深与浅：双调和方程上物理信息神经架构的基准测试
Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation
AI总结：
本文基准测试RBF-PIELM（一种径向基激活的极限学习机变体）在双调和方程上的性能，相比PINNs训练快350倍、参数少10倍以上，但精度仍不及成熟网格法且在高振荡解上退化。
摘要：
偏微分方程（PDE）求解器是工程模拟的基础。经典的基于网格的方法（有限差分/体积/单元法）在高质量网格上快速且准确，但在处理高阶算子和复杂、难以网格化的几何形状时存在困难。最近开发的物理信息神经网络（PINNs）及其变体无需网格且灵活，但计算密集且通常精度较低。本文系统性地对RBF-PIELM（一种快速的PINN变体——采用径向基激活函数的极限学习机）在高阶PDE上的性能进行了基准测试。RBF-PIELM用单次最小二乘求解替代了PINNs耗时的梯度下降。我们在四阶双调和方程上测试了RBF-PIELM，使用了两个基准：顶盖驱动空腔流（流函数形式）和一个构造的振荡解。我们的结果表明，与PINNs相比，训练速度提升高达350倍，在相当的求解精度下参数数量减少超过10倍。尽管超越了PINNs，RBF-PIELM仍落后于成熟的基于网格的求解器，且其精度在高度振荡解上有所下降，这凸显了实际部署中仍存在的挑战。
英文摘要：
Partial differential equation (PDE) solvers are fundamental to engineering simulation. Classical mesh-based approaches (finite difference/volume/element) are fast and accurate on high-quality meshes but struggle with higher-order operators and complex, hard-to-mesh geometries. Recently developed physics-informed neural networks (PINNs) and their variants are mesh-free and flexible, yet compute-intensive and often less accurate. This paper systematically benchmarks RBF-PIELM, a rapid PINN variant-an extreme learning machine with radial-basis activations-for higher-order PDEs. RBF-PIELM replaces PINNs' time-consuming gradient descent with a single-shot least-squares solve. We test RBF-PIELM on the fourth-order biharmonic equation using two benchmarks: lid-driven cavity flow (streamfunction formulation) and a manufactured oscillatory solution. Our results show up to $(350\times)$ faster training than PINNs and over $(10\times)$ fewer parameters for comparable solution accuracy. Despite surpassing PINNs, RBF-PIELM still lags mature mesh-based solvers and its accuracy degrades on highly oscillatory solutions, highlighting remaining challenges for practical deployment.
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