This paper introduces Physics-Informed Deep Equilibrium Models (PIDEQs) for solving initial value problems (IVPs) of ordinary differential equations (ODEs). Leveraging recent advancements in deep equilibrium models (DEQs) and physics-informed neural networks (PINNs), PIDEQs combine the implicit output representation of DEQs with physics-informed training techniques. We validate PIDEQs using the Van der Pol oscillator as a benchmark problem, demonstrating their efficiency and effectiveness in solving IVPs. Our analysis includes key hyperparameter considerations for optimizing PIDEQ performance. By bridging deep learning and physics-based modeling, this work advances computational techniques for solving IVPs, with implications for scientific computing and engineering applications.

翻译：本文提出了一种基于物理信息的深度平衡模型（PIDEQs），用于求解常微分方程（ODEs）的初值问题（IVPs）。该方法结合了深度平衡模型（DEQs）的隐式输出表示与物理信息神经网络（PINNs）的训练技术，充分利用了DEQs和PINNs的最新进展。我们以范德波尔振荡器为基准问题验证了PIDEQs的有效性，证明了其在求解初值问题方面的高效性和优越性。分析中重点探讨了优化PIDEQ性能的关键超参数设置。通过融合深度学习与基于物理的建模方法，本研究推进了求解初值问题的计算技术，对科学计算与工程应用具有重要价值。