Climate change presents one of the greatest challenges of our time, and accurate modeling of its dynamics is crucial for understanding and mitigating its impacts. This paper investigates the quantitative properties of weak solutions to the Budyko–Sellers climatological energy balance model, employing Physics-Informed Neural Networks (PINNs). The Budyko–Sellers model, which describes the Earth’s surface temperature through the balance of absorbed solar radiation and emitted thermal energy, poses challenges in the form of nonlinear differential systems with non-unique solutions. Traditional numerical methods often struggle to handle such complexities, underscoring the need for innovative approaches like PINNs, which leverage machine learning techniques to approximate generalized solutions. Given the increasing urgency of climate studies, our research reformulates the climatological model as an infinite-dimensional stochastic optimization problem, applying a deep learning Galerkin method to transform it into a solvable finite-dimensional framework. This allows for more efficient and accurate climate modeling. Through this approach, we demonstrate the remarkable efficacy of PINNs in approximating weak solutions, offering detailed numerical simulations that verify the accuracy and computational efficiency of the proposed method. This work contributes to the rapidly expanding field of applying AI and machine learning techniques to complex dynamical systems, particularly in the context of climatological modeling. By integrating AI-driven methods like PINNs into climate science, we open new avenues for understanding climate processes, which is critical for informing effective climate action and policy decisions.