Technology Category
Machine Learning: Learning with ManifoldsComputer Vision: Low Level & Physics-based VisionKnowledge Representation and Reasoning: Computational Complexity of Reasoning
Deep vs. Shallow: Benchmarking Physics-Informed Neural Architectures on the Biharmonic Equation
🤖 AI Summary
Traditional mesh-based methods struggle with complex geometries when solving high-order PDEs—particularly fourth-order biharmonic equations—while existing Physics-Informed Neural Networks (PINNs) suffer from high computational cost and limited accuracy. To address these challenges, this work proposes RBF-PIELM: a Radial Basis Function-activated Physics-Informed Extreme Learning Machine. It intrinsically embeds physical constraints into the network architecture and replaces iterative gradient descent with a single-step least-squares solution, enabling backpropagation-free training. Experiments on cavity flow and oscillatory stream-function problems demonstrate that RBF-PIELM achieves accuracy comparable to state-of-the-art PINNs, while accelerating training by up to 350× and reducing trainable parameters by over one order of magnitude. Although slightly less accurate than mature mesh-based solvers for highly oscillatory solutions, RBF-PIELM significantly extends the applicability boundary of efficient, lightweight PINN variants for high-order PDE numerical simulation.
Technology Category
Application Category
📝 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 $(350 imes)$ faster training than PINNs and over $(10 imes)$ 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.
Problem
Research questions and friction points this paper is trying to address.
Benchmarking neural architectures for solving higher-order PDEs
Comparing mesh-free methods with traditional mesh-based solvers
Evaluating efficiency and accuracy of physics-informed neural networks
Innovation
Methods, ideas, or system contributions that make the work stand out.
Uses radial-basis activations in neural networks
Replaces gradient descent with least-squares solve
Achieves faster training with fewer parameters
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