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

2026-01-14

AI Summary

This work addresses the challenge of simultaneously achieving high accuracy, rapid convergence, and analytical interpretability in solving nonlinear differential equations by proposing a novel hybrid approach that integrates the Optimal Homotopy Analysis Method (OHAM) with physics-informed constraints. A physics-informed residual loss function—incorporating the governing equation, boundary conditions, and available data—is constructed to systematically optimize convergence-control parameters. Notably, the framework introduces, for the first time, the core idea of Physics-Informed Neural Networks (PINNs) into the homotopy analysis methodology, thereby unifying analytical transparency with numerical efficiency. Numerical experiments on the Blasius boundary-layer problem demonstrate that the proposed method converges faster and achieves higher accuracy than both standard HAM and PINNs, closely matching established numerical benchmarks while preserving an explicit analytical structure.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Hardware-aware ML

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• nonlinear differential equations
• analytical interpretability
• numerical stability
• physics-informed methods
• convergence control

Innovation

• Physics-Informed
• Optimal Homotopy Analysis Method
• Nonlinear Differential Equations
• Convergence-Control Optimization
• Analytical Interpretability