Physics-Informed Neural Networks and Functional Interpolation for Solving the Matrix Differential Riccati Equation

Kristofer Drozd, Roberto Furfaro (), Enrico Schiassi and Andrea D’Ambrosio
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Kristofer Drozd: System & Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA
Roberto Furfaro: System & Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA
Enrico Schiassi: System & Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA
Andrea D’Ambrosio: System & Industrial Engineering, University of Arizona, Tucson, AZ 85721, USA

Mathematics, 2023, vol. 11, issue 17, 1-24

Abstract: In this manuscript, we explore how the solution of the matrix differential Riccati equation (MDRE) can be computed with the Extreme Theory of Functional Connections (X-TFC). X-TFC is a physics-informed neural network that uses functional interpolation to analytically satisfy linear constraints, such as the MDRE’s terminal constraint. We utilize two approaches for solving the MDRE with X-TFC: direct and indirect implementation. The first approach involves solving the MDRE directly with X-TFC, where the matrix equations are vectorized to form a system of first order differential equations and solved with iterative least squares. In the latter approach, the MDRE is first transformed into a matrix differential Lyapunov equation (MDLE) based on the anti-stabilizing solution of the algebraic Riccati equation. The MDLE is easier to solve with X-TFC because it is linear, while the MDRE is nonlinear. Furthermore, the MDLE solution can easily be transformed back into the MDRE solution. Both approaches are validated by solving a fluid catalytic reactor problem and comparing the results with several state-of-the-art methods. Our work demonstrates that the first approach should be performed if a highly accurate solution is desired, while the second approach should be used if a quicker computation time is needed.

Keywords: differential Riccati equation; differential Lyapunov equation; functional interpolation; optimal control; physics-informed neural network (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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