Adaptive Physics-Informed Neural Networks for Singular Matrix Differential Systems with Applications to Optimal Control Synthesis

Singular matrix differential systems arising in optimal control present significant computational challenges through boundary layers, proportional delays, and structural constraints that classical numerical methods struggle to resolve effectively. We develop an adaptive physics-informed neural network framework that addresses three representative classes: singularly perturbed boundary value problems, pantograph delay differential equations, and matrix Riccati equations from linear–quadratic regulator design. The methodology integrates feedforward neural architectures with residual-based adaptive collocation strategies that automatically detect and resolve rapid variation regions. For matrix-valued problems, we introduce a Cholesky-type parameterization that algebraically enforces symmetry and positive definiteness by construction, eliminating the need for penalty tuning and post-processing corrections. Weak consistency is established through compactness arguments. Numerical experiments demonstrate mean absolute errors of 10⁻⁶ for boundary layer problems, approximately two orders of magnitude reduction in interpolation errors for delay equations compared to both fourth-order Runge-Kutta and MATLAB’s specialized dde23 solver, and guaranteed structure preservation for Riccati solutions with symmetry errors below machine precision (< 10⁻¹⁵). A hybrid refinement strategy combining physics-informed neural network predictions with classical integrators achieves improved point-wise accuracy while maintaining structural guarantees. The proposed framework provides a computational bridge between theoretical controllability analysis and practical implementations for numerical control synthesis, enabling reliable feedback controller design with structural certification for deployment in aerospace trajectory optimization and robotic control applications.