  Predicting the occurence of rare and extreme events in complex systems is a
well-known problem in non-equilibrium physics. These events can have huge
impacts on human societies. New approaches have emerged in the last ten years,
which better estimate tail distributions. They often use large deviation
concepts without the need to perform heavy direct ensemble simulations. In
particular, a well-known approach is to derive a minimum action principle and
to find its minimizers.
  The analysis of rare reactive events in non-equilibrium systems without
detailed balance is notoriously difficult either theoretically and
computationally. They are described in the limit of small noise by the
Freidlin-Wentzell action. We propose here a new method which minimizes the
geometrical action instead using neural networks: it is called deep gMAM. It
relies on a natural and simple machine-learning formulation of the classical
gMAM approach. We give a detailed description of the method as well as many
examples. These include bimodal switches in complex stochastic (partial)
differential equations, quasi-potential estimates, and extreme events in
Burgers turbulence.
