# A probabilistic framework for crystal structure denoising, phase classification, and order parameters

arXiv: https://arxiv.org/abs/2512.11077 · HTML: https://mlsanity.com/paper/2512.11077/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 45 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 10 of 10)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One](https://mlsanity.com/paper/1912.03263/) (2019)
- [Your Diffusion Model is Secretly a Zero-Shot Classifier](https://mlsanity.com/paper/2303.16203/) (2023)
- [Pre-training via Denoising for Molecular Property Prediction](https://mlsanity.com/paper/2206.00133/) (2022)
- [Score-based denoising for atomic structure identification](https://mlsanity.com/paper/2212.02421/) (2022)
- [Deep Learning for Automated Classification and Characterization of Amorphous Materials](https://mlsanity.com/paper/1909.04648/) (2019)
- [Self-supervised learning for crystal property prediction via denoising](https://mlsanity.com/paper/2408.17255/) (2024)

## Abstract (arXiv)

Atomistic simulations generate large volumes of noisy structural data, yet extracting phase labels and continuous order parameters (OPs) in a robust and general manner remains challenging. Existing tools are often specialized to a limited set of prototypes and split thermal-noise removal, phase classification, and OP construction into separate steps. Here we present a unified probabilistic framework for analyzing noisy atomic configurations with respect to known crystal prototypes. The model predicts per-atom, per-prototype logits and aggregates them into a scalar log-probability (logP) landscape over atomic coordinates. Its gradient defines a conservative denoising field, while the logits provide local phase labels, prototype-resolved OPs, and ambiguity measures through logit margins. We train on AFLOW-mapped crystalline structures from the Materials Project with synthetic positional and elastic perturbations, then test extrapolation to stronger noise, finite-temperature disorder, point defects, water--ice coexistence, binary polymorphs, and shock-compressed Ti. A single differentiable scalar model recovers prototype identity after denoising, tracks smooth transformations such as Bain and Burgers paths, and exposes low-confidence regions near defects and phase boundaries. This provides an integrated and extensible tool for analyzing complex atomistic simulations.