WIAS Preprint No. 2544, (2018)

Variational Monte Carlo -- Bridging concepts of machine learning and high dimensional partial differential equations


  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Trunschke, Philipp
  • Schneider, Reinhold
  • Wolf, Sebastian

2010 Mathematics Subject Classification

  • 35R60 47B80 60H35 65C20 65N12 65N22 65J10


  • Partial differential equations with random coefficients, tensor representation, tensor train, uncertainty quantification, stochastic finite element methods, log-normal, adaptive methods, ALS, low-rank, reduced basis methods




A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.

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