A deep branching solver for fully nonlinear partial differential equations
This work addresses a computational bottleneck in solving complex PDEs for fields like physics and finance, though it appears incremental as it builds on existing branching and deep learning techniques.
The authors tackled the numerical solution of fully nonlinear PDEs with gradient terms by developing a deep learning stochastic branching algorithm, which outperformed other deep learning methods like BSDE or Galerkin approaches in fully nonlinear examples.
We present a multidimensional deep learning implementation of a stochastic branching algorithm for the numerical solution of fully nonlinear PDEs. This approach is designed to tackle functional nonlinearities involving gradient terms of any orders, by combining the use of neural networks with a Monte Carlo branching algorithm. In comparison with other deep learning PDE solvers, it also allows us to check the consistency of the learned neural network function. Numerical experiments presented show that this algorithm can outperform deep learning approaches based on backward stochastic differential equations or the Galerkin method, and provide solution estimates that are not obtained by those methods in fully nonlinear examples.