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Integral equations and machine learning

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  • Keller, Alexander
  • Dahm, Ken

Abstract

As both light transport simulation and reinforcement learning are ruled by the same Fredholm integral equation of the second kind, reinforcement learning techniques may be used for photorealistic image synthesis: Efficiency may be dramatically improved by guiding light transport paths by an approximate solution of the integral equation that is learned during rendering. In the light of the recent advances in reinforcement learning for playing games, we investigate the representation of an approximate solution of an integral equation by artificial neural networks and derive a loss function for that purpose. The resulting Monte Carlo and quasi-Monte Carlo methods train neural networks with standard information instead of linear information and naturally are able to generate an arbitrary number of training samples. The methods are demonstrated for applications in light transport simulation.

Suggested Citation

  • Keller, Alexander & Dahm, Ken, 2019. "Integral equations and machine learning," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 161(C), pages 2-12.
    • Handle: RePEc:eee:matcom:v:161:y:2019:i:c:p:2-12
    DOI: 10.1016/j.matcom.2019.01.010
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    References listed on IDEAS

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    1. Keller, Alexander, 2001. "Hierarchical Monte Carlo image synthesis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 55(1), pages 79-92.
    2. David Silver & Julian Schrittwieser & Karen Simonyan & Ioannis Antonoglou & Aja Huang & Arthur Guez & Thomas Hubert & Lucas Baker & Matthew Lai & Adrian Bolton & Yutian Chen & Timothy Lillicrap & Fan , 2017. "Mastering the game of Go without human knowledge," Nature, Nature, vol. 550(7676), pages 354-359, October.
    3. David Silver & Aja Huang & Chris J. Maddison & Arthur Guez & Laurent Sifre & George van den Driessche & Julian Schrittwieser & Ioannis Antonoglou & Veda Panneershelvam & Marc Lanctot & Sander Dieleman, 2016. "Mastering the game of Go with deep neural networks and tree search," Nature, Nature, vol. 529(7587), pages 484-489, January.
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    Cited by:

    1. Parand, K. & Aghaei, A.A. & Jani, M. & Ghodsi, A., 2021. "A new approach to the numerical solution of Fredholm integral equations using least squares-support vector regression," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 180(C), pages 114-128.

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