| Abstract: |
| Estimating the average amount of time needed for a stochastic particle to reach the boundary of a bounded domain $\Omega$ is a classical problem with applications in numerous fields. As there are no explicit solutions for this Mean Escape Time (MET) outside of the simplest domain geometries, practicioners have to rely on Monte Carlo simulations, series expansion ansatz, or grid-based methods which generally scale poorly with problem size and lack strong theoretical guarantees. By considering the MET as the solution of a PDE induced by the stochastic dynamics of the particle, we propose to learn the solution operator of the PDE by applying dimension reduction tools and Deep Learning. Leveraging results from the approximation theory of Deep Neural Networks, we provide a rigorous theoretical analysis of the sample complexity associated with the MET problem in our framework for different families of stochastic dynamics and domain geometries. |
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