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Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation

Ameya Daigavane, Song Kim, Mario Geiger, Tess Smidt

TL;DR

Symphony, an E(3)-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments is presented, outperforming existing autore progressive models and approaching the performance of diffusion models.

Abstract

We present Symphony, an $E(3)$-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments. Existing autoregressive models such as G-SchNet and G-SphereNet for molecules utilize rotationally invariant features to respect the 3D symmetries of molecules. In contrast, Symphony uses message-passing with higher-degree $E(3)$-equivariant features. This allows a novel representation of probability distributions via spherical harmonic signals to efficiently model the 3D geometry of molecules. We show that Symphony is able to accurately generate small molecules from the QM9 dataset, outperforming existing autoregressive models and approaching the performance of diffusion models.

Symphony: Symmetry-Equivariant Point-Centered Spherical Harmonics for 3D Molecule Generation

TL;DR

Symphony, an E(3)-equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments is presented, outperforming existing autore progressive models and approaching the performance of diffusion models.

Abstract

We present Symphony, an -equivariant autoregressive generative model for 3D molecular geometries that iteratively builds a molecule from molecular fragments. Existing autoregressive models such as G-SchNet and G-SphereNet for molecules utilize rotationally invariant features to respect the 3D symmetries of molecules. In contrast, Symphony uses message-passing with higher-degree -equivariant features. This allows a novel representation of probability distributions via spherical harmonic signals to efficiently model the 3D geometry of molecules. We show that Symphony is able to accurately generate small molecules from the QM9 dataset, outperforming existing autoregressive models and approaching the performance of diffusion models.
Paper Structure (44 sections, 2 theorems, 31 equations, 17 figures, 6 tables, 2 algorithms)

This paper contains 44 sections, 2 theorems, 31 equations, 17 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Methods
  4. Building Molecules Via Sequences of Fragments
  5. Handling the Symmetries of Fragments
  6. The Design of Symphony
  7. Bypassing the Angular Frequency Bottleneck
  8. Training and Inference
  9. Relation to Prior Work
  10. Experimental Results
  11. Validity of Generated Structures
  12. Capturing Training Set Statistics
  13. Generalization Capabilities
  14. Molecule Generation Throughput
  15. Conclusion
  16. ...and 29 more sections

Key Result

Theorem B.1

Suppose Embedder produces $O(3)$-equivariant and translation-invariant features $h_{v,l} = \textsc{Embedder}(\mathcal{S}^n)_{v,l}$ for every atom $v$. Then, $p^\text{position}$ is $O(3)$-equivariant and translation-invariant (and hence, $E(3)$-equivariant):

Figures (17)

  • Figure 1: One iteration of the Symphony molecular generation process, in which one atom is sampled given the positions and atom types of an unfinished molecular fragment $\mathcal{S}^n$. The complete molecule after all iterations is shown in the bottom right of the figure.
  • Figure 2: CreateFragmentSequence
  • Figure 3: Symphony represents the atom position logits using radial shells of spherical signals. (A) illustrates an example angular distribution for a given radial shell prior to applying the softmax function. The softmax enhances signal peaks and prediction precision. (B) presents two ways to plot the signals for a single shell: as a colored sphere, or as a surface where distance from the origin represents signal magnitude, enhancing peak visibility. (C) breaks the pre-activated signal into contributions from $l=0$ to $l=4$ spherical harmonics. (D), (E), and (F) further break these contributions into the $2l+1$ spherical harmonics for $l=0, 1$ and $2$.
  • Figure 4: Bispectra of local environments of type C: C2,H2 and type C: C1,H3 respectively. Each row corresponds to a sample of the bispectrum (an array of length 15). Every entry of the bispectra is colored by value according to the colorbar on the right.
  • Figure 5: Histogram of bond lengths for the five most frequent bonds in QM9.
  • ...and 12 more figures

Theorems & Definitions (2)

  • Theorem B.1
  • Theorem B.2