Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation
This work addresses the computational inefficiency of PINNs for engineering simulations, though it remains incremental as it still lags behind traditional mesh-based solvers.
The paper tackled the problem of solving higher-order partial differential equations like the biharmonic equation by benchmarking RBF-PIELM, a fast variant of physics-informed neural networks, achieving up to 350x faster training and 10x fewer parameters compared to PINNs while maintaining similar accuracy.
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.