MD-NOMAD: Mixture density nonlinear manifold decoder for emulating stochastic differential equations and uncertainty propagation
This work addresses uncertainty propagation in stochastic differential equations for scientific computing applications, representing an incremental hybrid approach building on existing neural operator and mixture density methods.
The authors tackled the problem of emulating stochastic simulators by proposing MD-NOMAD, a neural operator framework that combines pointwise operator learning with mixture density methods to estimate conditional probability distributions for stochastic output functions. The results demonstrated performance improvements across various stochastic ordinary and partial differential equations, though no concrete numerical metrics were provided.
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.