E(3)-equivariant models cannot learn chirality: Field-based molecular generation
Authors: Alexandru Dumitrescu
Dani Korpela
Markus Heinonen
Yogesh Verma
Valerii Iakovlev
Vikas Garg
Harri Lähdesmäki
Venue: ICLR 2025 Poster (2025)
Keywords: deep generative models
molecule generation
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: FMG is a field-based generative model for drug-like molecules that achieves highly competitive performance while capturing all molecular geometric properties.

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.

Published as a conference paper at ICLR 2025
E(3) MODELS CANNOT LEARN CHIRALITY: FIELD-BASED MOLECULAR GENERATION
Alexandru Dumitreseu', Dani Korpela Markus Heinonent, Yogesh Verma',
Valerii Takovlevi, Vikas Gargi & Harri Lähdesmäki
"Department of Computer Science, Aalto University
Yai Ltd
Correspondence:
ABSTRACT
Obtaining the desired effect of drugs is highly dependent on their molecular ge- ometries. Thus, the current prevailing paradigm focuses on 3D point-cloud atom representations, utilizing graph neural network (GNN) parametrizations, with ro- tational 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 their mirror image by rotation and translation. Chi- rality plays key role in determining drug safety and potency. To address this glaring issue, we introduce novel field-based representation, proposing refer- ence 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. Code is available at https://dmirrrecu alw.xaaaduuuuu..e.eeaaaaaaaa wwt.ouuur.ittuiiooooooooo github.io/MMG-welee
INTRODUCTION
Developing generative models for specific data modalities requires carefully choosing the generative method, feature representation, and NN architecture. Field-based representations may offer uni- fying perspective, extending the usage of highly successful architectures from natural language or images other input modalities Zhuang et al. (2023). We develop Field-based Molecule Generation (FMG), generative model of the joint distribution of molecular graphs and 3D conformations.
field representations of molecules, which encode
Organic molecules generally contain restricted number of atom types, bound mostly through co- valent bonds, and they can be used as drugs, inhibiting or promoting biological processes. Their effectiveness erucially depends successfully binding and interacting with proteins or other small molecules in the organism. In turn, the interaction between two molecules can only be determined their 3D conformations for molccular conformations. Recently, the vast majority of work has been done on generative model parameterizations that are invariant the Euclidean group of isometries in 3D, E(3) (Hoogeboom et al., 2022; Peng et al., 2023; Vignac et al., 2023b; Xu et al., 2023; Wu et al., 2022).
known. This motivated the development of many generative models
The B(3) group contains any reflections, rotations, and translations, whereas SE(3) E(3) (special E(3)) contains only proper rotations and translations. In general, molecules are not E(3) invariant. Certain molecules are sensitive reflections meaning that one cannot superimpose the molecule to its mirror-image by (proper) rotations and translations. This property is called chirality or handed- ness, and chiral molecules form enantiomer pairs (Figure 1).
Chiral centers are often tetrahedral carbon configurations containing four unique groups. In molec- ular biology, chirality is crucial to the potency and safety drugs. In ibuprofen, albuterol, and naproxen, one of the two enantiomers was shown to be more effective, while in thalidomide and propoxyphene, the incorrect enantiomer toxic and can lead to severe side effects (Smith, 2009). In proteins, out of 21 standard amino acids, only Glycine is not chiral. Although classically it was believed that only one chiral configuration is present in nature (Devinsky, 2021), patients with disor- ders such cataracts, macular degeneration, arteriosclerosis, and Alzheimer's disease present both

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R1
R1
R4
R4
Ry
R₃
R lbuprofen
5 lbuprofen
a) General enantiomers
b) lbuprofen enantiomers
Figure 1: E(3) invariant spatial features (e.g,, relative bond angles and distances) are not suf- ficient to represent chirality. Subfigure a) depicts a general chiral molecule pair, which cannot be superimposed by rotations. b) Chiral S-ibuprofen an efficient COX-inhibitor resulting pain and inflammation relief, while the mirror R-ibuprofen is not (Evans, 2001).
(Masters et al., 1977: Fujii al., 2018). the drug-like molecule dataset, GEOM, 27% of the molecules contain chiral centers.
In essence, reflections are distance preserving, and therefore parameterizations that utilize Euclidean distances will be unable to distinguish enantiomers. Later on, we prove theoretical results that show how designing enantiomer (reflection) variant features (SE(3) invariant) grows the complexity from (n²) (for pairwise Euclidean distances between N atoms) to O(n*), To account for chiral centers, we instead chose to replace rotation symmetry constraints, and propose reference rotations, which bring our performance to highly competitive results with E(3) invariant methods.
We organize our contributions as follows:
Theoretical: We prove that functions of E(3) invariant features fail properly model chiral molecules. We then link the parametrizing function's 's E(3) invariance to the resulting generative E(3) equivariance. Finally, we prove that SE(3)-invariant, chiral aware features can only be func- tions of least four R vectors, and the number of such features scales in Onn*).
We provide solutions to the challenges of handling 3D fields. We present a method of reorienting molecules to deterministic references, introduce bond fields, describe our diffusion process design, and complete our pipeline with molecular graph optimization based on fields.
Practical:
Empirical: Put together, our technical contributions result in SOTA or highly competitive perfor- mance compared point-cloud approaches, while keeping chiral-awareness.
RELATED WORKS
large number of generative models for molecules have been proposed. We distinguish them pri- marily by their molecular representation instead of generative method families.
SMILES SMILES is a string-based representation in which molecular graphs can be sufficiently encoded (Weininger, 1988), and has been employed in autoregressive (Segler et al., 2018) and vari- ational autoencoder (VAE) models (Gómez-Bar Boobaz- et al., 2018; Kusner et al., 2017). An im- portant limitation of SMILES used as input representations for machine learning is that similar molecular graphs may have very different SMILES encodings (Jin et al., 2018).
Graphs Alternative molecule representations consist of molecular graphs, where vertices encode atom types and their covalent bonds are represented through adjacency matrices. VAEs (Liu et al., 2018; Jin et al., 2018), flow-based models (Shi et al., 2020; Verma et al., 2022), reinforcement learning (Shi et al., 2020; You et al., 2018; Zhou et al., 2019) and diffusion- et al., 2023a; Mercatali et al., 2024) have been developed. The main drawback of only generating molecular graphs is that full molecular geometry specification is missing, and the functionality of various drug-like molecules depends on it.
methods (Vignac
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Point clouds The diree-dimensional configuration of molecule be encoded as R point clouds of atoms. Molecules reside in the I space, and their conformations are symmetric w.rt. global translations and rotations. Inspired by this, E(3) equivariant generative methods have been de- veloped. Autoregressive models (Gebauer et al., 2018; 2019), GNN parametrized diffusion (Hooge- boom et al., 2022; Peng al., 2023; Vignac al., 2023b; Xu et al., 2023; Wu et al., 2022), and transformer parametrized diffusion (Hua et al., 2023; Huang et al., 2024) methods have recently gained lot of interest. Although rotational invariance is well motivated, these methods also encode the undesired invariance to reflections, which are part of the E(3) group.
Fields The R coordinates of atoms in molecules sians, which fill the three-dimensional space. These densities can be interpreted as fields, where each molecule is represented by function on R3 with values given by the densities of the atoms. We crucially distinguish this approach from the previously developed methods that use point cloud formulations by the E(3) symmetries they adhere to. Namely, our approach only respects translation invariance through the CNN inductive biases, while point cloud approaches respect all E(3) sym- metries. At the cost of losing rotational invariance, our method can generate chiral conformations according to data statistics, while E(3) invariant formulations cannot. Similar input representations have been previously modeled using VAE (Ragoza et al., 2020) and score-based models (Pinheiro et al., 2023). The latter is the closest to work, which bases its method on the neural empirical Bayes framework and only generates atom voxels, based on which they use post- cost-generation bond extraction. Using diffusion models, show that field representations can reach SOTA performance while generating the full molecule specification (including bond fields).
be used the mean positions of e.g. Gaus-
Chiral-aware methods Prior, chiral-aware discriminative methods for molecular analysis have been developed in (Adams et al., 2022; Liu et al., 2022). Jing et al. (2022) also accounts for chiral- ity in a generative setting using diffusion model, but considers the conditional generation of 3D molecular conformations given a molecular graph p(C))), while the scope of this work is on the joint generation p(C, G). Finally, Luo & Ji (2022) uses (Liu al., 2022) NN parametrization of joint p(C,G) G) AR generative model that, in principle, is chiral-aware, but does not discuss chirality. AR methods need to define arbitrary node order generation, and whether they can reach diffusion model performance despite this fact remains an open question. In general, chiral-aware SE(3) diffu- sion models on point clouds are surprisingly difficult develop.
3 CHIRALITY AND ISOMETRY GROUPS
Most recent work on molecule generation model E(3) equivariant distributions. Although molecules respect rotation and translation symmetries, chiral molecules are not symmetric reflections. In such cases, the reflected version of therapeutic drug can become toxic (Smith, 2009).
Proposition 1. If ps is E(3) invariant probability distribution, then ps (m) = Po (m'), where m and m' are the B positions of enantiomer pairs of molecules with N atoms.
Proposition 1 is consequence of the fact that po(m) = p& (Rm). A R orthogonal, det R = +1. This implies that NN parameterizations of point-cloud diffusion models should be SE(3) invariant, but we argue that this becomes intractable.
Lemma 2. Let f: m. -> S. for some set S, a function of m n-dimensional vectors. Ifmm S n and f is SE(n) invariant, then is E(n) invariant.
For example, in R3, the reflection of planar object (m 3) can be recovered by rotations and translations. This implies that there is no SE(3) invariant function that, applied on triplets of atom positions, would distinguish enantiomer pairs.
Proposition 3. Let n be the number of atoms of a molecule. The number of features that encode all chiral configurations of random point clouds scules in O
The result implies that SE(3) (and not E(3)) invariant features scale prohibitively with the number of atoms.
Proofs can be found in Appendix B. There, also concretely describe the model types refer to in Proposition 3, which include all SOTA diffusion models (Hoogeboom et al., 2022; Peng et al., 2023; Vignac al., 2023b; Xu et al., 2023; Wu al., 2022).
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a)
1.00
b)
Atoms Bonds
0.75
9:
0.0
0.5
1.0
(tom pmd
bund: 2-1.5
0.25
0.15
0.1
0.05
0.0
(u, (te)
d)
Bond detection (Ug, m) + Bond optimization (ue, b')
Penk detection Posétion uptimizsion (M,)
fend
Suni
Figure 2: a) Atom u, and bond fields noise schedules. b) Sampling illustration. A subset of 200 3D locations with the highest values are shown for all u, and u, channels. c) Extracting atoms and bonds from the resulting and u, fields at = 0. d) Visualizing the optimized atoms and bonds.
4 MOLECULAR FIELD DIFFUSION
In this Section, we propose a diffusion model to generate molecular fields.
4.1 FIELD REPRESENTATION
We consider a molecule as a set of atoms {a,, m and a set of bonds {big, my J. with atom types a, A (H, C, N, O, P, ... bond types bıj E B {br, b2, b3), and assume locations m.+m; in Angströms.
m; E 3R3 and mid-point bond locations m; = 2
We start by representing molecule using collection of continuous 3D fields, one for each atom and bond type. We define the field for atom type a, Ug as a mixture of radial basis functions (RBFs) positioned the atoms' centers
N Ya Σ
1x- mill?
(x)
I(a
a] exp
20
1m
where N is the number of atoms, [[-] is the indicator function, the variance of represents the size of atom type a, and Ya normalises the channels to unit interval ua ux(x) [0, 1] (see Appendix for field parameter values). We create bond fields similarly
us (xxx) 7b
Σ 1b. (i,J)
b) exp (- 203
(~20²
mmsl²)
We evaluate molecule's s atom and bond fields uar uy at fixed set of H x W x (height, width, and depth) locations, and create vector-valued field based which multi-channel 4D tensors derived
(Ug, ugy...; Ub Ub, Ugg) B KxHwwxD
where = ||||+B|| is the number of channels, We use tensor u as the input molecule representation for the diffusion model.
4.2 DIFFUSION MODEL
We consider a diffusion generative model to generate molecular fields u. diffusion models, the data is incrementally noised by forward process, and our goal is to learn denoising reverse process
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parameterised by 9. We consider the field tensors u, over time t [0, 1], where uy at time = 0 are the observed or generated tensors. Figure 2b depicts an example of our denoising reverse process.
We follow the denoising diffusion probabilisticc model (DDPM) proposed in (Sohl-Dickstein et al., 2015; Ho et al., 2020; Kingma et al., 2021), and refer the reader to the original publications for full derivations. We note that tensor fields effectively multi-channel 3D images, and thus image-based diffusion models translate to molecular voxel grid fields a straightforward manner.
Forward process
We begin by assuming Gaussian forward process q(u,/ub) A (u T)
(4)
where Or E [0, 1] monotonically reduces the original signal uo. while o, E [0, 1] monotonically increases noise. We choose prrtance-preserving noise specification, where
04 V
(5)
At / we obtain pure noise g(ui [no) = IO.
Forward posterior The forward process admits a tractable conditional posterior distribution for an intermediate state u, if we know both the original state Ug and further noised state u, for
Otts 0
a, uo,
(u, ur un) = =N
+
6
of
where 0t, = as/a, and of, = of of.0? (See cf. Kingma et al. (2021)). Due to Gaussianity of the forward, the posterior interpolates linearly between the endpoints while adding variance.
Reverse process Our goal is to learn a generative process that reverses the forward by matching their marginals. The reverse process can be framed as the posterior of the forward
pe (u, u.) = 4 us u, ug == ú (o(u,,)) f)):
where we replace the observation conditioning un with fully denoised prediction úæ(u, t) from neural network parameterised by 0.
Parameterisation In practice, it is often more efficient to train neural network to predict the added noise ~ 1) of
u, = a, + o,e, than to predict the signal ug. We then train neural network (u,) to predict the noise c, and follow the above equations with a substitution
(8)
ús u/ar oz f)
Loss The diffusion model admits variational lower bound (Kingma et al., 2021), but earlier works have argued for a simplified loss (Ho et al., 2020)
Limple Eu. r,e [lle ( [)]
(10)
where we learn to denoise arbitrary combinations of observations ug noises e and noise scales 1 U(O, 1). We also opt for the simple loss Lsimple.
4.3 NOISE SCHEDULES
To further emphasize the stability of generated molecules, Vignac et al. (2023b) and Peng et al. (2023) employed separate noise schedulers for the bonds, atom types, and their positions using modified cosine and a sigmoid scheduler, respectively. We also observed increased performance using a similar approach Vignac et al. (2023b), which adds parameter v the original cosine schedule from (Nichol & Dhariwal, 2021):
π (t + 8)
a₂ = cos
(11)
2 1
where de determines the signal amplitude for each time in Equation 4, and set to 1 and 1.5 for atom and bond channels, respectively. The schedule a, and effect on generation can be seen in Figure 2, where bonds are denoised faster than atoms, increasing the generated molecules' stability.
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4.4 CLASSIFIER FREE GUIDANCE
Ho & Salimans (2022) empirically showed that using an approximation of the implicit classifier p(y (u)) y(u:)))/((u) achieves similar conditional generation performance to (Dhariwal & Nichol, 2021), without needing a separately trained classifier Po (u,) for multiple noise levels During sampling, the conditional and unconditional noise estimators Eø y) and Eg (u,) are linearly interpolated:
Eo; = (1 + (0) Seo (ut):
(12)
where we note that ey y) and Eg (ur) are up to a constant equal to the conditional and marginal score estimates Vu log pa (u and V log (u.).
Point-cloud approaches always strictly enforce the generated atom counts. We determine the effect of atom count conditioning using classifier-free guidance generated molecule statistics. While not strictly necessary for our field-based formulation, we show that atom count conditioning yields molecule graph quality improvements. Further details can be found in Appendix G.
4.5 ARCHITECTURE CHOICE, ROTATION INVARIANCE AND SCALING
We parametrize the denoiser 60 with a three-dimensional U-Net architecture, which consists of ResNet blocks, scaling layers, and attention layers. CNN layers provide useful translational invari- ance inductive biases but lack rotation invariance.
We propose simple technique that side-steps the issue by orienting each molecule wrt its principal components V = (v1, V2, Vs) of their atom positions M = (mı..... my). Oriented molecules M = M have a canonical frame of reference V, where the axis of the largest variation is placed on the horizontal axis, the aecond-larges on the vertical axis, and the smallest on the depth axis. This ensures high overlap between the large values of fields from different molecules.
Compared to previous methods that use E(3) invariant NN and can generate molecules with any ro- tation, our method (on optimality) will only generate molecules according to this frame of reference. NN hyperparameters can be found in Appendix D.
Our method's s scaling is dependent upper limit of our method's scaling and show that it differs substantially from empirical values, even surpassing the generation speed of (Hoogeboom al., 2022) for the largest molecules.
the evaluated field locations G. In Appendix J, we derive the
4.6 EXTRACTING MOLECULAR GRAPHS
We briefly illustrate the molecular graph extraction method and give additional details in Ap- pendix E. To extract molecular graph from the generated fields (ug, uc..... Ub, Uby Ubg ) we first initialize the number of atoms N, their positions (m;), and their types {a;), using a com- putationally efficient peak picking algorithm which retrieves sets M, of atom positions m,, for each atom type. We then optimize the mean-squared error loss between the values of the generated fields ug and the corresponding atom RBF function values determined by m, Ma by matching the generated atom fields against the RBF mixtures induced by {m:} across the grid points G
ΣΣ arg min [m,}] REAXES
[a(x) Ya Σ 1a; = a) exp (- 20 x m. m..lp)]]
(13)
Similarly, given the fixed, optimized pairs (m: my) within certain distances define our initial bond set B. We then consider sets Bij.b of values u, around bond positions m,j (m; + my)/2, that are above predefined thresholds, for every (m, my) E B. For all B # Ø we assign Yay and optimize:
positions {m:} we initialize bonds. All atom position
arg min
ΣΣ (x)(x) BEB
Eny ny ny b] exp (-
203
1
Ixx
m.y m
(12)
(14)
(t.f)
where the small RBF component values Ti corresponding to erroneously detected bonds are re- moved.

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5 EVALUATION
We briefty present the benchmarking metrics and refer to Appendix H for details and definitions.
Basic molecule properties We use atom and molecule neutrality and validity. Neutrality is the same metric defined in (Hoogeboom et al., 2022) as "stability". but, since other work considered stable atoms even those without a formal charge of 0 (Vignac et al., 2023b), we denote it neutrality, to clarify and distinguish from their definitions Atom neutrality the percentage of atoms with formal charge of 0, while we consider molecules as neutral if all their atoms are neutral (note how the definition slightly differs from the chemical one). Unlike validity, neutrality cannot be trivially increased by e.g., generating only covalent type 1 bonds, other trivial solutions that may increase this metric at the expense of other distribution properties (Hoogeboom et al., 2022). We treat neutrality like any other distribution property, and we aim to match the generated neutral molecule fraction of the data.
Molecule graph quality We compute atom total variation (TVa) and bond total variation (TV)) between the datasets and generated distributions, and report the sum the distances over all atom and bond types, respectively.
Next, we report the average maximum similarity between the test dataset and generated molecules (MST). We compute the Morgan fingerprints (Zhong & Guan, 2023) of all the generated and test dataset molecular graphs and determine the maximum Tanimoto similarity between each generated molecule and the test set.
Molecular conformation metries To assess the quality of the generated 3D molecular conforma- tions, employ the Wasserstein metries introduced in (Vignac et al., 2023b), and compute bond angle (BA) and bond length (BL) W₁ distance between generated and data distributions. We addi- tionally compute conformational energies (CEMMFF and CE&TB) W, distances, based on the MMFF and GFN2-XTB (Bannwarth et al., 2019) energies, respectively, of generated and data conforma- tions. Since point-cloud methods may create large outliers and W, distances are outlier sensitive, we restrict generated angles and lengths to those generated within the data limits (see Appendix H).
EXPERIMENTS AND RESULTS
We report the basic molecule properties and molecule graph quality metrics in Tables 1 and 2, while molecule conformation and conditional generation results can be found in Appendix A.1 and A.2.
Previous methods have employed many different settings for molecule generation. While all of them may be valid approaches, it makes benchmarking challenging. We note in Appendix how explicit aromatic generation makes it difficult to compute and compare certain metric values against Kekule representations (used in this work). In Tables and 2, we annotate explicit aromatic formu- lations with (t). Methods that employ post- post-generation corrections or algorithmic bond extractions are annotated with (f). The methods we tested ourselves are marked with (*).
6.1 QM9
QM9 (Ramakrishnan al., 2014) a small molecule dataset, containing 134k molecules of at most 29 atoms (including hydrogen). We use the splits from (Hoogeboom et al., 2022), with 100k, 18k, and 10k molecules for training, validation, and testing respectively. We use 98522 molecules out of 100k, selected s.t. + 2o of all atom RBF components fit within the evaluated field positions G.
Basic molecular properties are reported in Table 1. We achieve state-of-the-art performance in terms of molecule and atom neutrality. In Table 2, the maximum test similarity and the bond distribution are competitive to the point-cloud approaches, while MiDi performs very well on TV,-
This is likely an effect of our model trading the correct atom count generation for improved neu- trality. Point cloud methods are hard-constrained to generate specific numbers of atoms, while our method may remove add atoms in favor of other properties. In Appendix G, report small shift in the generated atom count distribution compared to the desired conditioning, while increasing the strength of classification guidance 3 in Table 3 significantly decreases TV,.
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Basic molecule properties for molecules generated by models trained the QM9 and GEOM-Drug datasets, based on 10k samples. We report the mean and standard deviation of the metries for three different runs (each generating 10k samples) for QM9. The remark column (R) symbols represent: (*) re-tested by us (f) post-procesiing corrections or algorithmic bond extrac- tion; (i) explicit aromatic bonds. Results for methods with are in italics since they cannot be directly compared with purely generative models. An H column v denotes explicit hydrogen generation.
Table 1:
QM9
GEOM-Drugs
Neutrality
Neutrality
Model
H
R
Atom(%)
Mol(%)
Val(%)
R
Atom(%)
Mol(%)
Val(%)
MolDiff
G. f)
97.0
VoxMol
(1)
99.2
89.3
98.7
(i)
98.1
75.0
93.4
MiDi
(*)
98.3
84.5
96.7
(*.4)
96.8
31.3
62.7
JODO
98.5
88.2
98.1
(*4)
97.0
36.7
77.8
EDM
98.1
81.9
91.6
(*)
81.0
0.5
3.0
GDM-AUG
97.6
71.6
90.4
EDM-Bridge
98.8
84.6
92.0
G-Schnet
95.7
68.1
85.5
GeoLDM
98.9
89.4
93.8
84.4
99.3
FMG
99.4 0.1
91.5 1.4
98.8
94.7
16.5
64.9
Data
99.9
99.6
99.6
99.4
86.2
89.6
MolDiff
(+, f)
85.0
FMG
80.0
Table Molecule graph quality of generated samples from models trained the QM9 and GEOM datasets, based on 10k samples. We report the mean and standard deviation of the metrics for three different runs (each generating 10k samples) for QM9. See Table for description of symbols.
QM9
GEOM-Drugs
Model
H
R
MST (t)
TV.())
TV&(L)
R
MST (f)
TV.(I)
TV&(L)
MiDi
V
(*)
0.57
0.19
0.81
(*.2)
0.53
1.07
EDM
(*)
0.55
0.49
0.98
(*)
0.46
1.63
5.72
JODO
(*)
0.58
0.19
0.89
(*
0.58
1.07
FMG
0.54 0.00
0.52 =: 0.02
0.95 =: 0.01
0.49
4.78
MolDiff
(*))
FMG
0.50
0.63
4.58
Although our NN architecture not invariant to SE(3) transformations, our method achieves satis- factory conformational accuracy, with very low distance between generated and data distributions. We report the W, distances for bond angles, bond distances, conformation energy distributions in Appendix A. Table 4, and show the corresponding CDFs in Figures 4 and 5,
6.2 QM9 CONDITIONAL. GENERATION
We employ the same setting as in previous work (Hoogeboom et al., 2022), training molecule prop- erty discriminator models 50% of the QM9 data, and conditional generation models the maining 50%. Our generated molecules are then tested using the classifiers. Further details of the experiment and the results are reported in Appendix A.2.
We compare the difference between the highest and lowest occupied molecular energies Ae and the polarizability of the generated molecules, compared the true, conditional values. FMG outper-
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0.50
Density
0.25
0.00
-2
det R
data
EDM
GDM-AUG
FMG
Figure 3: Kernel density estimation of det R. resulted from the data and generated molecules. EDM unable distinguish and correctly generate the enantiomer distribution, while the E(3) variant method (GNN) generates improper, 0 volume conformations. FMG matches the enantiomer distri- bution.
forms EDM in terms of molecular stability, while EDM better in its fidelity to the conditioning variables
Additionally, our method generates similar fidelity to the conditioning variable c, even without classification-free guidance on the number of atoms N. p(Nalc). This could prove useful in situations where the conditional p( (N).c). cannot be accurately determined based on data statistics (e.g., for generating ligands that interact with specific proteins).
6.3 GEOM-DRUGS
The Geometric Ensemble Of Molecules (GEOM) dataset (Axelrod & Gómez-Bombarelli, 2022)
contains molecules of up to 181 atoms and 37 million conformations along with their corresponding energies. Following previous work (Hoogeboom et al., 2022; Peng et al., 2023), we select the 30 lowest energy conformers for each molecule graph. Similar (Peng et al., 2023), we also restrict the data points to only those molecules that contain atoms from {H. C. N, O, F, P, S, c1). We extract those molecules that can contained the pre-defined evaluated positions that determine the tensors u, and up. similar to QM9 experiments. After filtering, the data contains 96.7% of the training molecular conformations and 99.7% unique molecular graph coverage.
We report basic molecule properties in Table 1, where method achieves highly competitive neu- tral atoms and molecule percentages. Because of cest-generation bond extractions and corrections (marked with (f)) and other bond representation choices, it remains unfortunately difficult to reli- ably compare against all methods (please refer to Appendix for details). In Table 2, we see the same effect mentioned in the QM9 experiments, where the method has worse TV, metric, likely caused by our method favoring other metrics to correct atom counts. When H atoms are not explicitly generated, our generated atom distribution becomes significantly closer the data.
The conformational properties of the generated molecules are detailed in Appendix A.I. Our method performs well, even with significantly more complex molecules.
6.4 E(3) INVARIANCE AND CHIRALITY ANALYSIS
In this section, empirically confirm that E(3) invariant implementations lead invariance wrt enantiomers and show that our proposed method captures all molecular geometries. Our setting consists of synthetic dataset of 48 chiral molecules with skewed enantiomer distribution. Details of the experiment are provided in Appendix B.2.
Figure 3 shows the KDEs of generated tetrahedron determinants of R [m, -m,, my Mcg m. m=]" T k3x3 The E(3) equivariant generative method (EDM) has p. R) = ps det R)) which empirically attests Proposition 1.

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Table 3: FMG performance with various design choices.
Neutrality
Model
Val(%)
Atom(%)
Mol(%)
TV.(J)
TV&(J)
EDM
91.6
98.1
81.9
0.49
0.98
FMG aug
95.8
98.6
82.6
0.69
0.94
FMGg=0
98.6
99.3
92.2
1.02
1.16
FMG
99.4
91.5
0,95
FMGs=5
98.6
99.4
92,2
0.48
1.02
FMG
97.7
99.1
90.9
0.54
1.00
We also test point-cloud method that is variant to reflections. The work of Hoogeboom et al. (2022) (GDM-AUG in Figure 3) provides a variant of their method, where atom positions {m;} are concatenated to node features. Interestingly, although this method has Pa (det R) p det R), generates many unnatural conformations, with tetrahedrons of approximately 0 volume.
Finally, FMG captures the distribution well.
6.5 ABLATION
We determine the aspects of our method that contribute the most to our competitive performance. In Table 3, FMGaug receives augmented fields based on randomly rotated sets of atom coordinates {m;) during training, FMGg, 3 E {0, 2, 5) varies the degree of classification free guidance on the number of atoms (see Section 4.4), and FMG the performance of the molecular graphs, where bonds are extracted according to bond pairwise atom distances, without considering channels. The rotation alignment is the main factor that increases the neutrality metric to state-of-the-an per- formance, FMGnug performance drops significantly, while classification guidance improves TV,,-
For FMG we determine bonds based on the lookup table from (Hoogeboom et al., 2022). This confirms that our molecular conformations analyzed in Appendix A. are remarkably good since the model's neutrality remains SOTA even when bonds are solely based on the atom positions (m:).
CONCLUSION
We demonstrated that continuous field-based molecular representations can achieve highly competi- tive results. Through theoretical and empirical analysis, we confirmed that E(3) equivariant methods consistently generate enantiomers with equal probabilities. This is crucial, as incorrect specification of enantiomers can lead to loss of drug effectiveness or severe side effects. We also discuss in detail why developing corrected SE(3) invariant point-cloud method would be significantly more challenging and show that our alternative successfully generates correct chiral properties.
Limitations and future work. While point-cloud methods scale quadratically with the number of atoms, our method depends directly on size and shape of the molecules. For any new dataset, the dimensions of the field tensors will always dictate the method's computational feasibility and should be carefully considered.
Our tensor dimensions remain fairly large (even with coarse resolution). Adaptations of current solutions of improved scaling (Gu et al., 2024; Ramesh et al., 2022; Rombach et al., 2022) could, in principle, allow for much larger architectures. In turn, scaling the expense of group invariances may result in significant strides towards drug-like molecule design (Abramson et al., 2024).
We show significant performance gains using reference rotations, keeping general graph metrics competitive to E(3) formulations while obtaining chiral sensitivity. It remains, however, an open question whether method can be applied other 3D object generation tasks.
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8 ACKNOWLEDGEMENTS
We acknowledge the computational resources provided by the Aalto Science-IT Project and CSC-IT Center for Science, Finland. This work was supported by Research Council of Finland (grants 334600, 359135, 342077), the Jane and Aatos Erkko Foundation (grant 7001703), and the Cancer Foundation of Finland.
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A
ADDITIONAL EXPERIMENTS AND DETAILS
A.1 CONFORMATIONAL ANALYSIS
We compute the Wasserstein Wi distances for bond angle (BA), and bond lengths (BL) and re- spectively averaged over atom and bond types. We additionally compute conformational energies (CEMMFF and CEXTB) W, (definitions these metries can be found in Appndix H). We report the Wi distances in Tables and 5. In Figures 4 and 5, we also plot the CDF of the distributions, based on which the W, distances were computed and Figure and we report the histograms xTB energies.
Note that for the GEOM-Drugs dataset, MiDi and JODO use explicitly generated aromatic bond types, and thus has different bond type set B compared to ours. This also affects the distributions of the bond types and 2 and we therefore omit MiDi from the GEOM-Drugs bond distance metric BL.
Fields cannot enforce the number of atoms that will be generated, while point clouds do so, sampling N PDVN) during generation. While not enforcing the number of atoms provides ful flexibility (e.g.. in conditional generation, the conditioning may be p(-ly), instead of p(- ly. N), where the latter requires knowledge of the joint p(y, N)), find TV, (ablation Table and CFAm (Table 4) to gain significant improvements as result of atom count conditioning (from no con- ditioning, F0 FMG3=2 which was used in this paper), but in this work, we opted for classifier-free freef guidance on binned atom intervals (see Appendix G), rather than conditioning on specific atom count. The high impact of atom count conditioning on these metrics provides a strong indication that, if more Eme-grained conditioning is adopted, fields will achieve similar TV, and CETTB numbers as point-clouds.
Table Conformation results of molecules generated by models trained on the QM9 dataset, based on 10k samples. We report the mean and standard deviation of the metrics for three different runs (each generating 10k samples). The remark column (R) symbols represent: (*) re-tested by us. Reported bond length WI distances (BL) are multiplied by 100.
Model
R
BL (1W1)
BA (1W1)
CEMAEE (1W,))
CBAR (UW1)
EDM
0.47
0.68
3,45
0.30
MiDi
1.92
0.95
3.12
0.34
JODO
0.62
0.83
1.41
0.22
FMG
0.59
1.08 = 0.12
2.15 0,385
0.71
FMG
0.98
Table 5: Conformation results of molecules generated by models trained the GEOM- GEOM-Drugs Drugs, based on 10k samples. The remark column (R) symbols represent: (+) re-tested by us (t) post-processing corrections or algorithmic bond extraction; (f) explicit aromatic bonds. Reported bond length W
distances (BL) are multiplied by 10
Model
R
BL (LW1)
BA ((W1)
CEMsar (WW1)
CExT (LW1)
EDM
(*)
8.84
4.36
1.93
MiDi
(*.))
1,38
40.98
1.82
JODO
(..
0.351
7.59
1.21
FMG
2.74
1.00
53.14
3.22
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GEOM
QM9
Bond type
Bond tyge
Bond type
Bond type
Bond tyge 2
Bond type
CDF
1.0
1.5
1.2
1.4
Bond length
length (A)
length
Bond lengtth
Oxygen
Phosphorus
Sultur
Carbon
Oxygen
Nitrogen
CDF
150
100 150
100
Angle (degrees)
Angle (degrees)
data
FMG
EDM
MiDi
Figure 4: Cumulative distribution function plot 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.
QM9
GEOM
1
1
CDF
CDF
0
-284.0
247.0
777.0
-322.0
560.0
1442.0
data
MiDi
EDM
FMG
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 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).
W1 distance: 0.88
WL distance: 0.34
W1 ditance: 0.30
W1: distonoo: 0.22
data
dana
dats com
fmg..
.odo
Enargy (Eh)
Energy (8))
Enargy (Eh]
Energy (2h)
Figure 6: xTB energy conformations (Eh) for MiDi, EDM, JODO, and FMG for the QM9 dataset.
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3.2 1.62
distanne:
dede
date
ECO4
0.00
4.14
6:14
608
308 -0 100
-00 -10 -20 -02 +e -6
-20
100 100 -.0 -oo
-1
4000
-20
Energy
Energy
Energyy
Figure 7: xTB energy conformations (Eh) for MiDi, EDM, JODO, and FMG for the GEOM-Drugs dataset.
A.2 CONDITIONAL GENERATION
We train classifiers de for the two considered properties, and Ae. The property classifiers d. E(3) invariant graph neural networks that receive pairwise distances between atoms ||m, - m and atom types as categorical features, similar to the architecture used for parametrizing EDM. We follow the setting from (Hoogeboom et al., 2022), splitting the 100 thousand molecules into two equal subsets, and training our generative method and property classifiers on each of the two subsets, respectively.
To sample the properties e ~ PD. we split molecular properties in one thousand equal bins, and form, based on training data statistics, the conditional and marginal categorical distributions p(N[c, and p(c:). where c; is the nhh property bin and N the number of atoms. We test our conditional model by sampling two thousand pairs of [c,, N] ((C))p(NN c,), and further sample the actual conditional property e U((c,,o 1, with which we condition the generative model on Eg (fe, f, Na c).
We test our generative models by predicting the generated molecules' properties, and computing the L loss between those predictions and the conditioning variable used during generation.
In Table 6, FMG is the trained conditional method for the two properties, respectively, and FMG the same model, where we do not use classification guidances on the number atoms ac- cording to the property pcN]cc..
We also include additional baselines: FMG,.. is an unconditional model, where we sample molecules according to appropriate number of atoms p( (c). and FMG are completely random samples, from an unconditional model.
Table 6: Conditional generation results. Atom and Mol Neutrality are reported in percentages. The mean results of 2k samples generated by our methods are shown.
a
Ac
Neutrality
Neutrality
Model
LI())
Atom
Mol
L. (L)
Atom
Mol
Conditional
EDM
2.76
80.4
655
81.7
FMG
3.55
99.0
86.8
746
98.9
84.5
FMG
3.92
99.0
86.7
734
98.6
83.0
FMG
6.92
99.4
91.5
1259
99.4
91.5
PMG
9.03
99.4
91.5
1451
99.4
91.5
We implement property conditioning by shift-scaling the channels, presented in (Perez al., 2018), for all U-Net layers (similar to time and number of atoms conditioning). Note that we did not. use classifier-free guidance for molecular properties c.
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B CHIRALITY ANALYSIS AND E(3) INVARIANCE
In Appendix B. we develop our theoretical analysis and in Appendix B.2 we describe our chirality experiment details.
B.1 THEORETICAL ANALYSIS
In Appendix B.1.1, we draw the connection between NN's B(3) invariance and its resulting invari- ance to enantiomer pairs.
In Appendix B.1.2, we show how point-cloud Diffusion models parametrized by E(3) invariant NN become E(3) equivariant.
We give proofs of Propositionl, Lemma 2, and Proposition 3, in Appendix B.1.3, B.1.4, and B.1.5. In Appendix B.1.6 we discuss cases where Proposition 3 applies.
B.1.1 E(3) INVARIANCE OF NNS AND CHIRALITY
Let VI, E IR be two, three-dimensional vectors, and let d = V2. By definition, orthogonal matrices Q have an inverse equal Q-1 Q² Therefore, it is easy to see that ||Qa]| since
||d]
||Qd]l (Qd) ="""dddddd'''-""""""""" d ||d|
(15)
where I is the identity matrix, and is the Le norm. Since reflection and rotation matrices are orthogonal, it follows, that the Euclidean distances between any two vectors V1, are invariant to orthogonal matrix multiplication, and, therefore. to any rotation or reflection matrix.
Consider now the cosine of the angle between any two vectors V1, E R3 transformed by an orthogonal matrix Q We have that
cos( Qv1.Qv2) = Qv1 Q
(Qv1) (Qv2)
v V.'vv
V1 V2 v2
cos( V1, V2),
QQv Qvall
|| 1
| vell
|
(16)
where we have used equation 15 for ||Qv;|| = ||v.|]. All distances and angles between vectors in the R space are therefore preserved.
Any function o (c.g., neural network) that receives pairwise atom distances or angles will therefore be invariant orthogonal matrix multiplication and implicitly, to reflections.
Ra
R
R
R3
R2
R2
Figure 8: R and S configurations of general tetrahedral configuration with four, distinct residues (the 4th atom, perpendicular to the planar perspective was not annotated for simplicity). The deter- minant det/mal mc, m22 mc, mas mc] is constant under rotations, but changes sign after reflections. Equivalently. the sign changes when swapping m and m positions, which defines the clockwise (left) counterclockwise (right) orientation. The determinant sign equivalently, the orientation determines the enantiomer R or S configuration.
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Consider tetrahedral configuration, where the central carbon atom is positioned at ma, and is connected to four unique residues R1, R2, R3, and R4, initially positioned at mæ1, m22, m3s and mz4, respectively (see Figure 8). Observing the molecule along the R--C-axis, way of defining the chirality of the molecule is to specify the ordering of the remaining residues in clockwise or counterclockwise orientation. Assuming and R3 positions has the effect of reversing the rotation direction and creating a distinct chiral configuration. Now, considering the plane defined by mæ2, the central carbon position me, and m24, the same effect can be achieved by reflection about this plane.
start with R2 position fixed at m22 swapping RI
By Householder transformation, reflection about hyperplane (in this case, plane), can be defined by the plane's s normal vector (perpendicular to the plane and of norm 1), for any plane that passes the origin:
P= I- I-vv'
(17)
where the reflection matrix P leaves any vector on the plane unchanged. Translating the object by. e.g., - (m R2 + mc + mR, the plane now passes the origin, meaning we can determine such matrix P for the m22, mc. MAI, and swap mR, and mæ, with that reflection.
In general, a reflection about any plane can be achieved by specific translation, a rotation, and reflection, applied to the whole R space (after reflection, the molecule may be moved back with an opposite translation and rotation). As Euclidean distances and angles between any two vectors are preserved under each of these operations, it follows that Euclidean distances and angles become invariant to the chirality.
To our knowledge, all current point-cloud-hased methods use Fuclidean distances or relative angles between atom position vectors m,, my. e.g., for message passing in their GNN architectures. These methods cannot distinguish enantiomers. As such, their resulting E(3) invariant probability distri- butions will always produce them completely at random, and the molecular graphs will not change based on whether the or R enantiomer configuration is present in that molecule.
B.1.2 E(3) INVARIANCE OF THE GENERATIVE SETTING (DDPM)
The problem arising from F(3) invariant neural networks in molecule generation is the H(3) equiv- ariance of the resulting conditional probability distribution Po (mm m,). Concretely, for a set of positions m = {mı,.... mN). }, m, E IR3 the and S enantiomer configurations of molecules will be equally likely, since the marginal becomes E(3) invariant P& (m) Po (Rmm) VR orthogonal. We illustrate this in the context of EDM (Hoogeboom al., 2022). Consider set of R points {mı,... my) } of a molecule containing N atoms and a neural network om. At each layer in the GNN network parametrizing EDM, positions updated follows:
m dej
m
m
om (d))....),
(18)
where @m receives atom pair Euclidean distances and other features that are E(3) invariant.
Every layer is linear combination of atom positions m; with E(3) invariant coefficients that are equal to I have the form (dkj....).)))
dzy). The update layer ( [+ of atom is then given by ++1 = T (m²), and since T coefficients are E(3) invariant, the equality Te (Rm')= = RT. (m') holds for all orthogonal matrices R.
Consider the es parametrization of the reverse DDPM process
Psl (mmm/mm, t) Nmm ps (m,, 8, E(s,t),
(19)
where
1
o
Ms (s
(
e (m.,)) (),
(20)
O4ts
where Ots 01/a, and 0f- = 03. keeping the notation from Section 4.2. It follows that
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o cs (Rm, ()
Po (.Rma 8,
Rm;
0tts
0
di R
Rm,
Re
equivariance of e
Otts
0t
ois co (me, ())
-R(
M.
Oits
Since the covariance matrix of the reverse process does depend on positions m, and because of the μs equivariance wrt E(3), we have that
Ps (m, t) = Psl Rm, Rm, t), Vt.
(21)
We give a short version of the proof from (Xu et al., 2022; Hoogeboom et al., 2022), which shows how the conditional pol (m, (m,/m,, t) and marginal pa (m,) being, respectively, equivariant and invari- ant to symmetry group, results in the invariance of p& (mo). Following (Hoogeboom et al., 2022), have
pl Rm,) =
pl (RRm, (mmmmm))((mm))
p( PRRm,RRR "m)p(RR mm.)
p(m, R mm)p(( (R mm))
Invariance of p(m:) & equivariance of (mm.
(immmm")piie det R
Change of variables m' Rm
m'
-p(mg),
where note that the change of variables m' Rm, also changes the integration volume, and the equality stands for both det R +1 (rotation and reflection matrices).
(O, is invariant to orthogonal matrix multiplication, it follows that p(ma) = p( Rmo) for all orthogonal matrices R (induction). We have shown how reflection matri- R switch the and S chiral configurations of molecule the beginning of this section. This result can be empirically observed in the orange line of the main text Figure 3.
Observing that p(m,)
B.1.3 PROPOSITION PROOF.
Consider atom positions m E R3xN and m' E 3 from enantiomer and S configurations, and Pa an E(3) invariant probability distribution. E(3) invariance implies SE(3) invariance, since EE33) This gives
Ps (m) = Po t).
(22)
for all rotation matrices R, (orthogonal, with det R, 1) and is a translation vector. The E(3) invariance of P gives
Po R_ (R t)) = P& RR.m +t) = (m).
(23)
where R_ is orthogonal, with det R. =
We restate that swapping two residue positions m,z1 and m of atoms R1 and R3 connected to chiral center will change the clockwise counterclockwise orientation (see Figure 8). Swapping
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mgn and mas positions be achieved through reflection about the plane defined by the points [m mc, mg2). In turn, a reflection about plane can be expressed as a rotation, a translation, and reflection. This implies that aR+, R_ t, s.t. m' R. (R, m + t). which gives
Pp(m') = Pol (R (R + = pm(m).
(24)
B.1.4 LEMMA 2 PROOF.
Let M' = (mi...., mm). m I n. m, E R" be n--dimensional vectors, and f R + S be an SE(n) invariant function. Center the object creating
M = [m: m,
Mm
m)
(25)
where m 1 >E= m, Observe how the vectors of M are linearly dependent, since m
Σ mm m Σ m - 72
m = 0
(26)
This means that (M) S m m- n- n- which gives dim( +) 2 1, since M contains at most linearly independent vectors. This gives
M = {x E (x, m) = 0, Vm E Span(M)) + (0}.
(27)
Let p E M be any M's hyperplane normal vectors (perpendicular to all vectors M and unit length). Using the Householder transformation, define reflection matrix about the hyperplane using
p= I- -pppp
(28)
Note how any vector in M unchanged when reflecting by P, since
Pm = (I 2pp" mm m 2ppp m m 0, Vm E M.
(29)
For all R, det(R) -1 and R orthogonal (reflection matrix), note how
f(M) f(RPM) = f(RM)
since RP proper rotation matrix and is SE(3) invariant
(30)
since PM = M
(31)
where RP is a proper rotation matrix since it is orthogonal and det(RP) det(R)det(()
-1 -1 1.
We have established that f is reflection invariant for O-centered objects of m < n I vectors. We will utilize the translation invariance of f to extend the proof to any A' object of m I RR vectors.
Let Ty translation operator applied to a set vectors:
T(M)) = (m, + v,.... mm + v).
(32)
Let R be reflection matrix. Then
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((RM)) (M))
from eq. 25, M' (M)
= =[(R(m: i + m.... mm i + mm)
=f([Rm, -Rm Rm..... Rm, Rm + Rm))
=f(TRm(RAN)
= =S(RM)
translation invariance equation 31
= =S(M)
(M))
translation invariance
=f(M),
which means that for any set of m S n n--dimensional vectors M', an SE(n) invariant function f has f (RRM' f(M).. Together with translation and rotation invariances of this implies that is B(n) invariant.
Remark 1. If RR + S, & is SE(n) invariant (equivalently E(n) invariant), and g Sm + P. then g(f (M.() with M2 E Rk*" E(n) invariant.
The remark is made by applying any translation, rotation, and reflections M2 which leaves unchanged. And will be useful for our next proof.
B.1.5 PROPOSITION 3 PROOF.
Proposition proof. Assume n vectors my..... m. in R., where n 2 5. Also, assume f Rmx3 - S is SE(3) invariant function, but not E(3) invariant. We know 2 4, by Lemma 2. Consider f(m1, m2, my, m4) = sign(det( m m4, m2 m4, m3 - ma))), and note that is SE(3) invariant (rotations and translations leave the function unchanged, whereas reflections change its sign).
Remark 2. Enantiomer configuration is local property, which can only be determined by at least vectors from its set of 5 atoms (4 residues and one central atom), and f can be used determine chirality. Throughout this proof, we refer to "chirality" the clockwise counterclockwise atom arrangements when looking down from one vertex of a tetrahedron.
Let M and M' be 4 atom subsets. Assume any triplet subset vectors. Then, f(M) conveys information any other set f fMM), unless M M'. Intuitively, choose any plane from M. The resulting determinant's 's sign, which distinguishes chiral centers, will depend what side of the plane choose the 4th point. The fact that can be used determine chirality stems from the equivalence between the classical chirality definition (using rotations) and determinant signs (see the beginning of Appendix B. All 4 sets of atoms need to evaluated to have all potential enantiomer states that can happen in n atom molecules.
has three linearly independent
We observe that do strictly need compute f(-) for all ordered sets of 4 atoms. For set of 4 atoms, observe how their permutations may, at most, the sign:
f(m m2 ma m,) = f (m,, m2 ma. m.)) = - f( (m1, m2, mg. m,)),
(33)
where we denote by He and m. even and odd permutation, respectively. Therefore, we need (*) to compute all unordered sets of 4 atoms, giving O(n+).
Remark 3. The (") features f(m;1 m,2, maa m.) need paired atom type information (a,1. a;2, ags, a,a) using function that is vadiant on atoms, meaning.
g(f(ma, mæ, m,3, ma), (a.), a,2, aga a;a)) # o(f(mm)) m m. ma), (I(a; a,2, a;3 a; ))=))), where is any permutation except the identity.
(34)
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B.1.6 CASES WHERE PROPOSITION 3 APPLIES
Consider the distributions Pl.... ... Pr. with p. (,(,) (0, I), Vi and random associated atom types {az} (also drawn from an uninformative distribution). The task is transforming these distributions into the conformer distribution of n atom molecules, some of which contain chiral centers. By the uninformative assumption, we need to compute the chirality of all sets of 5 atoms, since all sets have equal probability of being chiral. By remarks 1, and 3, this scales prohibitively, in On²)).
Chiral-aware, SE(3) invariance becomes intractable in point-cloud, diffusion models since the oc- currence of reflections needs to be assessed in local neighborhoods (we are specifically interested in potential reflections of chiral centers, but all atoms may be part of chiral centers equally likely). For diffusion methods that define uninformative and identically distributed vector positions implies that all sets of 4 atoms may be part of tetrahedrons at / 0.
Examples of E(3) invariant diffusion models operating on point-clouds are (Hoogeboom et al., 2022; Peng et al., 2023; Vignac et al., 2023b; Xu et al., 2023; Wu et al., 2022). All these methods imple- ment NN Φ activating on sets of (f(M)),... f(M,),. where M1 g2x3, and f are Euclidean norms f(M) = || M2,2112- They are E(3) invariant by Remark 1, and can only be made chiral-aware using Onn*) scaling, by Proposition 3.
B.2 EXPERIMENT DETAILS
We create dataset of molecules, each one having a central tetrahedral carbon that always connected to the set of four atoms C {c, H, N, O}. We create multiple unique molecules, by connecting the three atoms C,N,O E to other arbitrary residues. In Figure 9, we show half of the molecular graphs within our data. For each of them, create molecular conformations using RDKit. Note how in this experiment we are not necessarily interested in creating realistic molecules, but simply any molecules that contain chiral configurations.
0H
M1
hit
OH
AM
NK
N11
My
Figure 9: Molecular graphs of half of the chiral dataset.
We train FMG for 10k iterations, and EDM for 300k iterations, and test for multiple checkpoints. Since the dataset is small, we do not use mini-batched and set our batch size to the data length.
C FIELD PARAMETERS
For the QM9 dataset, we determined that a resolution of 0.33A and 323 voxels for each molecule is good compromise between the sample size and the percentage of the dataset's 's samples (the total size being u I x232x32232 for and atom and bond channels, respectively). We set the 0.176. The component values
RBF component's 's were determined as the maximum value of an RBF component having its positions m in one of the evaluated positions G. Although the field parameters do not reflect the actual, physical size of the atoms, we note how true physical fields may be recovered by appropriately scaling the variance of each atom RBF component, based on the channels a they belong to.
0.224, for all fields, and
0.176
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We used the same resolution for GEOM. After rotating all molecules using the principle components of the atom position vectors, the resulting tensors u I |I were enough to cover 96.66% of the total data points and 99.71 71% % unique molecular graphs.
The field hyperparameters were selected during early experimentation. We focused on the effect of field hyperparameters fitting uaa (2:) and (xxx) the generated fields u. and uy (Equations 13 and 14 and Appendix E). We compute the "extraction accuracy". which measures how many molecules have the same molecular graph extracted from the fields they create. We additionally report the average "distance from original", computed as the average Euclidean distance between the correct atom positions and the ones extracted. For both, we determine how robust they are for various levels of noise applied to the fields. Ablation of field hyperparameters for the resulting diffusion model's accuracy was not performed, due to the heavy computational resources required.
Field standard deviation σ: Intuitively, larger standard deviations o would facilitate smaller re- ceptive field CNN kernels to capture longer distance neighboring RBF components. At the same time, require that molecular graphs and conformations are correctly extracted. Figure 10 shows how increasing a worsens extraction accuracy, and our choice (a² = 0.05) prioritized molecular conformation (distance from original) and graph (Extraction accuracy) reliability over noise levels.
8:50
1.064
5,965
23300
3875
2000
6.95
Nolue
100.00
Figure 10: Noise robustness for the graph extraction procedure for various or field values.
Field component weight y: The component weights may be thought of as an input normalization parameter. Therefore, did not consider other values than 0.176, which should normalize inputs to roughly values between (0, 1).
Resolution In general, it is desired have as low resolution possible, this will influence the input dimensionality and, correspondingly. the depth of the parametrizing NN for limited com- putational resources. At the same time, the accurate extraction of conformers and graphs must be ensured. Figure 11 shows how lowering the resolution from 0.33 (our choice) to 0.44A worsens "ex- traction accuracy" and "distance from original" across noise levels. We note, however, that 0.44 may still be a valid choice, assuming the resulting NN would have low enough amounts of noise.
1.000
412
5.98
3.5
+.33
032
0.06
Nolse
No.5e
Figure 11: Noise robustness for the graph extraction procedure for two resolution levels.
D MODEL AND OPTIMIZATION PARAMETERS
Diffusion Model We use 100 timesteps t E [0, 1] for our diffusion processes, the parametriza-
tion. and the modified cosine scheduler for a,, presented in main Section 4. The conditional scale for the classifier-fres guidance was kept to B = during the generation phase, for our main results (unless explicitly mentioned otherwise).
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Optimization We optimize the Csimple objective using Adam optimizer with 8 10-6 learning rate and (0.9, 0.99) for the Adam's 3 parameters. The models are trained using batch sizes of 32 and 64, for GEOM- GEOM-Drugs and QM9 datasets, respectively.
U-Net We use a three-dimensional adaptation of the U-Net employed by (Ho et al., 2020). For both GEOM-Drugs and QM9 datasets, we use the same hyperparameters, except for the batch size. The architecture consists of three U-Net layers, with linear attention for each resolution level, except the last one, which uses full attention. Its initial convolution converts the and 11 channels (for QM9 and GEOM-Drugs datasets, respectively). into a hidden representation of R 128 xw which is halved for each the H, W, D dimensions, while its channel size is multiplied by 1, and 3.
Resources For the QM9 experiments, we use four A100 GPUs (40GB memory version) for 180 hours, and 1.2 million iterations (or about 780 epochs). On the GEOM- Drugs dataset, we used the same four A A100 GPUs for 330 hours and trained explicit H BE6M-Drugs model for 1.32 million iterations (6 epochs), and our implicit one for 1.2 million iterations (5.5 epochs).
E MOLECULE GRAPH EXTRACTION PARAMETERS AND ALGORITHMS
The molecular graph extraction has four components, used in the following order: deterministic peak extraction, atom positions optimization, deterministic bond extraction, and Viy parameter opti- mization that removes false bonds.
Deterministic peak extraction The algorithm works by defining simple, recursive flood-fill algorithm for retrieving the neighbors. Assuming the RBF functions have small a parameter and reasonably small amounts noise produced by the diffusion model, should never fail extract the correct number of atoms and their approximate positions from the fields.
Algorithm Peak extraction
Input: fields NNN,N,,,,, threshold t, 9. (x; t); Am repeat
p p= q-pop()
neigh +- get_neigh(y, 1,
Am (nsert(meann meennneiih,, neigh. p)) (neigh))
until q empty
return Am
Optimizing the atom positions The initial atom positions retrieved in the Am sets are further optimized using Equation 13. Note how both atom numbers and their types obtained using the peak extraction algorithm, and the optimization only fine-tunes their positions {m,}-
Bond detection The distances between any two bonds byj and bys half large the distances between the atoms and ag. meaning ||m:j - m2x1/2 significantly larger overlap between the bond RBF components than the atoms' RBF components. Therefore, we treat the bond assignment differently and do not reuse Algorithm 1. Using the op- timized atom positions Mg. we check for all the atom pairs m, my E M, that are within some predetermined threshold distance |m; - m5il2 < d1 and create sets
||m, m& 1/22.. This may create
Bij.6 [ug
m.ylle < d2, (x)x) t,,
where G consists of the evaluated field locations, to is an additional bond threshold, and de is the radius of the sphere centered m,j. around which check for u, values. A covalent bond type b added between atoms and j if B.j.b + Ø.
To ensure don't 't miss bonds due the low resolution, we select large sphere (determined by its radius dz) around the approximate bond positions m,j and small threshold values t,. This may then recover bonds positions where the model did not generate true RBF bond component.
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Bond optimization "Vy To remove erroneous bonds, extracted due to lenient bond-extraction parameters (d1, d2, ty), we optimize the "ij values for all Byj * assuming the mean bond positions my fixed, using equation 14. In Figure 12, analyze how robust the method is to various degrees of uniform noise. We choose 300 molecules from the QM9 dataset and gradually increase the noise amount to the voxelized field values
Accuracy
0.995
0.990
0.965
0.980
0.975
0.05
0.1 U(0, x)
0.2
with optimization
w/o optimization
Figure 12: Performance of the graph extraction method for various degrees of noise. The bond optimization step crucial and makes the method highly robust noise.
Figure 13 illustrates a bond that was initially extracted based on Big + Ø. but removed after optimizing and finding the low Vay value.
1.0
0.200
0.8
0.150
0.6
0.075
0.4
0.025
bond
0.000
Bond
Bond
removed bond
Figure 13: Upper left subfigure illustrates a noisy covalent bond type channel values üi = Ub +e (according the color bar), where has the same dimension as Up, and eijk 2/(0, 0.1). The bond marked with "x" on the left is removed since its optimized Yú value (right bar plot) is significantly lower than the rest.
Parameters We choose threshold parameter t 0.3 in Algorithm 1. For bond detection, we choose large distance d1 of 0.35A (for reference, the margin used to detect bond type 1 based on distance the ablation Table is OÀ). The distance d2 = 0.45 À will result the method checking for the closest midpoint voxel to m,j and few others directly in contact with it. The bond threshold is set to t, = 0.3. Both the mean positions m and bond RBF component factors Yiy optimized using gradient descent for 500 iterations and learning rate of 5. All parameters were selected according to robustness analyses, such as the one illustrated in Figure 12, based on the training data.
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F GENERATED SAMPLES
F.1 QM9 SAMPLES
Figures 14 and 15 depict set of random valid samples from a model trained on QM9.
N
H
Bond 2 Bond 3
Bond 1
Figure 14: Random valid samples from FMG trained on QM9. To the left of each molecule, the top 200 largest u. and U, values are shown along their corresponding extracted atoms and bonds.
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N
H
Bond 2 Bond 3
Bond 1
Figure 15: Random valid samples from FMG trained on QM9, To the left of each molecule, the top 200 largest u, and u, values shown along their corresponding extracted atoms and bonds.
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F.2 GEOM-DRUGS SAMPLES
Figures 16 and 17 depict a set of random valid samples from a model trained on GEOM.
Cl H
Bond 2 Bond
Bond 1
Figure 16:
Random valid samples from FMG trained GEOM. To the left of each molecule, the top 200 largest u, and u, values are shown along their corresponding extracted atoms and bonds.
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Bond Bond 3
H
Bond 1
Figure 17: Random valid samples from FMG trained on GEOM. To the left of each molecule, the top 200 largest u, and u, values are shown along their corresponding extracted atoms and bonds.
G CLASSIFICATION-FREE GUIDANCE ANALYSIS
We create 17 and 91 bins for QM9 and GEOM-Drugs data sets, respectively, and assign them indices that we use as categorical conditioning variables. This led to a better match between the generated and data atom distributions (measured using TV,) (see Table and we deem it relevant when com-
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paring point-cloud-brat approaches since these methods strictly enforee the number of generated atoms.
We determine how accurately the classification free guidance results in molecules containing the conditioned number of atoms. We compute the accuracy as the percentage of generated molecules in the correct conditioning bin N, the L1 distance between the binned number of atoms generated by the model N, and the conditioning bin N min(INNigh N N), and "L.1 lower" is the fraction of the total L1 distance attributed to incorrectly generated bins because strictly lower number of atoms generated N N (as measure of distribution skewness).
NNowl, [-Vnggh NNowl) (or 0, when
In Figure 18, the generated bin accuracy is relatively low but is generally off by approximately atom (measured by the 1.1 distance) for the QM9 dataset. In terms of the distribution shift, 1.1 lower suggests that the QM9 FMG model tends to generate more atoms than the conditioning variable.
Similar results in terms of accuracy and L1 distance can be noted in the GEOM-Drugs dataset (relative to the much larger possible number of atoms), but now shift in the opposite direction. This could be artifact of the number of training iterations: early on in the QM9 dataset training experiments, we observed that the model tends to generate a lower number of atoms and bonds, which likely explains this opposite shift of the 150M-DDrugs method. For reference, GEOM- Drugs model has been trained for about 6 epochs, compared to its QM9 counterpart (about 780).
QM9. Accuracy/L1 dist/L1 lower:
GEOM. Accuracy/L1 dist/L1 lower: 0.14/2.78/0.77
0.
0.15
% of samples
% samples
0.04
0.10
0.03
0.02
0.05
0.01
0.00
0.00
5
10
15
20
40
60
80
Class
Class
generated
required
Figure 18: Histogram of the generated number of atom classes N and the conditioning used N. Small and more significant atom number distribution shifts occur for QM9 and GEOM-Drugs datasets, respectively. Together with the result of Table 3 which shows how critical it is en- force the correct number of atoms for TV,, this result suggests that our method may remove atoms for stability, inducing an atom distribution shift.
H PERFORMANCE METRIC DETAILS
Validity: Percentage of valid molecules, computed using RDKit's SanitizeMol function, com- puted using explicit H atoms.
Neutrality: Percentage of generated atoms that have a formal charge of 0, and percentage of molecules that contain all atoms with a formal charge of O.
Among our molecular quality metrics, the atom and bond total variation distances TV, and TV, are good measure of how well most of the data modes captured, whereas MST should better measure the quality of the molecules. For instance, if the method captures a single mode distribution well, and generates very high-quality molecules, may still have relatively bad TV, and TVp-
TV., TV, We define TV, and TV,
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TV, = Σ Σ aEA Na=i
(DD(NN) - PPo
TV,
(D((N) (N6))
BEB Ny=i
where Na, N₂ are the numbers of atoms and bonds of type and are the atom and bond types, A and B are the sets of atoms and bonds considered, PD and Pa are probability distributions based on data and generated statistics, and the absolute value.
MST:
Denoting the Tanimoto Similarity as TS= (v where v1 and V2 are Morgan Fingerprints based on molecular graphs, we compute the average maximum similarity to test (MST) as:
max TS(Vi, Vt), V.EVoe
where is the set of all test set Morgan Fingerprints.
BL, BA, CEMMEF, and CExTB: We determine the Wasserstein Wj distance between the generated and data bond angles, bond lengths, and Merck molecular force field energies (in kcal/mol).
The angle distributions are defined for all atom types that have two more bonds, and we consider all pair-wise angles between position vectors m; mg, and m, mmg for any two atoms i and connected the same central atom We consider the average W₁ distance over all atom types, and we denote it by BA.
Similarly, we create bond Euclidean distance distributions for all bond types b e: B and denote our averaged W, distance over all bond types as BL..
We use discretized bins of length lo 0.1 degrees for angles, la 0.01A for bond Euclidean distances, and Le = 0.1 (kcal/mol) for conformation energies, and approximate the generated and data cumulative distribution functions Fg and Fa and compute:
BA = lo NA
Σ
Σ [F)a(()) (2)((z))
aEAwEB.
BL = la NB BEB ZBB
Σ
Σ
[Fas(x) Fgb(z)),
where the bin upper limits B., Bb, and B. are chosen to contain all maximum and minimum values of the generated and training data samples, N, and Ng are the number of unique atoms and bonds considered (e.g., excluded and for BA computations, these atoms never have two bonds).
For both Merck molecular force field (CEMMFF) and xTB (CExTB) we use the same bin length 4e and compute the meties in the same way:
CE = Σ Fa.( (z))
J.
where Fa. may be either MMFF or xTB CDF of generated and data energies.
Outlier problems of W, distance: Compared to our method, point-cloud methods may generate molecule with arbitrarily long bond length (i.e.. + oc), making the BL W, metric arbitrarily high based on one or few bonds. Let pa and pr the generated and data bond length distributions. If their corresponding CDFs are Fg and Fo, the W, distance be equivalently written
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Wi Wi(po. [F) (b(1)
where the integration is over the generated and data molecule lengths I. In fields, bond lengths are constrained by the chosen evaluated points G. while in point clouds, may be arbitrarily high. In turn, this can theoretically incur arbitrarily high penalties due to few outliers. In practice, this does not happen (Appendix 1. Figure4, depicting the CDFs of bond lengths), and we further ensure we don't have such an outlier advantage by restricting the bond lengths and angles only those generated within the data limits when computing BA and EL metrics.
I EXPLICIT AROMATIC BONDS AND NEUTRALITY
Problem explanation In our settings, we have used the Kekulé molecule representations, which represent aromatic bonds alternating covalent bond types and 2. In Figure 19, show three different such aromatic molecules, using the Kekulé representation. Noting that the N and valen- cies are 3 and 4, respectively, all formal charges are 0 in these cases (all carbons are assumed to be implicitly connected to H).
Pyrrole
Pyrazole
Phenanthroline
NH
NH
Figure 19: Three aromatic molecules, with increasingly harder to determine neutrality.
Instead of the alternating bond types and 2, an equivalent representation denotes these bonds as separate "aromatic" bond. By considering an electron occupancy of 1.5 for aromatic bonds, Pyrrole has the problem that the N in its molecular graph is apparently (but not in actuality) ionized, having a formal charge -1. Furthermore, in Pyrrazole, only one of the atoms should have the (apparent) formal charge -1. Lastly, the C atoms connecting to two rings in Phenanthroline would have formal charge of -0.5 (as connects with three 1.5 bonds).
This induces ambiguity. One way solve this would be use RDKit and convert the explicit aromatic representation to Kekulé representations. However, in the case of invalid molecules (about 38% of the generated molecules by MiDi), becomes hard, if not impossible, determine exactly how many atoms connecting to a (potentially incorrect) aromatic ring have the formal charge 0.
Explaining the reported results for MiDi We have used the molecules reported in the MiDi github repository. We note that on the QM9 dataset, both our method and MiDi generate molecules in Kekulé form, and they are directly comparable Table (QM9 columns). For the GEOM- Drugs neutrality results in Table (GEOM-Drugs columns), we have reported an upper-bound for MiDi, by considering bond type "aromatic" " as 1.5 electron occupancy and allowing C to have both 0 and -0.5 formal charge. Since many N atoms in REOM-Drugs are (in actuality) ionized, it is not "allowed" to have a formal charge of 1. Regardless, the comparison is very hard to do, and is difficult to say which model generates more neutral atoms molecules.
GEOM-Drugs
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MTS bias We compute the maximum test similarity using valid molecules. This results in bias in the methods that generate aromatic bonds explicitly, as RDKit will require these molecular graphs to be converted to Kekulé representations (Table 2). In this case, this metric will not consider molecules containing incorrect aromatic rings (as determined by RDKit's SanitizeMol). However, in our case, since directly generate Kekulé molecule representations, this quality filter does not exist anymore, and samples that have incorrect aromatic rings (but are not identified as aromatic because of e.g., one circle atom being incorrectly hybridized) will be considered in the final metric. Note how this bias is unrelated to the performance of the generative model, as those unkekulizable molecules are still generated by explicit-aromatte generating methods, but are simply filtered out by this metric.
J COMPLEXITY ANALYSIS
J.O.1 TIME SCALING
We convolutional layers the main blocks of our UNet parametrization. As such, the complex- ity of our NN scales linearly with H W. D. We omit the channel term Ci Cout. as is equivalent the hidden activations node states from point-cloud methods. Linear attention layers are added after each UNet layer, giving O(H W D).
Comparatively, all layers of point-cloud DDPM methods scale quadratically in the number of atoms O(N*). The scaling of these two methods is not directly comparable, since the number of atoms does directly relate the necessary volume that encompasses all molecules of that size, which is highly dataset-dependent, Still, a theoretical upper limit can be determined.
Consider dataset that contains molecules of atoms, which, for simplicity, are all atoms bonded through covalent bond type I (approximately Á). We now discuss how the atoms should be arranged create the worst possible scenario (without considering physical constraints). The upper limit of our grid sizes would be achieved when all three of the following n-atom molecules are present:
One molecule has all atoms placed linearly, which will determine H N 1.35A, and W = 0.
One molecule is perfectly planar, having = = W N-1.35 and D=0. Note that W H is not possible, since our alignment method (Section 3.5) would rotate the molecule s.t. the longest axis is H.
One molecule has H W D
N-1.35
3
The scaling of our convolutional layers would be O( LL335 of our evaluated field locations. We compare the number floating point operations in terms of the number of atoms in Figure 20. For FMG, show both the theoretical maximum and the grids that encompass GEOM-Drugs molecules of various numbers of atoms. Figure 21, we determine the training and generation time (in seconds) of FMG and EDM.
135N
1.35.
where is the resolution
When close to the theoretical upper limit, our method becomes intractable, but this case is highly unrealistic (Figure 20, comparing "FMG max" with "FMG GEOM"). In Figure a). FMG has a comparable generation time but takes longer to train. Figure 2 b) depicts our training time for various molecule sizes, where 99.7% of the molecules are kept (used in this work). The molecule graphs were chosen s.t. they fit within a grid size of 64 40 32.
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7
6
10g1o(flop)
5
4
EDM
FMG max
3
FMG GEOM
2
0
10
20
30
40
100
110
120
130
140
150
160
170
180
190
Number of atoms
Figure 20: Number of floating point operations for EDM and FMG over the number of atoms in molecule. The theoretical maximum of FMG differs substantially from the real datasets, and its practicality relies on application specifications. For the axis lengths FMG 0.33A was used (same value as the one used in our model).
b)
generation
14
Time (zecones)
80
EDM training
FMG generation
5.0
Bo wOXXWXX D)
12
4.50
logso01 ()
training
10
EDM training
40
FMG tr training
4,00
20
3.5
3.50
0,0,0
0 9.
Number of atoms All data
Number of atoms 99.7% molecule coverage
Time elapsed (in seconds) for EDM and FMG models, over various molecule sizes (in number of atoms, and tensor field volumes, for EDM and FMG, respectively). We used 100 iterations with a batch four molecules to determine the training curves. For this experiment, we used Nvidia A 100 GPU. All GEOM-Drugs molecules considered. b) Only molecules used in the current work (99.7% of the molecules) that fit within our grid size of 64-40-32 are used the analysis.
Figure 21:
J.0.2 MEMORY SCALING
Increasing the molecule sizes has a significant impact on memory overhead. Similar to the analysis done in the time scaling analysis in Appendix J.O.1, we construct grids that can fit all GEOM- Drugs molecules with certain number of atoms N. We report the maximum memory allocated for EDM (Hoogeboom et al., 2022) and our method, when generating samples of various atom sizes (determined the label by the number of atoms N, dots denoting volume sizes), with 32 samples per batch.
Figure 22 a) shows that method performs better for larger molecules, but for smaller ones. We note here that GNN architecture batching done by considering a single large graph, where nodes from different batches are disconnected. Therefore, adjacency matrices of this implementation
34

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Published as a conference paper at ICLR 2025
have poor scaling for large batch sizes: (EbEB No)2), where B is the batch sizes and No is the number atoms in a molecule.
We find that increasing the number of unique channels (atoms bonds) is negligible and linear. The additional memory required stems from the input CNN layer's kernel dimension (the depth sizes of the initial 3D CNN kernels equal to |A| [B),, and the larger intermediate states. However, these are considerably smaller than the architecture's overhead (Figure 22 b). Channel scaling is performed with batch size of 1.
b)
EDM memory
547.5
FMG (channels scaling)
6000
memory
5.0
GRU Memory (MB)
5000
FMG memory
(99.7 7% coverage)
log WX XwXDD D3
(MB)
545.0
542.5
4000
540.0
3000
2000
4.0
537.5
1000
3.5
566
Number of atoms
Number of channels
Figure 22: Memory scaling in terms of molecule size for generating molecules. a) We report the maximum GPU memory allocated when sampling for various grid sizes, which fit all molecules of the corresponding number of atoms. We additionally report the memory when using the grid sizes chosen for this work. b) Generation memory in terms of the number of channels.
35


Reviews:
[{"review_id": "bheRJgUlQe", "reviewer": "Reviewer_Gfde", "summary": "This paper tackles the limitations of E(3)-invariant features in modeling chirality, as mirror symmetry is inherently implied in such features. The authors first provide a theoretical motivation, illustrating why E(3)-invariant features are inadequate for representing chirality. They then propose using fields to model molecular graphs, learning these fields through a reverse diffusion process. The effectiveness of their approach is benchmarked on the QM9 and Geom datasets.", "questions": "Please see weakness.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Chiral awareness in E(3)-invariant architectures is essential for accurately modeling molecular properties. I appreciate the authors’ approach of representing molecules as collections of fields corresponding to atoms and bonds. These fields, as probability densities, closely mirror the quantum mechanics formalism than point clouds, where molecules are wavefunctions. This aligns with methods like Density Functional Theory (DFT), which models charge density and provides a more expressive representation than point clouds. However, the impact on memory overhead from this approach warrants further investigation. Overall, I found this paper to be well-written, clearly delineating previous work, the contributions of this study, and its limitations.", "weaknesses": "The impact of this approach on memory overhead requires further investigation. How does the memory overhead compare to point cloud representations? Could you provide insights on how memory usage scales with increasing molecule size and greater diversity of atom and bond types?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5ouzi1ROzS", "reviewer": "Reviewer_V7Qk", "summary": "This paper proposes to generate molecules by representing them as functions in 3D space. Each atom is represented as a gaussian distribution centered at the atom position. The bonds are represented similarly with the center of the gaussian being placed at the center of the bonds. Compared to  such a representation can learn the chirality since it bears the reflection invariance. Experiments are conducted on QM9 and GEOM.", "questions": "- What's the trade-off between losing rotation invariance and gaining the ability of learning chirality?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- Since most deep learning methods use GNNs for molecules, using CNNs is novel.\n- As stated by the authors, capturing chirality is important for representing molecules. And GNN-based methods may have difficulty in breaking  the reflections symmetry.", "weaknesses": "- Although alignment methods are employed, the proposed method cannot guarantee the rotation invariance.\n- As shown in table 2, the molecule graph quality metric is not good on the QM9 dataset.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "7E0Czk8Qg6", "reviewer": "Reviewer_QxY8", "summary": "The authors identify a critical limitation in current E(3)-equivariant models for 3D molecular generation: they cannot properly account for molecular chirality, which is crucial for drug efficacy and safety. To address this, they introduce a field-based representation approach that replaces rotational symmetry constraints with reference rotations, allowing their model to correctly capture chiral properties while maintaining competitive performance with existing E(3)-based methods. They provide theoretical proofs showing why E(3)-equivariant models must necessarily ignore chirality, and demonstrate empirically that their method (FMG) achieves state-of-the-art performance in molecular neutrality while successfully capturing chiral properties, though at the cost of increased computational complexity for larger molecules.", "questions": "1. Could adaptive grid resolutions or sparse representations help reduce memory requirements while maintaining accuracy?\n2. How sensitive is the model's performance to the choice of field parameters (σ, γ)? What's the theoretical basis for choosing these values?\n3. Why did the authors choose 0.33Å resolution for the grid, and how would different resolutions affect the quality-efficiency trade-off?\n4. How might this approach be extended to other molecular properties beyond chirality? \n5. Since chirality is a crucial property of molecules that are involved in drug discovery, is there any specific case that the author can provide on the FMG? Other than that, some properties that are not related to the chirality of the molecule (i.e. orbital energies), why FMG outperforms?\n6. Can this model be utilized in total synthesis and retrosynthesis?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The work identifies and addresses a fundamental limitation in current E(3)-equivariant models regarding chirality, which has been largely overlooked in previous research. Besides, it provides original theoretical proofs about the inherent limitations of E(3)-equivariant models in handling chirality.\n\nThe overall quality is high, reflected by the rigorous theoretical foundation with formal proofs about E(3) invariance and chirality (Propositions 1-3) and comprehensive empirical validation on standard benchmarks (QM9 and GEOM datasets). Well-structured presentation with clear motivation and problem statement. Ablation studies and", "weaknesses": "The first weakness is the method's computational complexity scales with the volume of the 3D grid (H×W×D), making it potentially intractable for larger molecules. As the author mentioned chirality in amino acids, will be a bottleneck for implementing the model in bio- and pharma-tasks. The current implementation requires high-resolution grids to maintain accuracy, leading to significant memory requirements. Another limitation is the analysis of field parameter sensitivity (σ, γ) is not very clear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qfWhOgXfy4", "reviewer": "Reviewer_hRsL", "summary": "The paper proposes a field-based diffusion method for molecule generation that is not invariant with respect to the chirality of the generated molecules. Recognizing that chirality is a critical property influencing the bioactivity of molecules—with several famous examples underscoring its impact on molecular toxicity and drug efficacy—the approach addresses an important aspect of molecular design.\n\nThe results demonstrate comparable performance to EDM and MiDi in terms of molecule validity and stability. The authors also highlight molecular neutrality as an important metric for generated molecules. Furthermore, they assess MMFF energies for the 3D molecules and compute the Wasserstein distance between the bond length distributions and bond angle distributions of the generated molecules and those of the dataset molecules.", "questions": "Questions to the Authors:\n\n1. Your model underperforms compared to JODO, EQGAT-Diff, and Semla-Flow but claims SOTA performance. Can you add a comparison to one of these models?\n\n2. Since non-equivariant models like MCF achieve high performance, can you check whether it still cannot distinguish chiral conformers?\n\n3. Your neutrality metric seems inaccurate (or at least description) (e.g., nitromethane is neutral despite charged atoms). Can you address this issue?\n\n4. Is the strong focus on chirality awareness justified when environmental factors often determine enantiomer generation in practice?", "limitations": "", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "The authors proposed a novel approach to molecular generation using field representations and developed a required theoretical background for the approach. The authors prove challenges to incorporate the same restriction in current E(3) equivariant models.", "weaknesses": "1. Overstated Claims of State-of-the-Art Performance: Although the paper claims to achieve state-of-the-art (SOTA) performance, it falls significantly behind previously published models such as JODO, EQGAT-Diff, and the more recent Semla-Flow—all of which were published before ICLR 2025. The performance gap is quite substantial. For larger and more realistic molecules in the GEOM Drugs dataset, only 16% have all valid atoms according to the paper's model, whereas other models are approaching perfect scores on these metrics.\n\n2. Equivariance Requirement Not Essential for High Performance: Some models that operate directly in E(3) space are eliminating the need for equivariance. For instance, Simplified Scalable Conformer Generation (MCF) demonstrates significant performance improvements in conformer generation without the requirement for equivariance. While MCF tackles a slightly different task, it illustrates that we can develop equivariance-unaware models for molecular modeling that achieve genuine SOTA performance.\n\n3. Incorrect Basis for Neutrality Metrics: The neutrality metrics in the paper are based on inaccurate assertions. Specifically, the statement: \"Atom neutrality is the percentage of atoms with a formal charge of 0, while molecules are neutral if all their atoms are neutral.\" is incorrect. For example, nitromethane has charged oxygen and nitrogen atoms but is an overall neutral molecule. Atoms within a molecule can carry formal charges that balance out, resulting in a neutral molecule despite individual atoms being charged.\n\n4. Inadequate and Incomplete 3D Structure Evaluation: The evaluation of the 3D structures is poor and incomplete. The paper only compares distribution differences for bond lengths, angles, and energies. These benchmarks are difficult to interpret and can be significantly affected by a single outlier result. Depending on the implementation—for example, in MiDi's implementation—one problematic molecule can drastically alter the results. Even with these limitations, the model does not outperform MiDi on these benchmarks.\n\n5. Overstated Importance of Chirality Awareness in the Model: The emphasis on chirality awareness in the model may be overstated. 3D molecule generation serves as a foundational model for structure-based drug design or molecular docking. In these applications, the provided environment (such as a protein or part of a protein) creates asymmetric restrictions on the generated molecule, naturally leading to the production of the appropriate enantiomer. Additionally, in conditional generation, the classifier itself can learn chirality in a simpler way.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)