The rapid urbanization makes the understanding of the evolution of urban environments of utmost importance to steer societies towards better futures. Many studies have focused on the emerging properties of cities, leading to the discovery of scaling laws mirroring the dependence of socio-economic indicators on city sizes. However, few efforts have been devoted to the modelling of the dynamical evolution of cities, as reflected through the mutual influence of socio-economic variables. Here, we fill this gap by presenting a maximum entropy generative model for cities written in terms of a few macro-economic variables, whose parameters (the effective Hamiltonian, in a statistical-physical analogy) are inferred from real data through a maximum-likelihood approach. This approach allows for establishing a few results. First, nonlinear dependencies among indicators are needed for an accurate statistical description of the complexity of empirical correlations. Second, the inferred coupling parameters turn out to be quite robust along different years. Third, the quasi time-invariance of the effective Hamiltonian allows guessing the future state of a city based on a previous state. Through the adoption of a longitudinal dataset of macro-economic variables for French towns, we assess a significant forecasting accuracy.

快速的城市化使人们对城市环境的演变至关重要，这对于引导社会走向更美好的未来至关重要。许多研究集中在城市的新兴特性上，导致发现了反映社会经济指标对城市规模的依存关系的定标法。但是，通过社会经济变量的相互影响可以看出，很少有人致力于城市动态演变的建模。在这里，我们通过提出一个针对城市的最大熵生成模型来填补这一空白，该模型是根据一些宏观经济变量来写的，其参数（在统计物理比喻中是有效的哈密顿量）是通过最大似然从真实数据中得出的方法。这种方法可以建立一些结果。第一，指标之间的非线性相关性对于经验相关性的复杂性的准确统计描述是必需的。其次，推断出的耦合参数在不同年份中非常稳健。第三，有效哈密顿量的准时不变性允许根据先前状态猜测城市的未来状态。通过采用法国城镇宏观经济变量的纵向数据集，我们评估了重要的预测准确性。