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

Abstract

A statistical learning approach for high-dimensional 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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Acknowledgments

R. Schneider was supported by the DFG project ERA Chemistry and through Matheon by the Einstein Foundation Berlin. M. Eigel was supported by the DFG SPP1886.

Funding

Research of S. Wolf was funded in part by the DFG Matheon project SE10. P. Trunschke acknowledges support by the Berlin International Graduate School in Model and Simulation based Research (BIMoS).

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Correspondence to Philipp Trunschke.

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Communicated by: Anthony Nouy

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Eigel, M., Schneider, R., Trunschke, P. et al. Variational Monte Carlo—bridging concepts of machine learning and high-dimensional partial differential equations. Adv Comput Math 45, 2503–2532 (2019). https://doi.org/10.1007/s10444-019-09723-8

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  • DOI: https://doi.org/10.1007/s10444-019-09723-8

Keywords

  • Machine learning
  • Uncertainty quantification
  • Partial differential equations
  • Statistical learning
  • Tree tensor networks

Mathematics Subject Classification (2010)

  • 68Q32
  • 65D15
  • 62J02
  • 15A69
  • 41A63
  • 65N30
  • 35R60
  • 65C20
  • 65N22