Physics Informed Optimal Homotopy Analysis Method (PI-OHAM): A Hybrid Analytical Computational Framework for Solving nonlinear Differential Equations

TL;DR

PI-OHAM is a hybrid analytical-computational method combining homotopy analysis and physics-informed neural networks to efficiently solve nonlinear differential equations with high accuracy, stability, and interpretability.

Contribution

It introduces a novel hybrid framework that integrates HAM with physics-informed residuals, enhancing convergence and interpretability over existing methods.

Findings

Faster convergence than standard HAM and PINNs.

High accuracy and stability in solving nonlinear boundary-layer problems.

Close agreement with established numerical solutions.

Abstract

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.