# Coherent energy and force uncertainty in deep learning force fields

arXiv: https://arxiv.org/abs/2312.04174 · HTML: https://mlsanity.com/paper/2312.04174/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 16 references that are in the map.
- First posted on arXiv 7 Dec 2023.

## Builds on (most cited first, 13 of 13)

- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Evaluating Scalable Bayesian Deep Learning Methods for Robust Computer Vision](https://mlsanity.com/paper/1906.01620/) (2019)
- [On the Pitfalls of Heteroscedastic Uncertainty Estimation with Probabilistic Neural Networks](https://mlsanity.com/paper/2203.09168/) (2022)
- [Evaluating and Calibrating Uncertainty Prediction in Regression Tasks](https://mlsanity.com/paper/1905.11659/) (2019)
- [Less is more: sampling chemical space with active learning](https://mlsanity.com/paper/1801.09319/) (2018)
- [Reliable training and estimation of variance networks](https://mlsanity.com/paper/1906.03260/) (2019)
- [Transition1x -- a Dataset for Building Generalizable Reactive Machine Learning Potentials](https://mlsanity.com/paper/2207.12858/) (2022)
- [Deep Ensembles from a Bayesian Perspective](https://mlsanity.com/paper/2105.13283/) (2021)
- [Calibrated Uncertainty for Molecular Property Prediction using Ensembles of Message Passing Neural Networks](https://mlsanity.com/paper/2107.06068/) (2021)
- [Graph Neural Network Interatomic Potential Ensembles with Calibrated Aleatoric and Epistemic Uncertainty on Energy and Forces](https://mlsanity.com/paper/2305.16325/) (2023)

## Abstract (arXiv)

In machine learning energy potentials for atomic systems, forces are commonly obtained as the negative derivative of the energy function with respect to atomic positions. To quantify aleatoric uncertainty in the predicted energies, a widely used modeling approach involves predicting both a mean and variance for each energy value. However, this model is not differentiable under the usual white noise assumption, so energy uncertainty does not naturally translate to force uncertainty. In this work we propose a machine learning potential energy model in which energy and force aleatoric uncertainty are linked through a spatially correlated noise process. We demonstrate our approach on an equivariant messages passing neural network potential trained on energies and forces on two out-of-equilibrium molecular datasets. Furthermore, we also show how to obtain epistemic uncertainties in this setting based on a Bayesian interpretation of deep ensemble models.