Neural approximation of PDE solutions: An application to reachability computations

We consider the problem of computing viability sets for nonlinear continuous systems. Our main goal is to deal with the "curse of dimensionality", the exponential growth of the computation in the dimension of the state space. The viability problem is formulated as an optimal control problem whose value function is known to be a viscosity solution to a particular type of Hamilton Jacobi partial differential equation. We propose a trial solution based on a feed-forward neural network for the Hamilton Jacobi equation with the given boundary conditions. We use random extractions from the state space to generate training points and then employ the r-algorithm for non smooth optimization to train the network. We illustrate the method on a 2 dimensional example from aerodynamic envelope protection