# Instanton based importance sampling for rare events in stochastic PDEs

## TL;DR

This paper introduces an instanton-based importance sampling method for efficiently estimating rare events in stochastic PDEs, demonstrated on the Burgers equation, outperforming traditional simulation techniques.

## Contribution

The paper develops a novel instanton-based importance sampling approach that constrains dynamics around extreme events, improving rare event probability estimation in stochastic PDEs.

## Key findings

- Outperforms direct numerical simulations in estimating tail probabilities.
- Accurately quantifies the full probability density function of velocity gradients.
- Demonstrates effectiveness on the one-dimensional Burgers equation.

## Abstract

We present a new method for sampling rare and large fluctuations in a non-equilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the so-called instanton formalism that corresponds to a saddle-point approximation of the action in the path integral formulation of the underlying SPDE. The crucial step in our approach is the formulation of an alternative SPDE that incorporates knowledge of the instanton solution such that we are able to constrain the dynamical evolutions around extreme flow configurations only. Finally, a reweighting procedure based on the Girsanov theorem is applied to recover the full distribution function of the original system. The entire procedure is demonstrated on the example of the one-dimensional Burgers equation. Furthermore, we compare our method to conventional direct numerical simulations as well as to Hybrid Monte Carlo methods. It will be shown that the instanton-based sampling method outperforms both approaches and allows for an accurate quantification of the whole probability density function of velocity gradients from the core to the very far tails.