{"ID":2859697,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.04490","arxiv_id":"2510.04490","title":"Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation","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.","short_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...","url_abs":"https://arxiv.org/abs/2510.04490","url_pdf":"https://arxiv.org/pdf/2510.04490v1","authors":"[\"Akshay Govind Srinivasan\",\"Vikas Dwivedi\",\"Balaji Srinivasan\"]","published":"2025-10-06T04:54:04Z","proceeding":"cs.CE","tasks":"[\"cs.CE\",\"cs.ET\",\"cs.LG\"]","methods":"[]","has_code":false}