A general theorem shows that neural networks inherit the absence of the curse of dimensionality from any discrete Monte Carlo scheme they can emulate, with applications to Kolmogorov PDEs.
Convergence of the Deep BSDE Method for Coupled FBSDEs
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abstract
The recently proposed numerical algorithm, deep BSDE method, has shown remarkable performance in solving high-dimensional forward-backward stochastic differential equations (FBSDEs) and parabolic partial differential equations (PDEs). This article lays a theoretical foundation for the deep BSDE method in the general case of coupled FBSDEs. In particular, a posteriori error estimation of the solution is provided and it is proved that the error converges to zero given the universal approximation capability of neural networks. Numerical results are presented to demonstrate the accuracy of the analyzed algorithm in solving high-dimensional coupled FBSDEs.
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Deep neural network approximations for Monte Carlo algorithms
A general theorem shows that neural networks inherit the absence of the curse of dimensionality from any discrete Monte Carlo scheme they can emulate, with applications to Kolmogorov PDEs.