  Machine learning methods for solving nonlinear partial differential equations
(PDEs) are hot topical issues, and different algorithms proposed in the
literature show efficient numerical approximation in high dimension. In this
paper, we introduce a class of PDEs that are invariant to permutations, and
called symmetric PDEs. Such problems are widespread, ranging from cosmology to
quantum mechanics, and option pricing/hedging in multi-asset market with
exchangeable payoff. Our main application comes actually from the particles
approximation of mean-field control problems. We design deep learning
algorithms based on certain types of neural networks, named PointNet and
DeepSet (and their associated derivative networks), for computing
simultaneously an approximation of the solution and its gradient to symmetric
PDEs. We illustrate the performance and accuracy of the PointNet/DeepSet
networks compared to classical feedforward ones, and provide several numerical
results of our algorithm for the examples of a mean-field systemic risk,
mean-variance problem and a min/max linear quadratic McKean-Vlasov control
problem.
