Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials

Risi Kondor

TL;DR

The paper introduces N-body networks, a covariant, hierarchical neural architecture for learning atomic potentials by enforcing rotational symmetry through SO(3) representations. It constructs a DAG-based network where each node (subsystem) carries a spatial position and an internal state that transforms covariantly under rotations; nonlinearities arise from tensor products followed by Clebsch–Gordan decompositions computed in Fourier space. Covariant aggregation rules are formalized, with zeroth and first order interaction gates providing scalable, symmetry-preserving building blocks, and learnable mixing matrices that respect irreducible fragment structure. This framework aims to bridge quantum accuracy and classical efficiency in molecular dynamics and is positioned for broad applicability to other many-body systems.

Abstract

We describe N-body networks, a neural network architecture for learning the behavior and properties of complex many body physical systems. Our specific application is to learn atomic potential energy surfaces for use in molecular dynamics simulations. Our architecture is novel in that (a) it is based on a hierarchical decomposition of the many body system into subsytems, (b) the activations of the network correspond to the internal state of each subsystem, (c) the "neurons" in the network are constructed explicitly so as to guarantee that each of the activations is covariant to rotations, (d) the neurons operate entirely in Fourier space, and the nonlinearities are realized by tensor products followed by Clebsch-Gordan decompositions. As part of the description of our network, we give a characterization of what way the weights of the network may interact with the activations so as to ensure that the covariance property is maintained.

N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials

TL;DR

The paper introduces N-body networks, a covariant, hierarchical neural architecture for learning atomic potentials by enforcing rotational symmetry through SO(3) representations. It constructs a DAG-based network where each node (subsystem) carries a spatial position and an internal state that transforms covariantly under rotations; nonlinearities arise from tensor products followed by Clebsch–Gordan decompositions computed in Fourier space. Covariant aggregation rules are formalized, with zeroth and first order interaction gates providing scalable, symmetry-preserving building blocks, and learnable mixing matrices that respect irreducible fragment structure. This framework aims to bridge quantum accuracy and classical efficiency in molecular dynamics and is positioned for broad applicability to other many-body systems.

Abstract

We describe N-body networks, a neural network architecture for learning the behavior and properties of complex many body physical systems. Our specific application is to learn atomic potential energy surfaces for use in molecular dynamics simulations. Our architecture is novel in that (a) it is based on a hierarchical decomposition of the many body system into subsytems, (b) the activations of the network correspond to the internal state of each subsystem, (c) the "neurons" in the network are constructed explicitly so as to guarantee that each of the activations is covariant to rotations, (d) the neurons operate entirely in Fourier space, and the nonlinearities are realized by tensor products followed by Clebsch-Gordan decompositions. As part of the description of our network, we give a characterization of what way the weights of the network may interact with the activations so as to ensure that the covariance property is maintained.

Paper Structure

This paper contains 11 sections, 1 theorem, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Fixed vs. learned representations.
  3. Representing structured objects with neural nets
  4. Compositional networks
  5. Compositional models for atomic environments
  6. Group representations and $N$--body networks
  7. Covariant aggregation rules
  8. Zeroth order interaction gates
  9. First order interaction gates
  10. Clebsch--Gordan transforms
  11. Conclusions

Key Result

Proposition 1

The output of the aggregation function (eq: aggreg1) is a $\hbox{\boldmath$\tau$}$--covariant vector if and only if $\mathcal{L}$ is of the form Equivalently, collecting all $\phi^\ell_{m'}$ fragments with the same $\ell$ into a matrix ${\tilde{F}^\ell\space\in\space \mathbb{C}^{(2\ell\space+\space1)\times \tau'_\ell}}$, all $(w^\ell_{m',m})_{m',m}$ weights into a matrix ${W^\ell\space\in\space \

Figures (2)

  • Figure 1: (a) A composition scheme for an object $\mathcal{X}$ is a DAG in which the leaves correspond to the elementary parts of $\mathcal{X}$, the internal nodes correspond to sets of elementary parts, and the root corresponds to the entire object. (b) A compositional network is a composition scheme in which each node $\mathfrak{n}_i$ also carries a feature vector (activation) $f_i$, which is computed from the feature vectors of the children of $\mathfrak{n}_i$.
  • Figure 2: In a comp-net for learning atomic force fields, the output of each "part" $\mathcal{P}_i$ is $(\hbox{\boldmath$r$}_i, \psi_i)$, where $\hbox{\boldmath$r$}_i$ is the position vector of the corresponding physical subsystem, and $\psi_i$ is a vector describing its internal state.

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1