# Multimodal Scientific Learning Beyond Diffusions and Flows

arXiv: https://arxiv.org/abs/2602.00960 · HTML: https://mlsanity.com/paper/2602.00960/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 56 references that are in the map.
- First posted on arXiv 1 Feb 2026.

## Builds on (most cited first, 28 of 28)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Recurrent World Models Facilitate Policy Evolution](https://mlsanity.com/paper/1809.01999/) (2018)
- [Mixture-of-Experts with Expert Choice Routing](https://mlsanity.com/paper/2202.09368/) (2022)
- [High-Performance Large-Scale Image Recognition Without Normalization](https://mlsanity.com/paper/2102.06171/) (2021)
- [A Neural Representation of Sketch Drawings](https://mlsanity.com/paper/1704.03477/) (2017)
- [What Are Bayesian Neural Network Posteriors Really Like?](https://mlsanity.com/paper/2104.14421/) (2021)
- [On Neural Differential Equations](https://mlsanity.com/paper/2202.02435/) (2022)
- [Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data](https://mlsanity.com/paper/2302.07194/) (2023)
- [Dynamical Regimes of Diffusion Models](https://mlsanity.com/paper/2402.18491/) (2024)
- [SMD-Nets: Stereo Mixture Density Networks](https://mlsanity.com/paper/2104.03866/) (2021)
- [Epistemic Neural Networks](https://mlsanity.com/paper/2107.08924/) (2021)
- [On the difficulty of learning chaotic dynamics with RNNs](https://mlsanity.com/paper/2110.07238/) (2021)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Discovering and forecasting extreme events via active learning in neural operators](https://mlsanity.com/paper/2204.02488/) (2022)
- [Mixture Dense Regression for Object Detection and Human Pose Estimation](https://mlsanity.com/paper/1912.00821/) (2019)
- [Near-optimal-sample estimators for spherical Gaussian mixtures](https://mlsanity.com/paper/1402.4746/) (2014)
- [Variational Autoencoding Neural Operators](https://mlsanity.com/paper/2302.10351/) (2023)
- [Gaussian Mixture Solvers for Diffusion Models](https://mlsanity.com/paper/2311.00941/) (2023)
- [Deep Learning for Bayesian Optimization of Scientific Problems with High-Dimensional Structure](https://mlsanity.com/paper/2104.11667/) (2021)
- [Canonical normalizing flows for manifold learning](https://mlsanity.com/paper/2310.12743/) (2023)
- [Deep Learning Alternatives of the Kolmogorov Superposition Theorem](https://mlsanity.com/paper/2410.01990/) (2024)
- [Composite Bayesian Optimization In Function Spaces Using NEON -- Neural Epistemic Operator Networks](https://mlsanity.com/paper/2404.03099/) (2024)
- [Equivariant Flow Matching for Symmetry-Breaking Bifurcation Problems](https://mlsanity.com/paper/2509.03340/) (2025)
- [MD-NOMAD: Mixture density nonlinear manifold decoder for emulating stochastic differential equations and uncertainty propagation](https://mlsanity.com/paper/2404.15731/) (2024)
- [Active search for Bifurcations](https://mlsanity.com/paper/2406.11141/) (2024)

## Built on by (most cited first, 1 of 1)

- [Bi-FORK: Generative Modeling of High-Dimensional Bifurcating Systems](https://mlsanity.com/paper/2610.12449/) (2026)

## Abstract (arXiv)

Scientific machine learning (SciML) increasingly requires models that capture multimodal conditional uncertainty arising from ill-posed inverse problems, multistability, and chaotic dynamics. While recent work has favored highly expressive implicit generative models such as diffusion and flow-based methods, these approaches are often data-hungry, computationally costly, and misaligned with the structured solution spaces frequently found in scientific problems. We demonstrate that Mixture Density Networks (MDNs) provide a principled yet largely overlooked alternative for multimodal uncertainty quantification in SciML. As explicit parametric density estimators, MDNs impose an inductive bias tailored to low-dimensional, multimodal physics, enabling direct global allocation of probability mass across distinct solution branches. This structure delivers strong data efficiency, allowing reliable recovery of separated modes in regimes where scientific data is scarce. We formalize these insights through a unified probabilistic framework contrasting explicit and implicit distribution networks, and demonstrate empirically that MDNs achieve superior generalization, interpretability, and sample efficiency across a range of inverse, multistable, and chaotic scientific regression tasks.