Reduced-order models are indispensable for multi-query or real-time problems. However, there are still many challenges to constructing efficient ROMs for time-dependent parametrized problems. Using a linear reduced space is inefficient for time-dependent nonlinear problems, especially for transport-dominated problems. The non-linearity usually needs to be addressed by hyper-reduction techniques, such as DEIM, but it is intrusive and relies on the assumption of affine dependence of parameters. This paper proposes and studies a non-intrusive reduced-order modeling approach for time-dependent parametrized problems. It is purely data-driven and naturally split into offline and online stages. During the offline stage, a convolutional autoencoder, consisting of an encoder and a decoder, is trained to perform dimensionality reduction. The encoder compresses the full-order solution snapshots to a nonlinear manifold or a low-dimensional reduced/latent space. The decoder allows the recovery of the full-order solution from the latent space. To deal with the time-dependent problems, a high-order dynamic mode decomposition (HODMD) is utilized to model the trajectories in the latent space for each parameter. During the online stage, the HODMD models are first utilized to obtain the latent variables at a new time, then interpolation techniques are adopted to recover the latent variables at a new parameter value, and the full-order solution is recovered by the decoder. Some numerical tests are conducted to show that the approach can be used to predict the unseen full-order solution at new times and parameter values fast and accurately, including transport-dominated problems.

翻译：降阶模型对于多查询或实时问题不可或缺。然而，针对时间依赖参数化问题构建高效的ROM仍面临诸多挑战。使用线性降阶空间对于时间依赖非线性问题效率低下，尤其在输运主导问题中。非线性通常需要借助超降阶技术（如DEIM）处理，但这类方法具有侵入性且依赖仿射参数假设。本文提出并研究了一种面向时间依赖参数化问题的非侵入式降阶建模方法。该方法完全基于数据驱动，自然划分为离线与在线两个阶段。离线阶段训练由编码器和解码器组成的卷积自编码器以实现降维：编码器将全阶解快照压缩至非线性流形或低维降阶/潜空间，解码器则能从潜空间恢复全阶解。针对时间依赖问题，采用高阶动态模式分解对每个参数在潜空间中的轨迹进行建模。在线阶段首先利用HODMD模型获取新时刻的潜变量，随后通过插值技术恢复新参数值下的潜变量，最终由解码器重建全阶解。数值实验表明，该方法能够快速准确地预测未见时间点与参数值下的全阶解，包括输运主导问题。