Mathematical Modeling and Dynamical Analysis of a Nonlinear Coupled Stress-Mitigation System with Signed Threshold-Relative Policy Feedback and Physics-Informed Neural Network Simulation

This study develops and analyzes a four-state nonlinear policy–feedback dynamical system that couples a system stressor, an accumulated burden, a signed mitigation–response variable, and a signed policy-pressure variable. The proposed model represents governance response through a smooth threshold-centered feedback mechanism, in which the policy-pressure dynamics depend continuously on the deviation of the stressor from a prescribed reference threshold. Unlike reduced-order formulations with purely exogenous interventions, the present framework generates endogenous interactions among stress accumulation, burden evolution, mitigation response, and policy adjustment. The qualitative analysis establishes local well-posedness in the admissible phase domain, conditional nonnegativity of the accumulated burden, and boundedness of trajectories on admissible intervals. An autonomous effective system is then derived to characterize quasi-stationary mean behavior of the periodically forced dynamics. For this effective system, local stability is investigated using Gershgorin estimates and Routh–Hurwitz criteria, leading to explicit analytical conditions for local asymptotic stability and a critical policy-responsiveness threshold associated with possible Hopf-type oscillatory transitions. The analysis highlights the stabilizing role of mitigation damping and cubic saturation in regulating the feedback loop. To approximate the nonlinear system, a Physics-Informed Neural Network (PINN) surrogate is constructed by embedding the governing equations into a differentiable residual loss while enforcing the initial conditions analytically. The accumulated burden is represented through an admissible neural-network ansatz to preserve the well-definedness of the logarithmic coupling term, while the mitigation–response and policy-pressure variables remain signed in accordance with the model formulation. Numerical validation against reference ode45 solutions across two governance regimes shows maximum absolute errors of order 10−3, indicating that the PINN provides a reliable differentiable surrogate for the coupled policy–feedback dynamics. The resulting framework offers a foundation for future inverse modeling, parameter estimation, and data-assimilation studies involving policy responsiveness, intervention thresholds, and burden- suppression effects.