Optimal Control of Propagating Fronts by Using Level Set Methods and Neural Approximations

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.