Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation

• URL: http://arxiv.org/abs/2510.04490v1
• Date: Mon, 06 Oct 2025 04:54:04 GMT
• Title: Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation
• Authors: Akshay Govind Srinivasan, Vikas Dwivedi, Balaji Srinivasan,
• Abstract summary: This paper systematically benchmarks RBF-PIELM, a rapid PINN variant-an extreme learning machine with radial-basis activations-for higher-order PDEs.<n>Our results show up to $(350times)$ faster training than PINNs and over $(10times)$ fewer parameters for comparable solution accuracy.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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