Partial Differential Equations (PDEs) with high dimensionality are commonly encountered in computational physics and engineering. However, finding solutions for these PDEs can be computationally expensive, making model-order reduction crucial. We propose such a data-driven scheme that automates the identification of the time-scales involved and can produce stable predictions forward in time as well as under different initial conditions not included in the training data. To this end, we combine a non-linear autoencoder architecture with a time-continuous model for the latent dynamics in the complex space. It readily allows for the inclusion of sparse and irregularly sampled training data. The learned, latent dynamics are interpretable and reveal the different temporal scales involved. We show that this data-driven scheme can automatically learn the independent processes that decompose a system of linear ODEs along the eigenvectors of the system's matrix. Apart from this, we demonstrate the applicability of the proposed framework in a hidden Markov Model and the (discretized) Kuramoto-Shivashinsky (KS) equation. Additionally, we propose a probabilistic version, which captures predictive uncertainties and further improves upon the results of the deterministic framework.

翻译：计算物理和工程领域中常遇到高维偏微分方程（PDE），但求解这些方程的计算成本高昂，因此模型降阶至关重要。我们提出了一种数据驱动方案，能够自动识别所涉及的时间尺度，并可在训练数据未包含的不同初始条件下实现向前时间的稳定预测。为此，我们将非线性自编码器架构与复数空间中潜在动力学的时间连续模型相结合，该模型可轻松处理稀疏和不规则采样的训练数据。学习到的潜在动力学具有可解释性，并能揭示所涉及的不同时间尺度。研究表明，该数据驱动方案可自动学习沿系统矩阵特征向量分解线性常微分方程系统的独立过程。此外，我们展示了该框架在隐马尔可夫模型和（离散化）Kuramoto-Shivashinsky（KS）方程中的适用性。同时，我们提出了一个概率版本，该版本可捕获预测不确定性并进一步改进确定性框架的结果。