We present the Physics-Informed Optimal Homotopy Analysis Method (PI-OHAM) for solving nonlinear differential equations. PI-OHAM, based on classical HAM, employs a physics-informed residual loss to optimize convergence-control parameters systematically by combining data, boundary conditions, and governing equations in the manner similar to Physics Informed Neural Networks (PINNs). The combination of the flexibility of PINNs and the analytical transparency of HAM provides the approach with high numerical stability, rapid convergence, and high consistency with traditional numerical solutions. PI-OHAM has superior accuracy-time trade-offs and faster and more accurate convergence than standard HAM and PINNs when applied to the Blasius boundary-layer problem. It is also very close to numerical standards available in the literature. PI-OHAM ensures analytical transparency and interpretability by series-based solutions, unlike purely data-driven or data-free PINNs. Significant contributions are a conceptual bridge between decades of homotopy-based analysis and modern physics-inspired methods, and a numerically aided but analytically interpretable solver of nonlinear differential equations. PI-OHAM appears as a computationally efficient, accurate and understandable alternative to nonlinear fluid flow, heat transfer and other industrial problems in cases where robustness and interpretability are important.

翻译：本文提出了用于求解非线性微分方程的物理信息最优同伦分析方法（PI-OHAM）。该方法基于经典的同伦分析方法（HAM），通过引入物理信息残差损失，以类似于物理信息神经网络（PINNs）的方式，系统性地结合数据、边界条件和控制方程来优化收敛控制参数。PI-OHAM融合了PINNs的灵活性与HAM的解析透明度，从而具有高数值稳定性、快速收敛性以及与经典数值解的高度一致性。在应用于Blasius边界层问题时，PI-OHAM相比标准HAM和PINNs具有更优的精度-时间权衡特性，收敛速度更快且精度更高，与文献中现有数值标准解非常接近。与纯数据驱动或无数据的PINNs不同，PI-OHAM通过级数形式的解保证了分析的透明度和可解释性。该方法的重要贡献在于：搭建了连接数十年基于同伦的分析方法与现代物理启发方法的概念桥梁，并提供了一个数值辅助且解析可解释的非线性微分方程求解器。在鲁棒性和可解释性至关重要的场景下，如非线性流体流动、传热及其他工业问题，PI-OHAM可作为一种计算高效、精确且易于理解的替代求解方案。