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. To the best of our knowledge, it is the largest benchmark with a diverse and comprehensive evaluation that will undoubtedly foster further research in PINNs.

翻译：尽管物理信息神经网络（PINNs）已取得显著进展，但目前仍缺乏对这些方法在各类偏微分方程（PDEs）上的全面比较。本研究推出PINNacle这一基准工具以填补该空白。PINNacle提供了包含20余种不同PDEs的多样化数据集，这些方程涵盖热传导、流体动力学、生物及电磁学等多个领域，并系统性地包含了实际问题的关键挑战，如复杂几何、多尺度现象、非线性和高维性。该工具还配备用户友好的工具箱，集成了约10种前沿PINN方法用于系统评估与对比。我们利用这些方法开展了大量实验，揭示了其性能优劣。除提供标准化评估手段外，PINNacle还通过深入分析为未来研究指明方向，特别聚焦于域分解方法和损失重加权技术以应对多尺度问题与复杂几何。据我们所知，这是规模最大且评估维度最全面的基准测试，必将进一步推动PINNs领域的深入研究。