  In this paper we establish a connection between non-convex optimization
methods for training deep neural networks and nonlinear partial differential
equations (PDEs). Relaxation techniques arising in statistical physics which
have already been used successfully in this context are reinterpreted as
solutions of a viscous Hamilton-Jacobi PDE. Using a stochastic control
interpretation allows we prove that the modified algorithm performs better in
expectation that stochastic gradient descent. Well-known PDE regularity results
allow us to analyze the geometry of the relaxed energy landscape, confirming
empirical evidence. The PDE is derived from a stochastic homogenization
problem, which arises in the implementation of the algorithm. The algorithms
scale well in practice and can effectively tackle the high dimensionality of
modern neural networks.
