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.

翻译：变分物理信息神经网络利用变分损失函数求解偏微分方程，其技术方法借鉴了有限元分析。传统的hp-VPINNs虽能有效处理高频问题，但计算开销大且随单元数量增加扩展性差，限制了其在复杂几何中的应用。本文提出FastVPINNs，一种基于张量的改进方法，显著降低了计算开销并提升了扩展性。通过优化张量运算，FastVPINNs相较于传统hp-VPINNs将每个训练周期的中位时间降低了100倍。在超参数选择得当的情况下，FastVPINNs在求解高频解问题时，其速度与精度均超越传统PINNs。在复杂域上求解逆问题的有效性验证了FastVPINNs在科学与工程挑战中的广泛应用潜力，为科学机器学习的实际应用开辟了新路径。