用方差项改进神经网络损失函数，让误差更均匀、减少局部错误。
Improved Physics-informed neural networks loss function regularization with a variance-based term
- 在损失函数中加入误差的标准差，使误差分布更均匀。
- 在多个物理方程上测试，最大误差显著降低，计算开销几乎不变。
- 适合需要高精度解的物理模拟场景，尤其对梯度突变区域有效。
在机器学习和统计建模中，均方误差或绝对误差常被用作损失函数。虽然能有效降低平均误差，但可能忽略局部异常值，导致在梯度陡峭或不连续区域出现显著误差。这一问题在物理信息神经网络（PINNs）中尤为突出。为此，本文提出一种新损失函数，将选定误差度量的均值与标准差结合。通过最小化该联合损失，可实现更均匀的误差分布，降低局部高误差区域的影响。该方法易于实现，并在不同复杂度的问题上进行了测试：一维泊松方程、非稳态伯格斯方程、二维线弹性固体力学以及二维稳态纳维-斯托克斯方程。结果表明，相比传统基于均值的损失函数，新方法提升了求解质量，显著降低了最大误差，且对计算时间影响极小。
原文摘要 · Abstract (English)
In machine learning and statistical modeling, the mean square or absolute error is commonly used as an error metric, also called a "loss function." While effective in reducing the average error, this approach may fail to address localized outliers, leading to significant inaccuracies in regions with sharp gradients or discontinuities. This issue is particularly evident in physics-informed neural networks (PINNs), where such localized errors are expected and affect the overall solution. To overcome this limitation, we propose a novel loss function that combines the mean and the standard deviation of the chosen error metric. By minimizing this combined loss function, the method ensures a more uniform error distribution and reduces the impact of localized high-error regions. The proposed loss function is easy to implement and tested on problems of varying complexity: the 1D Poisson equation, the unsteady Burgers' equation, 2D linear elastic solid mechanics, and 2D steady Navier-Stokes equations. Results demonstrate improved solution quality and lower maximum error compared to the standard mean-based loss, with minimal impact on computational time.
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