A Neural-Network Method for Solving the Optimal Control Model of Carbon Emissions

China has pledged to reach carbon neutrality by 2060 and peak its carbon emissions before 2030, making carbon emission control a pressing issue in environmental economics. In the Yangtze River Delta (YRD), optimal carbon-emission control poses significant computational challenges due to high dimensionality, nonlinear dynamics, and complex boundary conditions. Traditional solution methods—including linear–quadratic regulators, finite-difference Hamilton–Jacobi–Bellman (HJB) solvers, spectral collocation, and reinforcement-learning–based approaches—have succeeded in low-dimensional or simplified models but often struggle with scalability and accurate enforcement of boundary and terminal constraints. Recent progress in triangular basis neural networks (TBNNs) highlights their ability to embed partial differential equation (PDE) residuals into network training and overcome these limitations. Therefore, we propose a unified residual-based collocation TBNN framework to solve the high-dimensional HJB equation governing stochastic carbon-emission control in Shanghai, Jiangsu, and Zhejiang. Our network parameterizes the value function in (t, I) space, and training minimizes a composite loss comprising the squared HJB residual, boundary-condition penalties, and terminal-condition error, using stochastic mini-batches and adaptive learning rates to ensure stable, rapid convergence without grid discretization. Applied to 2000–2022 regional emissions, GDP, and population data, the TBNN yields value-function surfaces bounded within approximately ±10⁵, exhibits sharp peaks at high emission intensities, and captures subtle temporal undulations without clear trends. Simultaneously, the cross-entropy loss remains below 10¹, falling under 0.1 after 5,000 epochs, which reflects high signal-to-noise fidelity. From these surfaces we derive actionable, region-specific investment guidelines—CNY 20–60/ton for Jiangsu, CNY 50–120/ton for Shanghai, and CNY 250–650/ton for Zhejiang—providing a robust, data-driven tool for real-time, adaptive emission-reduction planning across heterogeneous regional dynamics.