Physics-Informed Neural Networks (PINNs) are Neural Network architectures trained to emulate solutions of differential equations without the necessity of solution data. They are currently ubiquitous in the scientific literature due to their flexible and promising settings. However, very little of the available research provides practical studies that aim for a better quantitative understanding of such architecture and its functioning. In this paper, we analyze the performance of PINNs for various architectural hyperparameters and algorithmic settings based on a novel error metric and other factors such as training time. The proposed metric and approach are tailored to evaluate how well a PINN generalizes to points outside its training domain. Besides, we investigate the effect of the algorithmic setup on the outcome prediction of a PINN, inside and outside its training domain, to explore the effect of each hyperparameter. Through our study, we assess how the algorithmic setup of PINNs influences their potential for generalization and deduce the settings which maximize the potential of a PINN for accurate generalization. The study that we present returns insightful and at times counterintuitive results on PINNs. These results can be useful in PINN applications when defining the model and evaluating it.

翻译：物理信息神经网络（Physics-Informed Neural Networks, PINNs）是一种无需解数据即可训练以模拟微分方程解的神经网络架构。由于其灵活且前景广阔的特性，当前在科学文献中应用广泛。然而，现有研究极少提供旨在更好定量理解此类架构及其运行机制的实践性分析。本文基于一种全新的误差度量以及训练时间等其他因素，分析了不同架构超参数和算法设置下PINNs的性能。所提出的度量与方法专门用于评估PINN在其训练域外点的泛化能力。此外，我们研究了算法设置对PINN在训练域内及域外预测结果的影响，以探索每个超参数的作用。通过本研究，我们评估了PINNs的算法设置如何影响其泛化潜力，并推导出能最大化PINN准确泛化潜力的设置。我们呈现的研究结果为PINNs带来了富有洞察力且有时反直觉的结论。这些结论在定义和评估PINN模型的应用场景中具有实用价值。