# Efficient mapping of phase diagrams with conditional Boltzmann Generators arXiv: https://arxiv.org/abs/2406.12378 ยท HTML: https://mlsanity.com/paper/2406.12378/ ## Key facts - Cited by 18 papers in the map (data through 14 Oct 2026). - Builds on 19 of its 54 references that are in the map. - First posted on arXiv 18 Jun 2024. ## Builds on (most cited first, 19 of 19) - [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017) - [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015) - [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016) - [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019) - [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014) - [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021) - [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020) - [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017) - [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019) - [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019) - [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021) - [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019) - [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020) - [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020) - [Normalizing flows for atomic solids](https://mlsanity.com/paper/2111.08696/) (2021) - [Learning Lattice Quantum Field Theories with Equivariant Continuous Flows](https://mlsanity.com/paper/2207.00283/) (2022) - [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023) - [Aspects of scaling and scalability for flow-based sampling of lattice QCD](https://mlsanity.com/paper/2211.07541/) (2022) - [Estimating Gibbs free energies via isobaric-isothermal flows](https://mlsanity.com/paper/2305.13233/) (2023) ## Built on by (most cited first, 18 of 18) - [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024) - [Scalable Equilibrium Sampling with Sequential Boltzmann Generators](https://mlsanity.com/paper/2502.18462/) (2025) - [Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models](https://mlsanity.com/paper/2506.17139/) (2025) - [Temperature-Annealed Boltzmann Generators](https://mlsanity.com/paper/2501.19077/) (2025) - [FALCON: Few-step Accurate Likelihoods for Continuous Flows](https://mlsanity.com/paper/2512.09914/) (2025) - [Scalable Boltzmann Generators for equilibrium sampling of large-scale materials](https://mlsanity.com/paper/2509.25486/) (2025) - [BoltzNCE: Learning Likelihoods for Boltzmann Generation with Stochastic Interpolants and Noise Contrastive Estimation](https://mlsanity.com/paper/2507.00846/) (2025) - [Generative Models for Crystalline Materials](https://mlsanity.com/paper/2511.22652/) (2025) - [Protein Language Model Embeddings Improve Generalization of Implicit Transfer Operators](https://mlsanity.com/paper/2602.11216/) (2026) - [Path Gradients after Flow Matching](https://mlsanity.com/paper/2505.10139/) (2025) - [Assessing generative modeling approaches for free energy estimates in condensed matter](https://mlsanity.com/paper/2512.23930/) (2025) - [Energy-Based Coarse-Graining in Molecular Dynamics: A Flow-Based Framework without Data](https://mlsanity.com/paper/2504.20940/) (2025) - [Estimating Solvation Free Energies with Boltzmann Generators](https://mlsanity.com/paper/2512.18147/) (2025) - [Boltzmann Generators for Condensed Matter via Riemannian Flow Matching](https://mlsanity.com/paper/2602.18482/) (2026) - [Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning models](https://mlsanity.com/paper/2604.09769/) (2026) - [Latent Thermodynamic Flows: Unified Representation Learning and Generative Modeling of Temperature-Dependent Behaviors from Limited Data](https://mlsanity.com/paper/2507.03174/) (2025) - [Generative Nested Sampling of Atomistic Thermodynamic Landscapes](https://mlsanity.com/paper/2609.03193/) (2026) - [Correlation-Free Transition Path Sampling through Shooting Point Generation Guided by Committor Learning](https://mlsanity.com/paper/2609.20461/) (2026) ## Abstract (arXiv) The accurate prediction of phase diagrams is of central importance for both the fundamental understanding of materials as well as for technological applications in material sciences. However, the computational prediction of the relative stability between phases based on their free energy is a daunting task, as traditional free energy estimators require a large amount of simulation data to obtain uncorrelated equilibrium samples over a grid of thermodynamic states. In this work, we develop deep generative machine learning models based on the Boltzmann Generator approach for entire phase diagrams, employing normalizing flows conditioned on the thermodynamic states, e.g., temperature and pressure, that they map to. By training a single normalizing flow to transform the equilibrium distribution sampled at only one reference thermodynamic state to a wide range of target temperatures and pressures, we can efficiently generate equilibrium samples across the entire phase diagram. Using a permutation-equivariant architecture allows us, thereby, to treat solid and liquid phases on the same footing. We demonstrate our approach by predicting the solid-liquid coexistence line for a Lennard-Jones system in excellent agreement with state-of-the-art free energy methods while significantly reducing the number of energy evaluations needed.