We propose a spectral learning method for stochastic nonlinear dynamical systems represented with embedded latent transfer operators in deep feature spaces. We instantiate the method as Deep Spectral Encoder (DSE), an operator-based latent state-space model in which a time-invariant neural encoder implements learnable nonlinear feature maps from observations, and these features define Markovian latent states whose temporal evolution and observation mapping are described by the transfer and observation operators, respectively. Functional canonical correlation analysis in a learnable Galerkin-projected feature space provides state coordinates from past and future observations, and the two linear operators are estimated on the state coordinates as ridge-regularized closed-form solutions that coincide with Galerkin projections of the associated covariance operators. On this representation, we generalize sequential Bayesian filtering and Koopman spectral mode decomposition in feature space. Experiments on several scenarios show stable and superior performance with sequential Bayesian filtering and dynamic mode decomposition baselines even under noise and partial observability.

翻译：我们提出了一种针对随机非线性动力系统的谱学习方法，该方法在深度特征空间中利用嵌入的潜在传输算子进行表征。我们将该方法实例化为深度谱编码器（Deep Spectral Encoder, DSE），这是一种基于算子的潜在状态空间模型，其中时不变神经编码器从观测数据中实现可学习的非线性特征映射，这些特征定义了马尔可夫潜在状态，其时间演化与观测映射分别由传输算子和观测算子描述。在可学习的伽辽金投影特征空间中，通过函数典型相关分析从过去和未来的观测中提取状态坐标，并以岭正则化的闭式解估计这两个线性算子，该解恰好对应于相关协方差算子的伽辽金投影。在此表征基础上，我们在特征空间中推广了序列贝叶斯滤波和库普曼谱模态分解。在多种场景下的实验表明，即使在噪声和部分可观测性条件下，该方法在序列贝叶斯滤波和动态模态分解基线中仍展现出稳定且优越的性能。