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

TL;DR

This paper tackles the challenge of solving high-order PDEs with mesh-free methods by presenting RBF-PIELM, a physics-informed extreme learning machine that replaces gradient-based PINN training with a single-shot least-squares solve using Gaussian radial basis functions. It introduces geometry-aware initialization to align RBF centers with domain features and boundary layers, and validates the approach on the biharmonic equation through a lid-driven cavity test and a Method of Manufactured Solutions (MMS) with oscillatory exact solutions. The results show substantial speedups and reduced model size compared to PINNs (up to $350\times$ faster and $13.2\times$ fewer parameters) while achieving comparable accuracy on smoother problems, though accuracy degrades on highly oscillatory fields and mesh-based solvers still outperform in robustness. The work highlights potential hybridizations (FEM-PIELM) and residual adaptive basis refinement as promising directions for practical deployment and broader applicability.

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.

Figures (7)

• Figure 1: Centerline Velocity Profiles: RBF-PIELM vs. PINNs. RBF-PIELM attains better accuracy while training $350\times$ faster than PINNs.
• Figure 2: Manufactured biharmonic solution ($k_1=k_2=10$): (a) RBF-PIELM solution, (b) Exact solution and (c) Distribution of absolute error. RBF-PIELM closely matches the oscillatory exact solution with small errors.
• Figure 3: Lid-driven cavity benchmark: moving top wall ($u=1$) and stationary side/bottom walls.
• Figure 4: Collocation/boundary points and RBF width visualization for Test Case 1.
• Figure 5: Manufactured biharmonic solution ($k_1=k_2=20$): (a) RBF-PIELM solution, (b) Exact solution, and (c) Distribution of absolute error. RBF-PIELM breaks down on this highly oscillatory case.