Physical systems whose dynamics are governed by partial differential equations (PDEs) find numerous applications in science and engineering. The process of obtaining the solution from such PDEs may be computationally expensive for large-scale and parameterized problems. In this work, deep learning techniques developed especially for time-series forecasts, such as LSTM and TCN, or for spatial-feature extraction such as CNN, are employed to model the system dynamics for advection-dominated problems. This paper proposes a Convolutional Autoencoder(CAE) model for compression and a CNN future-step predictor for forecasting. These models take as input a sequence of high-fidelity vector solutions for consecutive time steps obtained from the PDEs and forecast the solutions for the subsequent time steps using auto-regression; thereby reducing the computation time and power needed to obtain such high-fidelity solutions. Non-intrusive reduced-order modeling techniques such as deep auto-encoder networks are utilized to compress the high-fidelity snapshots before feeding them as input to the forecasting models in order to reduce the complexity and the required computations in the online and offline stages. The models are tested on numerical benchmarks (1D Burgers’ equation and Stoker’s dam-break problem) to assess the long-term prediction accuracy, even outside the training domain (i.e. extrapolation). The most accurate model is then used to model a hypothetical dam break in a river with complex 2D bathymetry. The proposed CNN future-step predictor revealed much more accurate forecasting than LSTM and TCN in the considered spatiotemporal problems.

中文翻译：

---

用于时间相关流问题的外推预测的深度卷积架构

动力学由偏微分方程 (PDE) 控制的物理系统在科学和工程中有着广泛的应用。对于大规模参数化问题，从此类偏微分方程获得解的过程可能在计算上非常昂贵。在这项工作中，采用专门为时间序列预测（例如 LSTM 和 TCN）或空间特征提取（例如 CNN）开发的深度学习技术来对平流主导问题的系统动力学进行建模。本文提出了一种用于压缩的卷积自动编码器（CAE）模型和一种用于预测的 CNN 未来步骤预测器。这些模型将从偏微分方程获得的连续时间步的高保真向量解序列作为输入，并使用自回归预测后续时间步的解；从而减少获得这种高保真解所需的计算时间和功率。利用深度自动编码器网络等非侵入式降阶建模技术来压缩高保真快照，然后将其作为预测模型的输入，以降低在线和离线阶段的复杂性和所需的计算量。这些模型在数值基准（一维伯格斯方程和斯托克溃坝问题）上进行测试，以评估长期预测精度，甚至在训练域之外（即外推法）。然后，使用最准确的模型通过复杂的二维测深对河流中假设的溃坝进行建模。在所考虑的时空问题中，所提出的 CNN 未来步骤预测器比 LSTM 和 TCN 表现出更准确的预测。