# A Multimodal Conditional Mixture Model with Distribution-Level Physics Priors

arXiv: https://arxiv.org/abs/2602.10451 · HTML: https://mlsanity.com/paper/2602.10451/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 58 references that are in the map.
- First posted on arXiv 11 Feb 2026.

## Builds on (most cited first, 18 of 18)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [A Physics-informed Diffusion Model for High-fidelity Flow Field Reconstruction](https://mlsanity.com/paper/2211.14680/) (2022)
- [Physics-Informed Diffusion Models](https://mlsanity.com/paper/2403.14404/) (2024)
- [CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems](https://mlsanity.com/paper/2312.10527/) (2023)
- [A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables](https://mlsanity.com/paper/2006.01789/) (2020)
- [A General Framework Combining Generative Adversarial Networks and Mixture Density Networks for Inverse Modeling in Microstructural Materials Design](https://mlsanity.com/paper/2101.10553/) (2021)
- [Micro-Macro Consistency in Multiscale Modeling: Score-Based Model Assisted Sampling of Fast/Slow Dynamical Systems](https://mlsanity.com/paper/2312.05715/) (2023)
- [Generative Learning for Slow Manifolds and Bifurcation Diagrams](https://mlsanity.com/paper/2504.20375/) (2025)
- [Physics-constrained Gaussian Processes for Predicting Shockwave Hugoniot Curves](https://mlsanity.com/paper/2601.06655/) (2026)

## Abstract (arXiv)

Many scientific and engineering systems exhibit intrinsically multimodal behavior arising from latent regime switching and non-unique physical mechanisms. In such settings, learning the full conditional distribution of admissible outcomes in a physically consistent and interpretable manner remains a challenge. While recent advances in machine learning have enabled powerful multimodal generative modeling, their integration with physics-constrained scientific modeling remains nontrivial, particularly when physical structure must be preserved or data are limited. This work develops a physics-informed multimodal conditional modeling framework based on mixture density representations. Mixture density networks (MDNs) provide an explicit and interpretable parameterization of multimodal conditional distributions. Physical knowledge is embedded through component-specific regularization terms that penalize violations of governing equations or physical laws. This formulation naturally accommodates non-uniqueness and stochasticity while remaining computationally efficient and amenable to conditioning on contextual inputs. The proposed framework is evaluated across a range of scientific problems in which multimodality arises from intrinsic physical mechanisms rather than observational noise, including bifurcation phenomena in nonlinear dynamical systems, stochastic partial differential equations, and atomistic-scale shock dynamics. In addition, the proposed method is compared with a conditional flow matching (CFM) model, a representative state-of-the-art generative modeling approach, demonstrating that MDNs can achieve competitive performance while offering a simpler and more interpretable formulation.