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Fixed-Dimensional Latent Flow for Generating Variable-Size 3D Molecules

Weichi Yao, Cameron Gruich, Bryan R. Goldsmith, Yixin Wang

arXiv:2609.08333v3physics.chem-phcs.LGq-bio.QM

TL;DR

Molecular generators often require a predefined atom count, although molecular size, composition, and structure must be determined jointly. EF-TALFM samples fixed-dimensional latents and autoregressively decodes variable-size molecules. On PCQM4Mv2, it reports strong unconditional generation and property-conditioned selection results.

  • Problem

    Many 3D molecular generators require a predefined atom count even though molecular size, composition, and structure must be determined jointly.

    Source: §1 · p. 1

  • Method

    EF-TALFM uses flow matching to sample a fixed-dimensional molecule-level latent, then an autoregressive decoder determines size and generates molecular structure.

    Source: §1 · pp. 2–3

  • Results

    87.9% of unconditional generations were unique, absent from training, and passed sanitization and PoseBusters checks; retaining the best 30% raised the DFT-verified hit rate within 0.1 eV from 25.0% to 52.4%.

    Source: §1 · p. 4

  • Takeaways & Limitations

    EF-TALFM demonstrates variable-size 3D molecular generation from a fixed-dimensional latent, with canonicalization and predicted chemical attributes supporting decoding without equivariant layers or learned dense bond outputs.

    Source: §4 · p. 17

    Results use a large molecular dataset and property targets within the training distribution; smaller datasets and out-of-distribution targets remain untested.

    Source: §4 · p. 17

Abstract

from arXiv · show

Molecular size is coupled to composition, structure, and function, yet most 3D molecular generators require a predefined atom count. We introduce Equivariant-Free Transformer-Autoencoded Latent Flow Matching, a two-stage framework that samples a fixed-dimensional latent vector using flow matching and uses an autoregressive Transformer to determine molecular size, atom types, coordinates, and chemical attributes. Canonical atom ordering and rigid-pose alignment enable Transformers without equivariant layers, while decoded attributes guide bond reconstruction. On PCQM4Mv2, unconditional generation yields 87.9\% unique, novel molecules passing sanitization and PoseBusters checks, exceeding baselines with lower end-to-end training and sampling time and higher end-to-end throughput. Across ten target HOMO-LUMO gaps, internal ranking retains 30\% of screened candidates and increases the density functional theory-verified hit rate within 0.1 eV from 25.0\% to 52.4\%, while largely preserving novelty and diversity. These results demonstrate fixed-dimensional latent generation with autoregressive decoding as a practical approach to molecular design without prespecifying size.

1 Introduction

Molecular generators often require atom count in advance, despite molecular size, composition, and structure being jointly determined. EF-TALFM samples a fixed-dimensional molecular latent and decodes variable-size molecules, with strong unconditional and property-conditioned results on PCQM4Mv2.

1 Introduction

  • EF-TALFM samples one size-independent molecule-level latent vector, then uses an EOM-terminated autoregressive decoder to determine molecular size and generate atom features and coordinates.The latent flow operates in a fixed-dimensional space without an atom count or presized atom-wise state; the decoder creates the atom-wise structure.Source: §1 · pp. 2, 4
  • Canonical atom ordering and proper rigid-pose alignment move permutation and pose handling into preprocessing, enabling standard Transformer and feed-forward backbones without equivariant layers.The alignment preserves chirality by excluding reflections.Source: §1 · pp. 3–4
  • 87.9% of generated molecules were unique, absent from training, and passed sanitization and PoseBusters checks, exceeding UAE-3D and FlowMol.Under the reported budgets on identical hardware, EF-TALFM also had shorter end-to-end training and sampling times and higher sampling throughput.Source: §1 · p. 4
  • 52.4% of retained candidates were within 0.1 eV of their target HOMO–LUMO gap, versus 25.0% before retaining the target-wise best 30%.The selected unique hits were 97.4% absent from corresponding property-matched training subsets, and selection largely preserved within-target structural diversity.Source: §1 · p. 4
  • Instead of predicting dense bond outputs, the decoder predicts chemical attributes that guide deterministic graph reconstruction; restrained refinement improves local geometry while preserving molecular identity.The attributes include formal charge, attached-hydrogen count, hybridization, chirality, aromaticity, conjugation, and ring membership.Source: §1 · p. 4
  • For property-conditioned generation, target-conditioned latent flow and optional autoencoder property supervision support internal-readout ranking that concentrates candidates near requested targets.Selection largely preserves the novelty and structural diversity of retained candidates.Source: §1 · p. 5

2 EF-TALFM: a fixed-dimensional molecule-level latent with autoregressive molecular decoding

EF-TALFM combines deterministic molecular canonicalization, a variable-length molecular autoencoder, and latent flow matching. Its decoder determines molecular size and predicts atom and molecule attributes, while flow matching samples the learned fixed-dimensional latent space.

2 EF-TALFM: a fixed-dimensional molecule-level latent with autoregressive molecular decoding

  • EF-TALFM preprocesses molecules, encodes each as one fixed-dimensional latent, trains flow matching with the autoencoder frozen, then decodes and post-processes generated molecule data.Figure 1 depicts preprocessing in panel (a) and the training and generation pipeline in panel (b).Source: Figure 1 · p. 5

2.1 Canonical molecular representation

  • Canonical atom ordering and rigid alignment provide reproducible sequences and poses, shifting permutation and rigid-motion handling to preprocessing without giving the network formal equivariance guarantees.Proper rotations are used, while reflections are excluded to preserve chirality.Source: §2.1 · pp. 6–7

2.2 Molecule autoencoding

  • EF-TAVAE maps canonicalized molecules to fixed-dimensional latents and reconstructs them with a causal Transformer that predicts molecule features and generates atom tokens until [EOS].Each atom token includes coordinates and categorical fields for atom type and chemical attributes; the representation also includes molecule-level categorical features.Source: §2.2 · pp. 7–8
  • Optional property supervision uses latent-only and molecule-conditioned readouts; re-encoded candidates are ranked by molecule-conditioned prediction error without evaluating their true property during selection.The property token is prevented from altering the autoregressive factorization.Source: §2.2 · p. 8

2.3 Latent flow matching

  • After the autoencoder is frozen, flow matching learns the latent distribution by transporting Gaussian noise toward data latents, and the terminal latent is decoded autoregressively.The flow model learns marginal latents for unconditional generation and conditional latents for property-conditioned generation.Source: §2.3 · pp. 9–10

3 Empirical studies

The empirical studies test whether EF-TALFM can generate variable-size molecules efficiently and whether internal-readout selection improves property targeting. Results show strong unconditional generation and more accurate property targeting while retaining much of the verified hits’ novelty and diversity.

3.1.1 Experimental setup

  • On PCQM4Mv2, EF-TALFM is compared with UAE-3D and FlowMol using a 90%/10% train/validation split and shared downstream reconstruction and refinement.Each method generates 10,000 molecules; evaluation uses PoseBusters-verified novel yield under the same H100 hardware setting.Source: §3.1.1 · pp. 10–11

3.1.2 Performance comparison against baselines

  • 87.9% PoseBusters-verified novel yield, versus 75.2% for UAE-3D and 69.6% for FlowMol; EF-TALFM also reduces end-to-end costs and raises throughput.The comparison generated 10,000 molecules per method under the reported budgets.Source: §3.1 · p. 10, Figure 2 · p. 11, §3.1.2 · pp. 11–12

3.1.3 Distributional fidelity and postprocessing ablation

  • EF-TALFM reproduces training-set trends in molecular size and global chemical attributes, with reconstruction shifts most evident in ring-related properties.Restrained relaxation leaves all five evaluated property distributions unchanged.Source: Figure 3 · p. 12, §3.1.3 · p. 13
  • Restrained refinement improves PoseBusters pass rates by up to 18.5% without changing unique sanitized or novel unique sanitized yields.The largest gains are in bond-length, bond-angle, and internal steric-clash checks; refinement leaves all five evaluated property distributions unchanged.Source: Figure 3 · p. 12, §3.1.3 · p. 13

3.2 Property-conditioned molecule generation

  • Across ten target HOMO–LUMO gaps, internal-readout selection narrows DFT-computed property distributions and retains most verified hits while largely preserving novelty and structural diversity.The study evaluates DFT-verified targeting, candidate retention, and the novelty and diversity of retained hits.Source: §3.2 · p. 13

3.2.1 Experimental setup and evaluation protocol

  • The property-conditioned protocol generates 10,000 samples at each of ten HOMO–LUMO gap targets spanning 4.1–7.8 eV and ranks screened candidates using the internal readout.Reference gaps are computed by single-point DFT at the B3LYP/6-31G(d) level without additional geometry optimization.Source: §3.2.1 · pp. 13–14

3.2.2 Reference-verified targeting and structural diversity

  • Selected verified hits remain largely novel and diverse: 97.4% of unique hits are unseen in the corresponding property-matched training subset by exact canonical-SMILES matching.The reference subset comprises training molecules with HOMO–LUMO gaps within 0.1 eV of the target.Source: §3.2.2 · pp. 14–15, Table 1 · p. 15
  • 25.0% to 52.4%: selecting the target-wise best 30% raises the DFT-verified hit rate within 0.1 eV, while largely preserving novelty and diversity.Selection also retains most available hits and improves hit rates across training-density bins.Source: §3.2 · p. 13, §3.2.2 · pp. 14–15, Figure 4 · p. 16

4 Discussion

EF-TALFM generates variable-size molecules from a fixed-dimensional latent and reports stronger generation and property-selection results on PCQM4Mv2. Its practical scope is qualified by heuristic reconstruction, limited formal symmetry guarantees, and evaluation on a large dataset with in-distribution property targets.

4 Discussion

  • EF-TALFM samples a fixed-dimensional molecule-level latent vector, leaving molecular size and atom-wise structure to autoregressive decoding.Canonicalization supports standard Transformers without equivariant layers, and predicted chemical attributes guide bond reconstruction without learned dense bond outputs.Source: §4 · p. 17
  • On PCQM4Mv2, EF-TALFM reports higher PoseBusters-verified novel yield and shorter end-to-end training and sampling times than evaluated baselines.Internal-readout selection doubles the fraction of structurally screened molecules within target property tolerances while largely preserving novelty and structural diversity.Source: §4 · p. 17
  • Canonicalization and post-decoding bond assignment simplify the architecture but do not provide formal equivariance guarantees, and bond assignment remains heuristic.The discussion identifies learned bond refinement and lightweight equivariant components as possible ways to improve robustness.Source: §4 · p. 17
  • The reported results use a large molecular dataset and property targets within the training distribution.Smaller datasets and targets beyond the training distribution remain to be studied.Source: §4 · p. 17

A Training and sampling algorithms of EF-TALFM

This section describes EF-TALFM’s staged training and sampling pipeline, including canonical-frame decoding, graph reconstruction using generated chemical attributes, and geometry refinement. It also covers the setup and baseline choices for unconditional-generation evaluation and reports improved UAE-3D exact SMILES reconstruction with generated bond types.

A Training and sampling algorithms of EF-TALFM

  • Training proceeds in two stages: a variational autoencoder is trained first, then its parameters are fixed while a latent flow is trained to estimate a vector field.During sampling, the flow starts from a standard-normal latent and the decoder emits atoms autoregressively until an EOS atom-type label is generated.Source: A Training and sampling algorithms of… · pp. 23–24

B Chemically informative atom-level and molecule-level state variables

  • Atom-wise chemical features encode structural context without explicit bonds, while molecule-wide descriptors add further chemical context for generation.RDKit features are encoded as categorical integer labels using the OGB convention [36]; listed molecule-level descriptors include total charge, spin multiplicity, ring count, and fused-ring status.Source: B Chemically informative atom-level… · pp. 24–25

C Canonical atom position alignment

  • Canonical pose alignment lets a non-equivariant model use consistent coordinates for a fixed atom ordering, making SE(3)-related inputs align identically up to numerical precision.For graphs with nontrivial automorphisms, the ordering is deterministic for a fixed input but is not claimed to be a mathematically unique labeling.Source: C Canonical atom position alignment · pp. 25–26

D Learning and sampling from posterior mean

  • The flow targets posterior means whitened by a fitted invertible affine map, while posterior scales and the KL penalty regularize the autoencoder latent geometry.The flow learns molecule-level latent codes; VAE regularization encourages a smoother latent space and decoder stability under local perturbations.Source: D Learning and sampling from posterior… · p. 26

E Latent regularization

  • A variance floor discourages inactive latent dimensions, while a covariance penalty discourages redundant dimensions in minibatch posterior means.The supplied passage describes these as the effects of the two latent-regularization terms.Source: E Latent regularization · p. 27

F Practical molecular graph reconstruction and geometry refinement

  • Sampling generates atom types, coordinates, and chemical attributes but no bond labels; a separate fixed-topology, heavy-atom-restrained refinement pipeline converts decoded states into molecular structures.The decoded representation may contain heavy atoms only or all atoms, including explicit hydrogens.Source: F Practical molecular graph… · p. 28, Figure 5 · p. 28

F.1 Molecular graph reconstruction

  • Heavy-atom graph reconstruction infers connectivity from distances and chemistry-aware constraints, assigns valence-aware bond orders, and corrects overlong multiple bonds without deleting edges.If reconstruction yields multiple components, the largest heavy-atom component is retained; a multiple bond is downgraded only when the resulting graph passes RDKit sanitization.Source: §F.1 · p. 29
  • All-atom reconstruction uses Open Babel to infer bonds, conditionally repairs hydrogen attachments without reconnecting heavy atoms, and refines geometry while preserving topology.Trailing hydrogens with implausible bond lengths or severe clashes are initialized before relaxation; the all-atom branch returns explicit hydrogens.Source: §F.1 · p. 31

F.2 Restrained geometry refinement

  • Restrained refinement uses a 0.15Å unpenalized displacement radius, a 200kcalmol−1 Å−2 force constant, and at most 200 minimization iterations for local cleanup.These are practical defaults, not formally optimized hyperparameters; refinement is evaluated separately from graph reconstruction.Source: §F.2 · p. 31

G Experimental setup for unconditional generation

G.1 Baseline selection

  • The unconditional-generation comparison selects UAE-3D [9] and FlowMol [5] as complementary strong bond-aware baselines with different generative settings.UAE-3D models variable-length atom-wise latents with latent diffusion, while FlowMol uses equivariant flow matching to generate structures including bond types.Source: §G.1 · p. 32
  • ADiT [8] is treated as an architectural reference, not a main quantitative baseline, because its coordinate-based graph recovery may miss chemically annotated states.The section excludes atom-type-and-coordinate-only methods because connectivity, bond orders, aromaticity, and formal charges can be ambiguous, especially for larger or heavy-atom-rich molecules.Source: §G.1 · p. 32
  • 99.5% versus 76.8%: adding generated bond types raises UAE-3D exact SMILES reconstruction while leaving sanitization nearly unchanged.The values compare UAE-3D without and with generated bond types, respectively.Source: Table 2 · p. 32

G.2 Details of PCQM4Mv2 dataset preprocessing

The protocol uses PCQM4Mv2 heavy-atom structures, compares methods under a common evaluation setting while retaining their native training and sampling procedures, and reports generation quality using PoseBusters-related and geometry checks.

G.2 Details of PCQM4Mv2 dataset preprocessing

  • PCQM4Mv2 preprocessing uses approximately 3.4M DFT-optimized structures, a 90%/10% train-validation split, heavy-atom scaffolds, and multiplicities up to four.The multiplicity filter retains singlet through quartet states and excludes the extremely sparse high-spin tail.Source: §G.2 · p. 33

G.3 Training, sampling, and comparison protocol

  • Because methods use different representation spaces, the comparison standardizes evaluation and hardware while reporting native optimizer-step budgets, parameter counts, update units, and wall-clock time.UAE-3D and EF-TALFM use two-stage training; FlowMol trains a direct molecular graph flow model.Source: §G.3 · p. 33
  • Training uses single-node H100 GPUs, with EF-TALFM's fixed molecule-level latent dimension set to 32 and UAE-3D's per-atom latent dimension set to 4.FlowMol is trained for 100K steps; UAE-3D and EF-TALFM each train their autoencoder for 400K steps and latent generator for 1M steps.Source: §G.3 · p. 33
  • Sampling benchmarks generate 10,000 molecules per method with batch size 5,000, using each method's specified sampler and shared downstream reconstruction and refinement stages.EF-TALFM uses an adaptive Dormand–Prince solver, while UAE-3D uses 100 DDPM steps and FlowMol uses 100 Euler steps.Source: §G.3 · p. 33

G.4 Evaluation metrics

  • The primary PoseBusters-verified novel yield counts unique training-novel molecules that pass sanitization and six checks, divided by all generated samples.For fragmented outputs, SMILES and PoseBusters metrics use the largest connected component; missing internal-energy evaluations count as failures.Source: §G.4 · p. 34
  • Connectivity, sanitization, uniqueness, and novelty are reported as percentages of generated outputs meeting their respective single-fragment, strict-sanitization, uniqueness, and training-novelty criteria.These metrics use all generated samples as the denominator.Source: §G.4 · p. 34
  • Geometry checks assess internal energy, steric clashes, bond lengths, bond angles, aromatic-ring flatness, and double-bond flatness against stated thresholds.The criteria include a maximum energy ratio of 7, a 0.8-times distance-geometry lower bound for nonbonded pairs, 20% bond-geometry tolerances, and 0.1 Å planarity tolerances.Source: §G.4 · p. 34

H Detailed unconditional-generation results

EF-TALFM produces a high yield of valid, unique, training-set-novel molecules without predicting bond labels, while retaining strong reconstruction fidelity through a fixed-dimensional latent.

H Detailed unconditional-generation results

  • EF-TALFM outperforms UAE-3D on every reported molecule-level metric and nearly matches FlowMol in sanitization and unique sanitized yield, with moderately lower connectivity.Sanitization is 98.25% versus 98.72%, unique sanitized yield is 98.21% versus 98.27%, and connectivity is 92.98% versus 98.72% for EF-TALFM and FlowMol, respectively.Source: H Detailed unconditional-generation… · p. 34
  • 94.33% of EF-TALFM molecules are sanitized, unique, and training-set-novel, versus 76.03% for UAE-3D and 69.83% for FlowMol.This end-to-end metric counts candidates that are chemically valid, nonduplicate, and absent from training.Source: Table 3 · p. 35
  • EF-TALFM passes all six PoseBusters checks individually at rates of 97.22–99.93%, with 93.13% passing all checks jointly.Rates are evaluated among sanitized, unique, training-set-novel molecules.Source: Table 3 · p. 35, H Detailed unconditional-generation… · p. 35
  • EF-TAVAE compresses each molecule into one fixed-dimensional vector yet retains 99.89% joint atom-feature recovery, 100.00% joint molecule-feature recovery, and 0.041Å coordinate MAE.The results support high-validity, high-novelty generation from the chemical and geometric information retained in that latent.Source: H Detailed unconditional-generation… · pp. 35–36, Table 4 · p. 36

I Additional performance comparison for unconditional generation

Additional comparisons report training and sampling efficiency, checkpoint behavior, and the measurement conventions used for comparing training exposure.

I Additional performance comparison for unconditional generation

  • Longer UAE-3D training yields only modest validity gains, while longer FlowMol training slightly improves validity but reduces novelty.The comparison uses additional checkpoints.Source: I Additional performance comparison… · p. 36, Table 5 · p. 37
  • 882 min of total training gives EF-TALFM a 1.86× time reduction versus UAE-3D's 1M-step budget and a 3.20× reduction versus its 2M-step checkpoint.EF-TALFM also completes training in 882 min versus 40 h for FlowMol's 100K-step checkpoint.Source: I Additional performance comparison… · p. 38
  • EF-TALFM has the shortest reported training time despite not using the smallest update or exposure budget, and achieves higher measured molecule throughput than UAE-3D under equal optimizer updates.The methods' exposure units are not directly interchangeable: FlowMol's edge-level figure is a data-volume proxy, not total compute.Source: I Additional performance comparison… · pp. 36–38, Table 6 · p. 37
  • 4 sec to sample 10,000 molecules makes EF-TALFM 7.50× faster than UAE-3D at 100 denoising steps, 37.50× faster at 500 steps, and 35.25× faster than FlowMol.These are generator-only sampling times; end-to-end sampling also includes reconstruction and pose relaxation.Source: Figure 6 · p. 38, I Additional performance comparison… · p. 38

J Detailed postprocessing ablation for unconditional generation

The postprocessing ablation distinguishes geometry improvements from largely unchanged molecular identity and population-level chemical statistics; generated examples illustrate structural variety.

J Detailed postprocessing ablation for unconditional generation

  • Overlong multiple-bond correction provides only modest geometric-sanity gains; restrained refinement is the dominant geometric improvement while preserving molecular identity and population-level chemical statistics.The ablation compares raw reconstructed, pre-relaxed, and relaxed molecules.Source: J Detailed postprocessing ablation for… · pp. 38–39, Table 7 · p. 39
  • 29.11 percentage points: restrained geometry refinement increases the joint PoseBusters pass rate, with the largest gains in bond lengths, angles, and steric clashes.Internal energy improves by 5.83 points, while molecular validity, uniqueness, and novelty remain essentially unchanged across stages.Source: J Detailed postprocessing ablation for… · pp. 38–39, Table 7 · p. 39

K Example unconditional generated molecules from EF-TALFM

L Experimental setup for property-conditioned generation

The section documents the data preparation, reference calculations, screening, and projection procedures used to evaluate property-conditioned generation.

L.1 Dataset preprocessing and conditioning targets

  • Training uses all-atom PCQM4Mv2 molecules so explicit hydrogen atoms and coordinates are available for DFT inputs; radical-containing graphs are excluded to maintain a consistent electronic-state domain.The resulting partition has 3,000,981 training molecules and 115,302 validation molecules.Source: §L.1 · pp. 40–41
  • Ten HOMO–LUMO gap targets are evenly spaced from the training-set 10th to 90th percentiles, rounded to 0.1 eV, with 10,000 samples generated per target.The targets range from 4.1 to 7.8 eV.Source: §L.1 · p. 41

L.2 Reference calculations and candidate populations

  • Generated molecules receive single-point Psi4 DFT reference gaps at B3LYP/6-31G(d), then radical-containing or PoseBusters-failing structures are filtered before target-wise ranking by internal-readout error.The best 30% of screened candidates per target form the retained population; the text also defines DFT-verified hit populations and hit-rate normalization.Source: §L.2 · p. 41

L.3 Training-density analysis in UMAP space

  • Training density is estimated in a shared UMAP projection of property representations, and generated candidates are binned by local-density percentile to compare DFT-error-threshold hit fractions.The mapping is fitted to 100,000 training molecules, while density uses the full training population.Source: §L.3 · pp. 41–42

M Additional evaluation metrics for property-conditioned generation results

The section defines measures for target-conditioned novelty, similarity, and diversity, then uses PCA to show that selected hits occupy multiple regions and hit rates improve across training-density deciles.

M Additional evaluation metrics for property-conditioned generation results

  • Exact-match novelty uses canonical SMILES after removing explicit hydrogens while retaining formal charges and bond orders, with uniqueness computed separately within each target.A molecule generated under two targets contributes once to each target-specific set.Source: M Additional evaluation metrics for… · p. 42
  • ECFP4 Tanimoto similarity measures analogue-level graph similarity independently of 3D pose, using radius-2 fingerprints with bond types included and chirality excluded.The maximum similarity is computed against the target-matched training reference.Source: M Additional evaluation metrics for… · p. 42
  • Scaffold novelty counts distinct nonempty Bemis–Murcko scaffolds target-wise, so acyclic molecules are excluded and scaffolds are not globally deduplicated across targets.The pooled rate describes novel target–scaffold pairs.Source: M Additional evaluation metrics for… · p. 43
  • Internal diversity compares only within-target molecular pairs, with training controls sampled without replacement and size-matched to each target’s unique generated molecules.Target-wise mean similarities are weighted by their within-target pair counts.Source: M Additional evaluation metrics for… · p. 43

N PCA visualization of the penultimate molecule-path property representation

  • PCA shows 0.1 eV hits across multiple projection regions, while internal-readout selection increases hit rates in every training-density decile.The first two principal components provide a linear view complementary to the nonlinear UMAP view.Source: N PCA visualization of the penultimate… · p. 44

O Ablation study on post-processing stages

The ablation separates the roles of refinement, validity screening, and internal-readout selection: refinement improves structural reliability, while selection produces the largest targeting gain at a coverage trade-off.

O Ablation study on post-processing stages

  • Refinement primarily reduces large-error outliers and improves structural reliability rather than typical target fidelity; validity screening then modestly increases hit rates while retaining nearly all accurate candidates.The stages were ablated on the same initial population of 100,000 generated structures.Source: O Ablation study on post-processing… · pp. 45–46
  • Internal-readout error correlates with true error across 77,724 generated molecules (Spearman ρ = 0.754; Pearson r = 0.837), supporting low-cost candidate ranking across conditioning targets.The per-target correlations are also strong, with Spearman ρ from 0.728 to 0.770 and Pearson r from 0.798 to 0.842.Source: Figure 9 · p. 45, O Ablation study on post-processing… · p. 46, Figure 10 · p. 46
  • Readout-based selection yields the largest 0.1 eV hit-rate gain, with stricter retention trading candidate coverage for higher DFT-verified hit rates.The retention sweep uses screened relaxed molecules ranked within each target; Figure 9 shows the 30% retention setting.Source: Figure 9 · p. 45, O Ablation study on post-processing… · p. 46, Table 9 · p. 47

P Hyperparameters for UAE-3D training on PCQM4Mv2 for unconditional generation

  • The UAE-3D appendix lists hyperparameter setups for unconditional VAE and latent-diffusion training on PCQM4Mv2.The supplied table captions identify the setups but do not report their parameter values.Source: Table 11 · p. 49, Table 12 · p. 50

Q Hyperparameters for FlowMol training on PCQM4Mv2 for unconditional generation

  • The appendix lists a separate hyperparameter setup for unconditional FlowMol training on PCQM4Mv2.The supplied table caption does not report parameter values.Source: Table 13 · p. 51

R Hyperparameters for EF-TALFM training on PCQM4Mv2 for unconditional generation

The unconditional EF-TALFM appendix reports EF-TAVAE and latent-flow training configurations, including loss weights and regularization settings.

R Hyperparameters for EF-TALFM training on PCQM4Mv2 for unconditional generation

  • The reported unconditional EF-TAVAE configuration clamps posterior scales to [0.01,1.5], uses 0.02 free nats per latent dimension, and masks 10% of teacher-forced atom-type inputs.The coordinate, categorical-sequence, charge, and spin-multiplicity loss weights are 10, 1, 1, and 1.Source: Table 14 · p. 52, R Hyperparameters for EF-TALFM… · p. 53
  • The reported EF-TAVAE configuration uses βKL = 10−3, λvar = 5×10−3, and λcov = 5×10−4, with posterior scales clamped to [0.01,1.5] and 0.02 free nats per latent dimension.Source: R Hyperparameters for EF-TALFM… · p. 53
  • A separate table reports the hyperparameter setup for unconditional latent flow matching training in EF-TALFM.The supplied caption identifies the setup but gives no parameter values.Source: Table 15 · p. 53

S Hyperparameters for EF-TALFM training on PCQM4Mv2 for property-conditioned generation

The property-conditioned appendix reports EF-TAVAE settings for molecule-condition and latent-only routes, alongside a separate conditional latent-flow configuration table.

S Hyperparameters for EF-TALFM training on PCQM4Mv2 for property-conditioned generation

  • With property supervision enabled, the EF-TAVAE activates the molecule-condition route 60% of the time and the latent-only route 40% of the time.Coordinate, categorical-sequence, charge, spin-multiplicity, and context loss weights are each 10, 1, 1, 1, and 1; route weights are 0.6 and 0.4.Source: Table 16 · p. 54, S Hyperparameters for EF-TALFM… · p. 55
  • A separate table reports the hyperparameter setup for property-conditioned latent flow matching training in EF-TALFM.The supplied caption identifies the setup but gives no parameter values.Source: Table 17 · p. 55
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