Quasi-Random Physics-informed Neural Networks
This addresses a key bottleneck in PINNs for solving PDEs, offering a practical improvement for computational science and engineering applications.
The paper tackled the sensitivity of physics-informed neural networks (PINNs) to point sampling by proposing Quasi-Random PINNs (QRPINNs), which use low-discrepancy sequences, resulting in proven better convergence rates and significant empirical outperformance over PINNs and adaptive methods, especially in high-dimensional PDEs.
Physics-informed neural networks have shown promise in solving partial differential equations (PDEs) by integrating physical constraints into neural network training, but their performance is sensitive to the sampling of points. Based on the impressive performance of quasi Monte-Carlo methods in high dimensional problems, this paper proposes Quasi-Random Physics-Informed Neural Networks (QRPINNs), which use low-discrepancy sequences for sampling instead of random points directly from the domain. Theoretically, QRPINNs have been proven to have a better convergence rate than PINNs. Empirically, experiments demonstrate that QRPINNs significantly outperform PINNs and some representative adaptive sampling methods, especially in high-dimensional PDEs. Furthermore, combining QRPINNs with adaptive sampling can further improve the performance.