# Neural Thermodynamic Integration: Free Energies from Energy-based Diffusion Models

arXiv: https://arxiv.org/abs/2406.02313 · HTML: https://mlsanity.com/paper/2406.02313/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 49 references that are in the map.
- First posted on arXiv 4 Jun 2024.

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [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)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Torsional Diffusion for Molecular Conformer Generation](https://mlsanity.com/paper/2206.01729/) (2022)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Flow-based generative models for Markov chain Monte Carlo in lattice field theory](https://mlsanity.com/paper/1904.12072/) (2019)
- [Riemannian Diffusion Models](https://mlsanity.com/paper/2208.07949/) (2022)
- [Machine learning for molecular simulation](https://mlsanity.com/paper/1911.02792/) (2019)
- [Two for One: Diffusion Models and Force Fields for Coarse-Grained Molecular Dynamics](https://mlsanity.com/paper/2302.00600/) (2023)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Learning Interpolations between Boltzmann Densities](https://mlsanity.com/paper/2301.07388/) (2023)
- [The Thermodynamic Variational Objective](https://mlsanity.com/paper/1907.00031/) (2019)
- [Estimating Gibbs free energies via isobaric-isothermal flows](https://mlsanity.com/paper/2305.13233/) (2023)

## Built on by (most cited first, 14 of 14)

- [RNE: plug-and-play diffusion inference-time control and energy-based training](https://mlsanity.com/paper/2506.05668/) (2025)
- [FEAT: Free energy Estimators with Adaptive Transport](https://mlsanity.com/paper/2504.11516/) (2025)
- [Complexity Analysis of Normalizing Constant Estimation: from Jarzynski Equality to Annealed Importance Sampling and beyond](https://mlsanity.com/paper/2502.04575/) (2025)
- [Solvation Free Energies from Neural Thermodynamic Integration](https://mlsanity.com/paper/2410.15815/) (2024)
- [First-Passage Approach to Optimizing Perturbations for Improved Training of Machine Learning Models](https://mlsanity.com/paper/2502.04121/) (2025)
- [Wavefunction Flows: Efficient Quantum Simulation of Continuous Flow Models](https://mlsanity.com/paper/2510.08462/) (2025)
- [Enhanced Sampling for Efficient Learning of Coarse-Grained Machine Learning Potentials](https://mlsanity.com/paper/2510.11148/) (2025)
- [Assessing generative modeling approaches for free energy estimates in condensed matter](https://mlsanity.com/paper/2512.23930/) (2025)
- [Estimating Solvation Free Energies with Boltzmann Generators](https://mlsanity.com/paper/2512.18147/) (2025)
- [Learning with springs and sticks](https://mlsanity.com/paper/2508.19015/) (2025)
- [CARD: Coarse-to-fine Autoregressive Modeling with Radix-based Decomposition for Transferable Free Energy Estimation](https://mlsanity.com/paper/2605.02657/) (2026)
- [Unsupervised Thermodynamics of Molecular Diffusion Models: Action-Operator Semantics and Auditable Free-Energy Readout](https://mlsanity.com/paper/2606.30687/) (2026)
- [AquaGen: Scaling generative models to molecular dynamics precision on thousands of atoms](https://mlsanity.com/paper/2607.03513/) (2026)
- [Grand Canonical Generators](https://mlsanity.com/paper/2610.00683/) (2026)

## Abstract (arXiv)

Thermodynamic integration (TI) offers a rigorous method for estimating free-energy differences by integrating over a sequence of interpolating conformational ensembles. However, TI calculations are computationally expensive and typically limited to coupling a small number of degrees of freedom due to the need to sample numerous intermediate ensembles with sufficient conformational-space overlap. In this work, we propose to perform TI along an alchemical pathway represented by a trainable neural network, which we term Neural TI. Critically, we parametrize a time-dependent Hamiltonian interpolating between the interacting and non-interacting systems, and optimize its gradient using a score matching objective. The ability of the resulting energy-based diffusion model to sample all intermediate ensembles allows us to perform TI from a single reference calculation. We apply our method to Lennard-Jones fluids, where we report accurate calculations of the excess chemical potential, demonstrating that Neural TI reproduces the underlying changes in free energy without the need for simulations at interpolating Hamiltonians.