PINNs Focusing on Structure Preservation
AI总结：
本文比较了基于Trefftz方法的PINN与标准PINN在保持磁场线结构和流体动力学流线结构方面的性能，发现Trefftz-PINN在结构保持上更具优势。
摘要：
在本研究中，我们通过将标准PINN与基于Trefftz的PINN（Trefftz-PINN）进行比较，探讨了物理信息神经网络（PINN）在保持全局物理结构方面的能力。目标问题是在螺旋状聚变反应堆配置中重现磁场线结构。使用相同的数据集，从精确解中采样，我们在匹配的均方误差（MSE）水平下进行比较。磁场线的可视化显示，标准PINN在磁场表面可能表现出结构崩溃，即使MSE足够小，而Trefftz-PINN成功地保持了磁场线的全局拓扑结构。此外，所提出的框架被扩展到计算流体动力学（CFD）问题中，分析了速度场的流线结构。观察到类似趋势，表明Trefftz-PINN相比标准PINN在结构保持方面更优。这些结果表明，仅最小化数值误差并不能保证物理一致性，而在学习之前约束解空间是一种有效的物理一致替代建模策略。
英文摘要：
In this study, we investigate the capability of physics-informed neural networks (PINNs) to preserve global physical structures by comparing standard PINNs with a Trefftz-based PINN (Trefftz-PINN). The target problem is the reproduction of mag-netic field-line structures in a helical fusion reactor configuration. Using identical training data sampled from exact solutions, we perform comparisons under matched mean squared error (MSE) levels. Visualization of magnetic field lines reveals that standard PINNs may exhibit structural collapse across magnetic surfaces even when the MSE is sufficiently small, whereas Trefftz-PINNs successfully preserve the global topology of magnetic field lines. Furthermore, the proposed framework is extended to computational fluid dynamics (CFD) problems, where streamline structures of veloc-ity fields are analyzed. Similar tendencies are observed, demonstrating that Trefftz-PINNs provide superior structure preservation compared to standard PINNs. These results indicate that minimizing numerical error alone does not guarantee physical consistency, and that constraining the solution space prior to learning is an effective strategy for physics-consistent surrogate modeling.
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