用深度特征空间的谱方法，高效建模随机非线性系统动态。
Deep Spectral Learning of Embedded Latent Transfer Operators for Stochastic Dynamical Systems
- 通过神经编码器提取可观测数据的深层特征，构建马尔可夫隐状态。
- 在特征空间中用正则化闭式解估计转移与观测算子，实现稳定预测。
- 适合处理噪声和不完全观测下的系统建模，尤其适合复杂动力系统。
我们提出一种针对嵌入在深度特征空间中的随机非线性动力系统所表示的嵌入潜在转移算子的谱学习方法。该方法具体实现为深度谱编码器（DSE），一种基于算子的潜在状态空间模型：时间不变的神经编码器实现从观测值到可学习非线性特征映射，这些特征定义了马尔可夫隐状态，其时间演化与观测映射分别由转移算子和观测算子描述。在可学习的伽辽金投影特征空间中，通过函数型典型相关分析从过去与未来观测中提取状态坐标，并以岭回归正则化的闭式解估计两个线性算子，其形式等价于相应协方差算子的伽辽金投影。在此表示基础上，我们推广了特征空间中的顺序贝叶斯滤波与柯普曼谱模式分解。在多个实验场景中，即使存在噪声与部分可观测性，该方法仍表现出稳定且优于顺序贝叶斯滤波与动态模式分解基线的性能。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。