A Triple Hybrid Framework Combining Neural ODEs, Thermodynamic Search Algorithm, and Adaptive Step Size Control for Solving Differential Equations

This paper introduces a novel triple hybrid framework for solving ordinary differential equations (ODEs) and differential-algebraic equations (DAEs) by synergistically combining three distinct machine learning and optimization paradigms. The proposed method integrates Neural Ordinary Differential Equations (Neural ODEs) with a Thermodynamic-Inspired Search Algorithm (TSA) and an adaptive machine learning-based step size controller. Neural ODEs provide a continuous-depth representation that naturally captures the differential structure of physical systems, while TSA---inspired by energy minimization, heat exchange, and entropy control---offers robust global exploration capabilities and prevents premature convergence. The third component employs a lightweight neural network that dynamically adjusts step sizes based on local error estimates, enhancing computational efficiency without sacrificing accuracy. This triple hybrid architecture addresses the limitations of existing methods: gradient instability in Physics-Informed Neural Networks, poor exploration in traditional metaheuristics, and the rigidity of fixed step size strategies. The method is evaluated on a diverse suite of benchmark problems including linear and nonlinear boundary value problems, stiff ODEs, systems of ODEs, and index-1 DAEs. Extensive statistical analysis over 50 independent runs demonstrates superior performance, with average improvements of 6.2\% in solution accuracy compared to state-of-the-art hybrid methods, 28\% reduction in standard deviation, and 35\% decrease in computational time through adaptive step sizing. The mesh-free nature of the approach provides differentiable closed-form solutions, making it particularly valuable for applications requiring sensitivity analysis or integration with downstream tasks.