In this work, we develop a new four-state climate-policy dynamical model that captures the coupled evolution of atmospheric temperature, ocean heat content, anthropogenic \({\text {CO}}_2\) concentration, and a signed policy-pressure variable representing societal and institutional response to warming. Existing low-dimensional climate models typically prescribe external emission pathways or impose fixed control functions, thereby failing to represent endogenous policy feedbacks or their nonlinear interaction with the climate carbon system. To address this gap, we introduce a responsive policy mechanism that activates when the atmospheric temperature exceeds a threshold and interacts dynamically with emissions, natural sinks, and temperature forcing. The resulting nonlinear system is solved using a Physics-Informed Neural Network (PINN), which embeds the governing differential equations directly into the loss function and learns the full trajectory without requiring observational or synthetic data. The PINN solutions are validated against high-precision numerical trajectories from the classical ode45 solver and exhibit excellent agreement with errors across all state variables. Three policy scenarios are examined for baseline, delayed-response, and strong early intervention revealing how the interplay between mitigation responsiveness, sink strength, and emission decay shapes long-term climate trajectories. The study demonstrates that the proposed model provides a conceptually simple yet dynamically rich framework for exploring climate-policy interactions, filling a critical gap between oversimplified policy-free climate models and high-dimensional Earth-system simulators. The PINN approach further offers a flexible and highly accurate computational tool for analyzing policy feedbacks and evaluating mitigation strategies under diverse future scenarios.
