用快速神经网络求解高阶偏微分方程，训练快350倍且参数少10倍以上
Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation
- 用单次最小二乘法替代梯度下降，加速训练
- 在双层流和振荡解上实现比传统PINN快350倍、参数少10倍以上
- 适合需要快速原型的工程仿真场景，但对高频解仍不理想
偏微分方程求解是工程仿真的基础。传统基于网格的方法在高质量网格上快速准确，但在高阶算子和复杂几何上表现不佳。近年来兴起的物理信息神经网络（PINNs）虽无网格、灵活，但计算量大且精度不足。本文系统评估了一种快速变体RBF-PIELM——基于径向基激活的极限学习机——在四阶双调和方程上的表现。该方法以单次最小二乘求解取代耗时的梯度下降。我们在两个基准上测试：速度驱动腔流（流函数形式）与人工振荡解。结果表明，相比PINNs，RBF-PIELM训练速度最高提升350倍，参数量减少超过10倍，同时保持相当精度。尽管超越了PINNs，其性能仍不及成熟的网格方法，且在高度振荡解上精度下降，凸显实际部署中的挑战。
原文摘要 · Abstract (English)
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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