We introduce the \emph{Symplectic Generative Network (SGN)}, a deep generative model that leverages Hamiltonian mechanics to construct an invertible, volume-preserving mapping between a latent space and the data space. By endowing the latent space with a symplectic structure and modeling data generation as the time evolution of a Hamiltonian system, SGN achieves exact likelihood evaluation without incurring the computational overhead of Jacobian determinant calculations. In this work, we provide a rigorous mathematical foundation for SGNs through a comprehensive theoretical framework that includes: (i) complete proofs of invertibility and volume preservation, (ii) a formal complexity analysis with theoretical comparisons to Variational Autoencoders and Normalizing Flows, (iii) strengthened universal approximation results with quantitative error bounds, (iv) an information-theoretic analysis based on the geometry of statistical manifolds, and (v) an extensive stability analysis with adaptive integration guarantees. These contributions highlight the fundamental advantages of SGNs and establish a solid foundation for future empirical investigations and applications to complex, high-dimensional data.

翻译：我们提出了一种名为辛生成网络（SGN）的深度生成模型，该模型利用哈密顿力学构建了潜在空间与数据空间之间的可逆、保体积映射。通过赋予潜在空间辛结构，并将数据生成建模为哈密顿系统的时间演化，SGN实现了精确的似然评估，而无需承担雅可比行列式计算的计算开销。在本工作中，我们通过一个全面的理论框架为SGN提供了严格的数学基础，包括：（i）可逆性与保体积性的完整证明，（ii）与变分自编码器和归一化流进行理论比较的形式化复杂度分析，（iii）带有定量误差界的强化通用逼近结果，（iv）基于统计流形几何的信息论分析，以及（v）具有自适应积分保证的广泛稳定性分析。这些贡献凸显了SGN的基本优势，并为未来对复杂高维数据的实证研究和应用奠定了坚实基础。