A Theory-guided Auto-Encoder (TgAE) framework is proposed for surrogate construction, and is further used for uncertainty quantification and inverse modeling tasks. The framework is built based on the Auto-Encoder (or Encoder–Decoder) architecture of the convolutional neural network (CNN) via a theory-guided training process. In order to incorporate physical constraints for achieving theory-guided training, the governing equations of the studied problems can be discretized by the finite difference scheme, and then be embedded into the training of the CNN. The residual of the discretized governing equations, as well as the data mismatch, constitute the loss function of the TgAE. The trained TgAE can be utilized to construct a surrogate that approximates the relationship between the model parameters and model responses with limited labeled data. Several subsurface flow cases are designed to test the performance of the TgAE. The results demonstrate that satisfactory accuracy for surrogate modeling and higher efficiency for uncertainty quantification tasks can be achieved with the TgAE. The TgAE also shows good extrapolation ability for cases with different correlation lengths and variances. Furthermore, inverse modeling tasks are also implemented with the TgAE surrogate, and satisfactory results are obtained.

提出了一种理论引导的自动编码器 (TgAE) 框架用于代理构建，并进一步用于不确定性量化和逆向建模任务。该框架是基于卷积神经网络 (CNN) 的自动编码器（或编码器-解码器）架构通过理论指导的训练过程构建的。为了结合物理约束以实现理论指导的训练，研究问题的控制方程可以通过有限差分格式离散，然后嵌入到 CNN 的训练中。离散控制方程的残差以及数据不匹配构成了 TgAE 的损失函数。经过训练的 TgAE 可用于构建一个代理，该代理通过有限的标记数据来近似模型参数和模型响应之间的关系。设计了几个地下流动案例来测试 TgAE 的性能。结果表明，使用 TgAE 可以实现令人满意的代理建模准确性和更高的不确定性量化任务效率。对于具有不同相关长度和方差的案例，TgAE 也显示出良好的外推能力。此外，还使用 ​​TgAE 代理执行了逆向建模任务，并获得了令人满意的结果。