High-fidelity numerical simulations of partial differential equations (PDEs) given a restricted computational budget can significantly limit the number of parameter configurations considered and/or time window evaluated for modeling a given system. Multi-fidelity surrogate modeling aims to leverage less accurate, lower-fidelity models that are computationally inexpensive in order to enhance predictive accuracy when high-fidelity data are limited or scarce. However, low-fidelity models, while often displaying important qualitative spatio-temporal features, fail to accurately capture the onset of instability and critical transients observed in the high-fidelity models, making them impractical as surrogate models. To address this shortcoming, we present a new data-driven strategy that combines dimensionality reduction with multi-fidelity neural network surrogates. The key idea is to generate a spatial basis by applying the classical proper orthogonal decomposition (POD) to high-fidelity solution snapshots, and approximate the dynamics of the reduced states - time-parameter-dependent expansion coefficients of the POD basis - using a multi-fidelity long-short term memory (LSTM) network. By mapping low-fidelity reduced states to their high-fidelity counterpart, the proposed reduced-order surrogate model enables the efficient recovery of full solution fields over time and parameter variations in a non-intrusive manner. The generality and robustness of this method is demonstrated by a collection of parametrized, time-dependent PDE problems where the low-fidelity model can be defined by coarser meshes and/or time stepping, as well as by misspecified physical features. Importantly, the onset of instabilities and transients are well captured by this surrogate modeling technique.

翻译：在有限计算预算下对偏微分方程（PDE）进行高保真度数值模拟，通常会显著限制所考虑的参数配置数量或/及评估给定系统模型的时间窗口。多保真度代理建模旨在利用计算成本较低但精度稍低的低保真度模型，以在高保真度数据有限或稀缺时提升预测准确性。然而，低保真度模型虽然常能呈现重要的定性时空特征，却难以准确捕捉高保真度模型中观测到的不稳定性起始点和关键瞬态过程，导致其作为代理模型实用性不足。为克服这一缺陷，我们提出了一种新的数据驱动策略，将降维与多保真度神经网络代理相结合。核心思想是通过对高保真度解快照应用经典本征正交分解（POD）生成空间基，并利用多保真度长短期记忆（LSTM）网络近似降维状态（即POD基中依赖于时间-参数的展开系数）的动力学过程。通过将低保真度降维状态映射至高保真度对应状态，所提出的降阶代理模型能以非侵入式方式高效恢复完整解场随时间和参数变化的信息。该方法通过一系列参数化、随时间变化的PDE问题验证了其普适性与鲁棒性，其中低保真度模型可通过粗网格和/或大步长时间步长定义，甚至可包含物理特征误设的情况。重要的是，该代理建模技术能良好捕捉不稳定性起始点及瞬态过程。