Neural networks-based backward scheme for fully nonlinear PDEs

Abstract : We propose a numerical method for solving high dimensional fully nonlinear partial differential equations (PDEs). Our algorithm estimates simultaneously by backward time induction the solution and its gradient by multi-layer neural networks, through a sequence of learning problems obtained from the minimization of suitable quadratic loss functions and training simulations. This methodology extends to the fully non-linear case the approach recently proposed in [HPW19] for semi-linear PDEs. Numerical tests illustrate the performance and accuracy of our method on several examples in high dimension with nonlinearity on the Hessian term including a linear quadratic control problem with control on the diffusion coefficient.
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https://hal.archives-ouvertes.fr/hal-02196165
Contributor : Huyen Pham <>
Submitted on : Tuesday, July 30, 2019 - 2:12:52 AM
Last modification on : Friday, August 2, 2019 - 3:01:33 AM

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  • HAL Id : hal-02196165, version 1
  • ARXIV : 1908.00412

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Huyen Pham, Huyên Pham, Xavier Warin. Neural networks-based backward scheme for fully nonlinear PDEs. 2019. ⟨hal-02196165⟩

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