Identification of groundwater contaminant sources in a highly-heterogenous geosystems results in a high-dimensional inverse problem and is often solved based on a surrogate model to alleviate the computational burden. Surrogate modeling through deep learning has a great potential for learning complex nonlinear relationships between model inputs and outputs. Most of the developed surrogates, however, can only estimate the contaminant concentration fields at a limit number of time-steps with a relatively large lag. In this paper, a transformer-based surrogate model is developed to provide a detailed release history of contaminant, which allows more accurate analyzing the distributions and planning. As such, a Koopman-operator-based convolutional autoencoder is trained and fixed prior to the training of transformer. Here, the encoder converts the concentration fields into a one-dimensional embedding space and the transformer is trained on this space to learn the system dynamics and predict the embedding feature at next time step, which is reconstructed back to the original space with the decoder. The proposed surrogate model is tested on a complex problem and the results demonstrate that the proposed transformer-based surrogate can efficiently provide an accurate estimation of the evolution of contaminant concentration field at a greater number of time-steps compared to the previous works.

在高度异质的地质系统中识别地下水污染源会导致高维逆问题，并且通常基于替代模型来解决以减轻计算负担。通过深度学习进行代理建模具有学习模型输入和输出之间复杂非线性关系的巨大潜力。然而，大多数已开发的替代方法只能在具有相对较大滞后的有限时间步长处估计污染物浓度场。在本文中，开发了一个基于变压器的替代模型，以提供污染物的详细释放历史，从而可以更准确地分析分布和规划。因此，基于 Koopman 算子的卷积自编码器在训练转换器之前被训练和固定。这里，编码器将浓度场转换为一维嵌入空间，并在该空间上训练转换器以学习系统动力学并预测下一个时间步的嵌入特征，然后用解码器将其重建回原始空间。所提出的替代模型在一个复杂问题上进行了测试，结果表明，与以前的工作相比，所提出的基于变压器的替代模型可以在更多的时间步长上有效地提供对污染物浓度场演变的准确估计。