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    This study optimizes level set control for the normal flow equation using neural networks. A novel approach based on the extended Kalman filter offers efficient and robust solutions, outperforming traditional methods.

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    Area of Science:

    • Computational Mathematics
    • Applied Mathematics
    • Numerical Analysis

    Background:

    • Optimal control problems involving level sets are crucial in various scientific and engineering fields.
    • The normal flow equation describes the evolution of interfaces, and its control is complex.
    • Analytical solutions for such problems are often intractable, necessitating approximation methods.

    Purpose of the Study:

    • To develop an effective method for the optimal control of level sets governed by the normal flow equation.
    • To propose and evaluate an approximation scheme for finding suboptimal solutions when analytical methods fail.
    • To investigate the efficacy of neural network structures for control law representation and optimization.

    Main Methods:

    • The study utilizes an approximation scheme based on the extended Ritz method.
    • A neural network structure is employed to represent the control law, with parameters tuned via optimization.
    • Two optimization approaches are compared: classical line-search descent and a quasi-Newton method (extended Kalman filter-based neural learning).

    Main Results:

    • The existence of a solution for the optimal control problem is established.
    • The extended Kalman filter-based neural learning approach demonstrates reduced computational effort compared to line-search methods.
    • This advanced method shows increased robustness against local minima, validated through 2D and 3D simulations.

    Conclusions:

    • The proposed neural network-based optimal control strategy provides an effective and efficient solution for the normal flow equation.
    • The extended Kalman filter optimization technique offers significant advantages in terms of computational cost and robustness.
    • This research contributes a powerful computational tool for problems involving level set evolution and control.