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

arXiv: https://arxiv.org/abs/2404.15731 · HTML: https://mlsanity.com/paper/2404.15731/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 41 references that are in the map.
- First posted on arXiv 24 Apr 2024.

## Builds on (most cited first, 17 of 17)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning](https://mlsanity.com/paper/2106.09004/) (2021)
- [Conditional Generative Moment-Matching Networks](https://mlsanity.com/paper/1606.04218/) (2016)
- [Learning the temporal evolution of multivariate densities via normalizing flows](https://mlsanity.com/paper/2107.13735/) (2021)
- [Variational Bayes Deep Operator Network: A data-driven Bayesian solver for parametric differential equations](https://mlsanity.com/paper/2206.05655/) (2022)
- [Replication-based emulation of the response distribution of stochastic simulators using generalized lambda distributions](https://mlsanity.com/paper/1911.09067/) (2019)
- [Emulation of stochastic simulators using generalized lambda models](https://mlsanity.com/paper/2007.00996/) (2020)
- [Deep Physics Corrector: A physics enhanced deep learning architecture for solving stochastic differential equations](https://mlsanity.com/paper/2209.09750/) (2022)
- [Stochastic simulators based optimization by Gaussian process metamodels -- Application to maintenance investments planning issues](https://mlsanity.com/paper/1512.07060/) (2015)
- [Generative modeling of time-dependent densities via optimal transport and projection pursuit](https://mlsanity.com/paper/2304.09663/) (2023)

## Built on by (most cited first, 2 of 2)

- [Towards Gaussian Process for operator learning: an uncertainty aware resolution independent operator learning algorithm for computational mechanics](https://mlsanity.com/paper/2409.10972/) (2024)
- [Multimodal Scientific Learning Beyond Diffusions and Flows](https://mlsanity.com/paper/2602.00960/) (2026)

## Abstract (arXiv)

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.