Hybrid Gaussian process regression-based harmony assessment in a water–land–energy–food–carbon-emission coupled system

Harmonization between production factors and the external environment is crucial for sustainable agricultural development. However, discussions on the harmony of water–land–energy–food–carbon-emission (WLEFC) coupled systems remain limited, resulting in its operational status and constraints being unclear. This study used harmony theory to examine WLEFC systems. We also developed a harmony analysis framework and enhanced its evaluation index system. An improved Gaussian process regression model (AOA-L-BFGS-GPR) combining an arithmetic optimization algorithm with a limited-memory BFGS algorithm was developed. This model evaluated the dynamic development of WLEFC harmony in Heilongjiang Province, China. We identified key challenges and analyzed the trends. The streamlined index system allowed for precise measurements. The AOA-L-BFGS-GPR model demonstrated improved accuracy, stability, and reasonability. From 1994 to 2023, the WLEFC system harmony in Heilongjiang increased from 3.085 to 3.527. The water resource subsystem consistently remained the main obstacle, with its barrier degree remaining above 10%. Later in the period, rural population decline and slow progress in energy conservation became significant hindering factors. Daxing’anling and Heihe experienced severe rural hollowing, with the agricultural population barrier degrees reaching 20.99% and 17.43%, respectively. Meanwhile, northeastern cities faced two challenges: low energy efficiency and high carbon emissions. Scenario analysis showed that the SSP126 pathway provided better potential for green agricultural development than SSP245, SSP370, and SSP585. Under the SSP126 pathway, the system harmony was expected to reach 3.923 by 2055. This study offers new insights and methods for analyzing WLEFC systems, supporting environmentally sustainable agriculture.