Galaxy evolution is commonly described through the time evolution of observational statistics such as luminosity functions and stellar mass functions. However, these quantities are projections of an underlying multivariate galaxy state space rather than fundamental dynamical variables. We develop a unified framework in which galaxy evolution is formulated as the time evolution of a probability measure on the galaxy manifold. Representing galaxy states by latent variables $θ\in\mathcal{M}$ and the population by a density $ρ(θ,t)$, the evolution is governed by a general equation containing continuous transport and nonlocal jump processes. By reinterpreting manifold learning as the pushforward of measures, we distinguish observational, representation, and physical measures, and emphasize that manifold coordinates themselves need not carry direct physical meaning. In this picture, luminosity functions and stellar mass functions arise as projected observables of a single underlying dynamics, and generally do not form closed equations in observational space. The framework contains existing models as limiting cases: reduction to a single mass variable yields continuity-equation models, while additive post-merger states recover the Smoluchowski coagulation equation. We further show that luminosity-function evolution is naturally described within the Schechter family, whose apparent stability is interpreted as an effective consequence of projection. Since observables are projections of measures, inference of galaxy evolution becomes a statistical inverse problem of recovering manifold dynamics from data. This framework shifts the focus from fitting observed statistics directly to inferring the underlying state-space dynamics, thereby bridging manifold learning and physical theory.

翻译：星系演化通常通过光度函数和恒星质量函数等观测统计量的时间演化来描述。然而，这些量是潜在多元星系状态空间的投影，而非基本动力学变量。我们构建了一个统一框架，将星系演化表述为星系流形上概率测度的时间演化。用潜在变量 $θ\in\mathcal{M}$ 表征星系状态，用密度 $ρ(θ,t)$ 表征星系种群，其演化由包含连续输运和非局部跳过程的一般方程控制。通过将流形学习重新解释为测度的前推，我们区分了观测测度、表征测度和物理测度，并强调流形坐标本身无需具有直接物理意义。在此图景中，光度函数和恒星质量函数作为单一底层动力学的投影观测量出现，且通常在观测空间中不构成封闭方程。该框架包含现有模型作为极限情况：简化为单一质量变量时得到连续方程模型，而可加性并合后状态可恢复Smoluchowski凝聚方程。我们进一步证明，光度函数演化可自然地在Schechter族内描述，其表观稳定性被解释为投影的有效结果。由于观测量是测度的投影，星系演化推断成为从数据中恢复流形动力学的统计逆问题。该框架将焦点从直接拟合观测统计量转移到推断底层状态空间动力学，从而架起了流形学习与物理理论之间的桥梁。