Deep-Learning-Based Adjoint State Method: Methodology and Preliminary Application to Inverse Modeling

Cong Xiao, Ya Deng, Guangdong Wang

Research output: Contribution to journalArticleScientificpeer-review


We present an efficient adjoint model based on the deep-learning surrogate to address high-dimensional inverse modeling with an application to subsurface transport. The proposed method provides a completely code nonintrusive and computationally feasible way to approximate the model derivatives, which subsequently can be used to derive gradients for inverse modeling. This conceptual deep-learning framework, that is, an architecture of deep convolutional neural network through combining autoencoder and autoregressive structure, efficiently produces an analogously analytical adjoint with the help of auto-differentiation module in the popular deep-learning packages. We intentionally retain training data at the specific time instances where the measurements are taken, the storage of the intermediate states and computation of their adjoint, therefore, are completely avoided. This proposed adjoint state method is tested on a synthetic two-dimensional model for parameter estimations. The preliminary results reveal the feasibility of the proposed adjoint state method in term of computational efficiency and programming flexibility.

Original languageEnglish
Article numbere2020WR027400
Pages (from-to)1-20
Number of pages20
JournalWater Resources Research
Issue number2
Publication statusPublished - 2021


  • adjoint
  • autoregressive neural network
  • deep learning
  • inverse modeling

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