Özalp Elise, Magri Luca
Department of Aeronautics, Imperial College London, South Kensington Campus, London, SW7 2BX UK.
The Alan Turing Institute, 96 Euston Road, London, NW1 2DB UK.
Nonlinear Dyn. 2025;113(11):13791-13806. doi: 10.1007/s11071-024-10712-w. Epub 2025 Feb 4.
Partial differential equations, and their chaotic solutions, are pervasive in the modelling of complex systems in engineering, science, and beyond. Data-driven methods can find solutions to partial differential equations with a divide-and-conquer strategy: The solution is sought in a latent space, on which the temporal dynamics are inferred ("latent-space" approach). This is achieved by, first, compressing the data with an autoencoder, and, second, inferring the temporal dynamics with recurrent neural networks. The overarching goal of this paper is to show that a latent-space approach can not only infer the solution of a chaotic partial differential equation, but it can also predict the stability properties of the physical system. First, we employ the convolutional autoencoder echo state network (CAE-ESN) on the chaotic Kuramoto-Sivashinsky equation for various chaotic regimes. We show that the CAE-ESN (i) finds a low-dimensional latent-space representation of the observations and (ii) accurately infers the Lyapunov exponents and covariant Lyapunov vectors (CLVs) in this low-dimensional manifold for different attractors. Second, we extend the CAE-ESN to a turbulent flow, comparing the Lyapunov spectrum to estimates obtained from Jacobian-free methods. A latent-space approach based on the CAE-ESN effectively produces a latent space that preserves the key properties of the chaotic system, such as Lyapunov exponents and CLVs, thus retaining the geometric structure of the attractor. The latent-space approach based on the CAE-ESN is a reduced-order model that accurately predicts the dynamics of the chaotic system, or, alternatively, it can be used to infer stability properties of chaotic systems from data.
偏微分方程及其混沌解在工程、科学及其他领域的复杂系统建模中普遍存在。数据驱动方法可以通过分而治之的策略找到偏微分方程的解：在一个潜在空间中寻找解，在该空间上推断时间动态（“潜在空间”方法）。这通过以下方式实现：首先，使用自动编码器压缩数据；其次，使用递归神经网络推断时间动态。本文的总体目标是表明，潜在空间方法不仅可以推断混沌偏微分方程的解，还可以预测物理系统的稳定性。首先，我们将卷积自动编码器回声状态网络（CAE-ESN）应用于混沌Kuramoto-Sivashinsky方程的各种混沌状态。我们表明，CAE-ESN（i）找到了观测值的低维潜在空间表示，并且（ii）在这个低维流形中针对不同吸引子准确推断了李雅普诺夫指数和协变李雅普诺夫向量（CLV）。其次，我们将CAE-ESN扩展到湍流，将李雅普诺夫谱与从无雅可比方法获得的估计值进行比较。基于CAE-ESN的潜在空间方法有效地产生了一个保留混沌系统关键属性（如李雅普诺夫指数和CLV）的潜在空间，从而保留了吸引子的几何结构。基于CAE-ESN的潜在空间方法是一种降阶模型，它可以准确预测混沌系统的动态，或者也可以用于从数据中推断混沌系统的稳定性。
