Numerically solving a large parametric nonlinear dynamical system is challenging due to its high complexity and the high computational costs. In recent years, machine-learning-aided surrogates are being actively researched. However, many methods fail in accurately generalizing in the entire time interval $[0, T]$, when the training data is available only in a training time interval $[0, T_0]$, with $T_0<T$. To improve the extrapolation capabilities of the surrogate models in the entire time domain, we propose a new deep learning framework, where kernel dynamic mode decomposition (KDMD) is employed to evolve the dynamics of the latent space generated by the encoder part of a convolutional autoencoder (CAE). After adding the KDMD-decoder-extrapolated data into the original data set, we train the CAE along with a feed-forward deep neural network using the augmented data. The trained network can predict future states outside the training time interval at any out-of-training parameter samples. The proposed method is tested on two numerical examples: a FitzHugh-Nagumo model and a model of incompressible flow past a cylinder. Numerical results show accurate and fast prediction performance in both the time and the parameter domain.

翻译：由于高度复杂性和高昂计算成本，大规模参数化非线性动力系统的数值求解极具挑战性。近年来，基于机器学习的辅助代理模型正受到广泛研究。然而，当训练数据仅存在于训练时间区间$[0, T_0]$（其中$T_0<T$）时，许多方法难以在整个时间区间$[0, T]$内实现精确泛化。为提升代理模型在整个时域的外推能力，本文提出一种新型深度学习框架：该框架采用核动态模态分解（KDMD）来演化由卷积自编码器（CAE）编码器生成的潜空间动力学。将KDMD-解码器外推数据加入原始数据集后，我们利用增强数据同时训练CAE与前馈深度神经网络。训练完成的网络能够对任意训练外参数样本预测训练时间区间之外的状态。所提方法在两个数值算例中进行了验证：FitzHugh-Nagumo模型与圆柱绕流不可压缩流动模型。数值结果表明，该方法在时间和参数域均展现出精确且快速的预测性能。