Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation
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arXiv
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| Format: | Preprint |
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2025
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| author | Srinivasan, Akshay Govind Dwivedi, Vikas Srinivasan, Balaji |
| author_facet | Srinivasan, Akshay Govind Dwivedi, Vikas Srinivasan, Balaji |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04490 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation Srinivasan, Akshay Govind Dwivedi, Vikas Srinivasan, Balaji Computational Engineering, Finance, and Science Emerging Technologies Machine Learning G.1.7; G.1.8; G.1.10; J.2 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. |
| title | Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation |
| topic | Computational Engineering, Finance, and Science Emerging Technologies Machine Learning G.1.7; G.1.8; G.1.10; J.2 |
| url | https://arxiv.org/abs/2510.04490 |