用梯度差优化物理神经网络，提升求解精度与速度
DiffGrad for Physics-Informed Neural Networks
- 引入DiffGrad优化器，利用当前与前一梯度差提升性能
- 在Burgers方程上，精度更高且训练时间更短
- 适合需要高效高精度求解的偏微分方程研究者
物理信息神经网络（PINNs）是基于偏微分方程求解高度非线性问题的前沿方法。尽管应用广泛，但其在效率、计算成本和精度方面仍面临挑战。以流体力学中的典型方程Burgers方程为例，传统Adam优化器因未考虑历史梯度而表现有限。本文提出将DiffGrad与PINNs结合的新策略，通过利用当前梯度与前一梯度的差值来增强优化效果。通过对比Adam、Adamax、RMSprop和DiffGrad等优化器的综合计算分析，结果表明，该方法不仅提升了求解精度，还显著缩短了训练时间。通过在不同时间步长下对空间解进行可视化，验证了网络解的准确性。
原文摘要 · Abstract (English)
Physics-Informed Neural Networks (PINNs) are regarded as state-of-the-art tools for addressing highly nonlinear problems based on partial differential equations. Despite their broad range of applications, PINNs encounter several performance challenges, including issues related to efficiency, minimization of computational cost, and enhancement of accuracy. Burgers' equation, a fundamental equation in fluid dynamics that is extensively used in PINNs, provides flexible results with the Adam optimizer that does not account for past gradients. This paper introduces a novel strategy for solving Burgers' equation by incorporating DiffGrad with PINNs, a method that leverages the difference between current and immediately preceding gradients to enhance performance. A comprehensive computational analysis is conducted using optimizers such as Adam, Adamax, RMSprop, and DiffGrad to evaluate and compare their effectiveness. Our approach includes visualizing the solutions over space at various time intervals to demonstrate the accuracy of the network. The results show that DiffGrad not only improves the accuracy of the solution but also reduces training time compared to the other optimizers.
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