MD-NOMAD: Mixture density nonlinear manifold decoder for emulating stochastic differential equations and uncertainty propagation

TL;DR

MD-NOMAD is a neural operator framework that combines mixture density models with nonlinear manifold decoders to efficiently emulate stochastic differential equations and propagate uncertainty.

Contribution

It introduces a novel neural operator approach that estimates complex probability distributions for stochastic outputs, enhancing scalability and accuracy.

Findings

Effective in modeling stochastic differential equations

Accurate uncertainty propagation demonstrated

Scalable to high-dimensional problems

Abstract

We propose a neural operator framework, termed mixture density nonlinear manifold decoder (MD-NOMAD), for stochastic simulators. Our approach leverages an amalgamation of the pointwise operator learning neural architecture nonlinear manifold decoder (NOMAD) with mixture density-based methods to estimate conditional probability distributions for stochastic output functions. MD-NOMAD harnesses the ability of probabilistic mixture models to estimate complex probability and the high-dimensional scalability of pointwise neural operator NOMAD. We conduct empirical assessments on a wide array of stochastic ordinary and partial differential equations and present the corresponding results, which highlight the performance of the proposed framework.