# GNN-Assisted Phase Space Integration with Application to Atomistics

arXiv: https://arxiv.org/abs/2303.16088 · HTML: https://mlsanity.com/paper/2303.16088/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 84 references that are in the map.
- First posted on arXiv 20 Mar 2023.

## Builds on (most cited first, 9 of 9)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/1812.08434/) (2018)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [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)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Blessing of dimensionality: mathematical foundations of the statistical physics of data](https://mlsanity.com/paper/1801.03421/) (2018)
- [Hamilton-Jacobi-Bellman Equations for Q-Learning in Continuous Time](https://mlsanity.com/paper/1912.10697/) (2019)

## Abstract (arXiv)

Overcoming the time scale limitations of atomistics can be achieved by switching from the state-space representation of Molecular Dynamics (MD) to a statistical-mechanics-based representation in phase space, where approximations such as maximum-entropy or Gaussian phase packets (GPP) evolve the atomistic ensemble in a time-coarsened fashion. In practice, this requires the computation of expensive high-dimensional integrals over all of phase space of an atomistic ensemble. This, in turn, is commonly accomplished efficiently by low-order numerical quadrature. We show that numerical quadrature in this context, unfortunately, comes with a set of inherent problems, which corrupt the accuracy of simulations -- especially when dealing with crystal lattices with imperfections. As a remedy, we demonstrate that Graph Neural Networks, trained on Monte-Carlo data, can serve as a replacement for commonly used numerical quadrature rules, overcoming their deficiencies and significantly improving the accuracy. This is showcased by three benchmarks: the thermal expansion of copper, the martensitic phase transition of iron, and the energy of grain boundaries. We illustrate the benefits of the proposed technique over classically used third- and fifth-order Gaussian quadrature, we highlight the impact on time-coarsened atomistic predictions, and we discuss the computational efficiency. The latter is of general importance when performing frequent evaluation of phase space or other high-dimensional integrals, which is why the proposed framework promises applications beyond the scope of atomistics.