The widespread use of neural networks across different scientific domains often involves constraining them to satisfy certain symmetries, conservation laws, or other domain knowledge. Such constraints are often imposed as soft penalties during model training and effectively act as domain-specific regularizers of the empirical risk loss. Physics-informed neural networks is an example of this philosophy in which the outputs of deep neural networks are constrained to approximately satisfy a given set of partial differential equations. In this work we review recent advances in scientific machine learning with a specific focus on the effectiveness of physics-informed neural networks in predicting outcomes of physical systems and discovering hidden physics from noisy data. We will also identify and analyze a fundamental mode of failure of such approaches that is related to numerical stiffness leading to unbalanced back-propagated gradients during model training. To address this limitation we present a learning rate annealing algorithm that utilizes gradient statistics during model training to balance the interplay between different terms in composite loss functions. We also propose a novel neural network architecture that is more resilient to such gradient pathologies. Taken together, our developments provide new insights into the training of constrained neural networks and consistently improve the predictive accuracy of physics-informed neural networks by a factor of 50-100x across a range of problems in computational physics. All code and data accompanying this manuscript are publicly available at \url{https://github.com/PredictiveIntelligenceLab/GradientPathologiesPINNs}.

本文回顾了科学机器学习的最新进展，特别关注物理启发式神经网络在预测物理系统结果和从噪声数据中发现隐藏的物理方面的有效性。我们提出了一个神经网络体系结构，该结构对梯度病理具有更强的鲁棒性，并提出了一个学习率退火算法，通过使用模型训练期间的梯度统计数据来平衡复合损失函数中不同项之间的相互作用。所有伴随本手稿的代码和数据都是公开可用的。

理解并减轻物理启示神经网络中的梯度病理