Mixture Density Nonlinear Manifold Decoder for Emulating Stochastic Simulators and One-Shot Uncertainty Propagation

We propose a neural operator framework, termed mixture density nonlinear manifold decoder (MD-NOMAD), for emulating the conditional response distribution of stochastic simulators. Our framework leverages an amalgamation of the pointwise operator learning neural architecture, the nonlinear manifold decoder (NOMAD), with versatile mixture density-based methods for approximating the conditional probability distributions of stochastic output functions. Furthermore, MD-NOMAD harnesses the ability of probabilistic mixture models to estimate complex probability distributions and inherits the high-dimensional scalability of NOMAD. A key feature of MD-NOMAD is its ability to perform one-shot uncertainty propagation, eliminating the need for Monte Carlo methods in statistical computations for uncertainty propagation tasks. 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.