While significant progress has been made on Physics-Informed Neural Networks (PINNs), a comprehensive comparison of these methods across a wide range of Partial Differential Equations (PDEs) is still lacking. This study introduces PINNacle, a benchmarking tool designed to fill this gap. PINNacle provides a diverse dataset, comprising over 20 distinct PDEs from various domains including heat conduction, fluid dynamics, biology, and electromagnetics. These PDEs encapsulate key challenges inherent to real-world problems, such as complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality. PINNacle also offers a user-friendly toolbox, incorporating about 10 state-of-the-art PINN methods for systematic evaluation and comparison. We have conducted extensive experiments with these methods, offering insights into their strengths and weaknesses. In addition to providing a standardized means of assessing performance, PINNacle also offers an in-depth analysis to guide future research, particularly in areas such as domain decomposition methods and loss reweighting for handling multi-scale problems and complex geometry. While PINNacle does not guarantee success in all real-world scenarios, it represents a significant contribution to the field by offering a robust, diverse, and comprehensive benchmark suite that will undoubtedly foster further research and development in PINNs.

翻译：尽管物理学-informed神经网络（PINNs）已取得显著进展，但目前仍缺乏对这些方法在广泛偏微分方程（PDEs）上的全面比较。本研究推出PINNacle，一个旨在填补这一空白的基准测试工具。PINNacle提供多样化数据集，涵盖热传导、流体动力学、生物学和电磁学等领域的20余种不同PDEs。这些PDEs囊括了真实世界问题固有的关键挑战，如复杂几何、多尺度现象、非线性和高维性。PINNacle还提供用户友好型工具箱，集成约10种最先进的PINN方法，用于系统评估与比较。我们利用这些方法开展了广泛实验，深入揭示了各自的优势与局限性。除提供标准化的性能评估手段外，PINNacle还通过深度分析指导未来研究，特别是在处理多尺度问题与复杂几何的领域分解方法和损失重加权方面。尽管PINNacle无法保证适用于所有真实场景，但作为一套稳健、多样且全面的基准套件，它无疑将推动PINNs领域的进一步研究与开发。