Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism

Qian Wang, Nicolò Ripamonti, Jan S. Hesthaven

Research output: Contribution to journalArticlepeer-review

60 Scopus citations


Closure modeling based on the Mori-Zwanzig formalism has proven effective to improve the stability and accuracy of projection-based model order reduction. However, closure models are often expensive and infeasible for complex nonlinear systems. Towards efficient model reduction of general problems, this paper presents a recurrent neural network (RNN) closure of parametric POD-Galerkin reduced-order model. Based on the short time history of the reduced-order solutions, the RNN predicts the memory integral which represents the impact of the unresolved scales on the resolved scales. A conditioned long short term memory (LSTM) network is utilized as the regression model of the memory integral, in which the POD coefficients at a number of time steps are fed into the LSTM units, and the physical/geometrical parameters are fed into the initial hidden state of the LSTM. The reduced-order model is integrated in time using an implicit-explicit (IMEX) Runge-Kutta scheme, in which the memory term is integrated explicitly and the remaining right-hand-side term is integrated implicitly to improve the computational efficiency. Numerical results demonstrate that the RNN closure can significantly improve the accuracy and efficiency of the POD-Galerkin reduced-order model of nonlinear problems. The POD-Galerkin reduced-order model with the RNN closure is also shown to be capable of making accurate predictions, well beyond the time interval of the training data.

Original languageEnglish
Article number109402
JournalJournal of Computational Physics
StatePublished - 1 Jun 2020
Externally publishedYes


  • Conditioned long-short term memory
  • Implicit-explicit Runge-Kutta
  • Memory closure
  • Model reduction
  • POD-Galerkin


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