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Pré-Publication, Document De Travail Année : 2019

Neural networks-based backward scheme for fully nonlinear PDEs

Résumé

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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Dates et versions

hal-02196165 , version 1 (30-07-2019)
hal-02196165 , version 2 (28-05-2020)
hal-02196165 , version 3 (10-12-2020)

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Huyen Pham, Huyên Pham, Xavier Warin. Neural networks-based backward scheme for fully nonlinear PDEs. 2019. ⟨hal-02196165v1⟩
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