A thermodynamic inspired AI based search algorithm for solving ordinary differential equations.

Solving Ordinary Differential Equations (ODEs) is an essential and very challenging computational problem in many areas of science and engineering like physics, biology, control system, and economics. However, traditional numerical methods such as Euler's method and the Runge-Kutta method generally suffer from problems like the grid dependency, propagation of errors and not all are applicable to nonlinear and systems that are complex. In return, metaheuristic algorithms have become promising alternatives that strongly transform the solution process into an optimization task. In this paper, we introduce a new search algorithm called Thermodynamic Inspired Search Algorithm (TSA) for approximate solution to linear ODEs (LODEs), nonlinear ODEs (NLODEs) and systems of ODEs (SODEs). Inspired by thermodynamic processes, heat exchange, energy minimization and entropy control, TSA employs thermodynamic on purpose for balancing global exploration and local exploitation throughout the search process. A mesh free ODE solver is constructed with an accurate approximation to the exact ODE using a Fourier periodic expansion basis function combined with the proposed algorithm and a weighted residual minimization method. The population of TSA is a population of candidate solutions that represent the coefficients of the Fourier expansion series. It comes up with energy levels based on energy associated with fitness values, enabling the adaptive exploration through energy exchange between solutions. Entropy is a diversity indicator that prevents the solutions from converging prematurely through the promotion of the solution diversity. The transformation from exploration to exploitation occurs gradually through a temperature reduction process. It is formulated as an optimization objective function with constraints, with the residuals of the ODEs and boundary conditions as the objective function and penalty functions ensuring constraint satisfaction. The performance of TSA is further evaluated on a diverse benchmark suite of twenty ODE problems which demonstrates that TSA is superior state-of-the-art ODE optimizers, namely ADE, PSO, and ABC in accuracy, convergence rate and robustness. Mesh free nature of TSA enables us to have the advantage of (i) not being grid dependent (iii) diversity preservation using entropy and (iii) Energy driven solution updates. TSA is shown through experimental results to achieve lower Root Mean Square Errors (RMSE) than existing algorithms on both IVPs and BVPs. In addition, the Fourier periodic expansion and the least-square weighted residual approach improve the precision of the approximations of TSA's performance. The proposed algorithm gives a robust, agile answer design framework for unraveling complicated ODE frameworks, with attainable investigation into PDE and hybrid local-global optimal enhancement plans.