Multilayer neural networks for solving a class of partial differential equations

Area of Science:

• Control Systems Engineering
• Computational Mathematics
• Artificial Intelligence

Background:

• Designing nonlinear control systems often involves solving complex first-order partial differential equations.
• Traditional methods for solving these equations can be computationally intensive and time-consuming.
• Input-to-state linearizable systems offer a structured approach but may require approximations.

Purpose of the Study:

• To present a novel method for solving first-order partial differential equations using neural networks.
• To apply this method to input-to-state linearizable or approximately linearizable systems.
• To facilitate the design of nonlinear control systems by providing efficient solution techniques.

Main Methods:

• Training the derivative of a feedforward neural network with an extended backpropagation algorithm.
• Utilizing the solutions of differential equations and Lie derivatives to find a change of coordinates.
• Designing a feedback control law based on the derived coordinate transformation.

Main Results:

• The proposed method efficiently finds approximate solutions for complicated first-order partial differential equations.
• Simulations demonstrate the ease and speed of obtaining these solutions.
• The method successfully yields a change of coordinates and enables feedback control law design.

Conclusions:

• The presented neural network approach offers a significant advantage in solving complex partial differential equations.
• This technique can streamline the design process for nonlinear control systems.
• The method is particularly beneficial when analytical solutions to partial differential equations are challenging to obtain.