Scalable conditional deep inverse Rosenblatt transports using tensor trains and gradient-based dimension reduction

Tiangang Cui, Sergey Dolgov, Olivier Zahm

Research output: Contribution to journalArticlepeer-review

4 Citations (SciVal)


We present a novel offline-online method to mitigate the computational burden of the characterization of posterior random variables in statistical learning. In the offline phase, the proposed method learns the joint law of the parameter random variables and the observable random variables in the tensor-train (TT) format. In the online phase, the resulting order-preserving conditional transport can characterize the posterior random variables given newly observed data in real time. Compared with the state-of-the-art normalizing flow techniques, the proposed method relies on function approximation and is equipped with a thorough performance analysis. The function approximation perspective also allows us to further extend the capability of transport maps in challenging problems with high-dimensional observations and high-dimensional parameters. On the one hand, we present novel heuristics to reorder and/or reparametrize the variables to enhance the approximation power of TT. On the other hand, we integrate the TT-based transport maps and the parameter reordering/reparametrization into layered compositions to further improve the performance of the resulting transport maps. We demonstrate the efficiency of the proposed method on various statistical learning tasks in ordinary differential equations (ODEs) and partial differential equations (PDEs).

Original languageEnglish
Article number112103
JournalJournal of Computational Physics
Early online date30 Mar 2023
Publication statusE-pub ahead of print - 30 Mar 2023

Bibliographical note

Funding Information:
TC acknowledges support from the Australian Research Council under the grant DP210103092 . SD is thankful for the support from the Engineering and Physical Sciences Research Council under the New Investigator Award EP/T031255/1 . OZ acknowledges support from the French National Research Agency JCJC project MODENA ( ANR-21-CE46-0006-01 ).


  • Approximate Bayesian computation
  • Dimension reduction
  • Generative models
  • Inverse problems
  • Markov chain Monte Carlo
  • tensor train
  • Transport maps

ASJC Scopus subject areas

  • Numerical Analysis
  • Modelling and Simulation
  • Physics and Astronomy (miscellaneous)
  • General Physics and Astronomy
  • Computer Science Applications
  • Computational Mathematics
  • Applied Mathematics


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