Physics-informed Neural Networks for Optimal Intercept Problem,

The novel Extreme Theory of Functional Connections (X-TFC) method is employed to solve the optimal intercept problem. With X-TFC, for the first time, Theory of Functional Connections (TFC) and shallow Neural Networks (NNs) trained via the Extreme Learning Machine (ELM) algorithm are brought together as a class of PINN methods and applied to solving a broad class of ODEs and PDEs. In particular, the unknown solutions (in strong sense) of the ODEs and PDEs are approximated via particular expressions, called constrained expression (CEs), defined within TFC. A CE is a functional that always analytically satisfies the specified constraints and has a free-function that does not affect the specified constraints. In the X-TFC method, the free-function is a single-layer NN, trained via ELM algorithm. According to the ELM algorithm, the unknown constant coefficients appear linearly and thus, a least-squares method (for linear problems) or an iterative least-square method (for nonlinear problems) is used to compute the unknowns by minimizing the residual of the differential equations. In this work, the differential equations are represented by the system arising from the indirect method formulation of optimal control problems, which exploits the Hamiltonian function and the Pontryagin Maximum/Minimum Principle (PMP) to obtain a Two-Point Boundary Value Problem. The proposed method is tested by solving the Feldbaum problem and the minimum time-energy optimal intercept problem. It is shown that the major advantage of this method is the comparable accuracy with respect to the state of the art methods for the solution of optimal control problems along with an extremely fast computational time. In particular, the low computational time makes the proposed method suitable for real-time applications.