用几何框架统一描述基因型与表型演化，揭示进化是适应性学习过程。
Geometric framework for biological evolution
- 构建协变演化模型，将进化视为在适应度景观上的梯度上升。
- 发现度量张量与噪声协方差存在根本关联，可推导出兰德方程。
- 提出实验需测量演化变化协方差，以确定微观动力学的函数关系。
我们发展了一种广义协变的演化动力学描述，适用于基因型和表型空间。最大熵原理揭示了逆度量张量与协方差矩阵之间的基本对应关系，使兰德方程成为协变梯度上升方程。这表明进化可建模为在适应度景观上的学习过程，具体学习算法由度量张量与微观动力学引起的噪声协方差之间的函数关系决定。尽管度量（或基因型协方差逆矩阵）已得到广泛经验刻画，但噪声协方差及其可观测量（演化变化的协方差）从未被直接测量，由此带来实验挑战：需确定度量与噪声协方差间的函数形式。
原文摘要 · Abstract (English)
We develop a generally covariant description of evolutionary dynamics that operates consistently in both genotype and phenotype spaces. We show that the maximum entropy principle yields a fundamental identification between the inverse metric tensor and the covariance matrix, revealing the Lande equation as a covariant gradient ascent equation. This demonstrates that evolution can be modeled as a learning process on the fitness landscape, with the specific learning algorithm determined by the functional relation between the metric tensor and the noise covariance arising from microscopic dynamics. While the metric (or the inverse genotypic covariance matrix) has been extensively characterized empirically, the noise covariance and its associated observable (the covariance of evolutionary changes) have never been directly measured. This poses the experimental challenge of determining the functional form relating metric to noise covariance.
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