E(3)-equivariant models cannot learn chirality: Field-based molecular generation

TL;DR

This work shows that $E(3)$-invariant, point-cloud diffusion models cannot distinguish molecular chirality, a crucial factor for drug safety and efficacy. To address this, it introduces Field-based Molecule Generation (FMG), which uses atom and bond density fields on a 3D grid and a diffusion model with reference rotations to generate chiral, geometry-rich molecules. Theoretical results prove the chirality limitation of $E(3)$-invariant parameterizations and demonstrate the impractical $(oldsymbol{O}(n^4))$ feature requirement for chiral-aware SE(3) invariants, while FMG achieves competitive state-of-the-art performance on QM9 and GEOM-Drugs and yields accurate enantiomer distributions. Empirically, FMG demonstrates strong neutrality, robust graph and conformational metrics, and explicit chirality awareness, offering a practical path toward chirality-correct drug-like molecular generation. The approach paves the way for scalable, chirality-sensitive 3D molecular generation by trading full $E(3)$ invariance for deterministic frame alignment and field-based representations.

Abstract

Obtaining the desired effect of drugs is highly dependent on their molecular geometries. Thus, the current prevailing paradigm focuses on 3D point-cloud atom representations, utilizing graph neural network (GNN) parametrizations, with rotational symmetries baked in via E(3) invariant layers. We prove that such models must necessarily disregard chirality, a geometric property of the molecules that cannot be superimposed on their mirror image by rotation and translation. Chirality plays a key role in determining drug safety and potency. To address this glaring issue, we introduce a novel field-based representation, proposing reference rotations that replace rotational symmetry constraints. The proposed model captures all molecular geometries including chirality, while still achieving highly competitive performance with E(3)-based methods across standard benchmarking metrics.

Figures (22)

• Figure 1: E(3) invariant spatial features (e.g., relative bond angles and distances) are not sufficient to represent chirality. Subfigure a) depicts a general chiral molecule pair, which cannot be superimposed by rotations. b) Chiral S-ibuprofen is an efficient COX-inhibitor resulting in pain and inflammation relief, while the mirror R-ibuprofen is not evans2001.
• Figure 2: a) Atom $\mathbf{u}_a$ and bond $\mathbf{u}_b$ fields noise schedules. b) Sampling illustration. A subset of 200 3D locations with the highest values are shown for all $\mathbf{u}_a$ and $\mathbf{u}_b$ channels. c) Extracting atoms and bonds from the resulting $\mathbf{u}_a$ and $\mathbf{u}_b$ fields at $t=0$. d) Visualizing the optimized atoms and bonds.
• Figure 3: Kernel density estimation of $\det {\bm{R}}$, resulted from the data and generated molecules. EDM is unable to distinguish and correctly generate the enantiomer distribution, while the E(3) variant method (GNN) generates improper, 0 volume conformations. FMG matches the enantiomer distribution.
• Figure 4: Cumulative distribution function plot of generated and training data for MiDi, EDM, and FMG for the GEOM-Drugs and QM9 datasets. All models capture these conformational properties very well on QM9, and reasonably well on GEOM.
• Figure 5: Cumulative distribution function plot of generated and training data energy conformations (kcal/mol) for MiDi, EDM, and FMG for the GEOM-Drugs and QM9 datasets. On GEOM, not enough valid samples were obtained to compute the energy distribution. Both models generate higher energies than the data on GEOM-Drugs (notably, MiDi was trained on a subset of lower energy molecules on GEOM).

Theorems & Definitions (6)

• Proposition 1
• Lemma 2
• Proposition 3
• Remark 1
• Remark 2
• Remark 3