In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

翻译：本文研究梯度下降算法在PINNs等物理启发式机器学习方法中的行为，这些方法用于最小化与偏微分方程相关的残差。我们的核心发现是：此类模型训练的困难程度与特定微分算子的条件数密切相关。该算子本质上对应于底层偏微分方程微分算子的厄米平方。若该算子呈现病态，将导致训练过程缓慢甚至不可行。因此，对该算子进行预处理至关重要。我们采用严格数学分析与实证评估相结合的方法，研究多种预处理策略，阐释它们如何改善该关键算子的条件数，并进而优化训练过程。