Variational Monte Carlo approach to partial differential equations with neural networks

Multilevel Monte Carlo method with applications to stochastic partial differential equations

In this work, the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to O(h−2) if h is the error of the approximation. The mult...

Generalizing Hamiltonian Monte Carlo with Neural Networks

We present a general-purpose method to train Markov chain Monte Carlo kernels, parameterized by deep neural networks, that converge and mix quickly to their target distribution. Our method generalizes Hamiltonian Monte Carlo and is trained to maximize expected squared jumped distance, a proxy for mixing speed. We demonstrate large empirical gains on a collection of simple but challenging distri...

A Multimodes Monte Carlo Finite Element Method for Elliptic Partial Differential Equations with Random Coefficients

This paper develops and analyzes an efficient numerical method for solving elliptic partial differential equations, where the diffusion coefficients are random perturbations of deterministic diffusion coefficients. The method is based upon a multimodes representation of the solution as a power series of the perturbation parameter, and the Monte Carlo technique for sampling the probability space...

Multi-Level Monte Carlo Finite Element Method for Elliptic Partial Differential Equations with Stochastic Data

Variational Sequential Monte Carlo

Many recent advances in large scale probabilistic inference rely on variational methods. The success of variational approaches depends on (i) formulating a flexible parametric family of distributions, and (ii) optimizing the parameters to find the member of this family that most closely approximates the exact posterior. In this paper we present a new approximating family of distributions, the v...