There has been some recent efforts in utilizing neural networks to approximate solutions of different types of PDEs. The key advantage of this approach is the possibility of tackling extremely high-dimensional problems, where traditional numerical approach involving discretization proves infeasible as the number of grid points scales exponentially with dimension.

Deep learning approaches sidestep the curse of dimensionality by converting the PDE problem into e.g. minimizing an energy functional https://arxiv.org/abs/1710.00211 or a stochastic optimal control problem https://arxiv.org/abs/2102.11379, then solve the related ERM problem via Monte Carlo sampling, whose convergence is independent of dimension.