FastVPINNs: Tensor-Driven Acceleration of VPINNs for Complex Geometries

TL;DR

FastVPINNs significantly accelerate VPINNs for complex geometries using tensor operations, achieving 100-fold faster training and better scalability, enabling practical high-frequency PDE solutions.

Contribution

Introduces FastVPINNs, a tensor-based method that reduces computational cost and enhances scalability of VPINNs for complex geometries.

Findings

Achieves 100-fold reduction in training time per epoch.

Surpasses traditional PINNs in speed and accuracy.

Effective in solving inverse problems on complex domains.

Abstract

Variational Physics-Informed Neural Networks (VPINNs) utilize a variational loss function to solve partial differential equations, mirroring Finite Element Analysis techniques. Traditional hp-VPINNs, while effective for high-frequency problems, are computationally intensive and scale poorly with increasing element counts, limiting their use in complex geometries. This work introduces FastVPINNs, a tensor-based advancement that significantly reduces computational overhead and improves scalability. Using optimized tensor operations, FastVPINNs achieve a 100-fold reduction in the median training time per epoch compared to traditional hp-VPINNs. With proper choice of hyperparameters, FastVPINNs surpass conventional PINNs in both speed and accuracy, especially in problems with high-frequency solutions. Demonstrated effectiveness in solving inverse problems on complex domains underscores FastVPINNs' potential for widespread application in scientific and engineering challenges, opening new avenues for practical implementations in scientific machine learning.