TS-GEN: ML Transition-State Predictor

• TS-GEN is a generative machine-learning model that directly predicts transition-state geometries from reactant and product structures, bypassing iterative optimization methods.
• It uses a conditional flow-matching formulation with an equivariant graph neural network to achieve sub-angstrom RMSD and high chemical accuracy.
• The model employs deterministic ODE integration for rapid inference, making it well-suited for high-throughput reaction network exploration and catalyst screening.

TS-GEN is a generative machine-learning model for direct prediction of chemical reaction transition-state geometries from reactant and product structures. It is formulated as a conditional flow-matching model that maps samples from a simple Gaussian prior to transition-state first-order saddle-point geometries in a single deterministic pass, replacing the iterative optimization procedures characteristic of nudged elastic band, string, dimer, and eigenvector-following workflows. The method conditions explicitly on both reactant and product conformations, uses an equivariant graph neural architecture for 3D atomic systems, and is reported to achieve a root-mean-square deviation of $0.004\ \mathrm{\mathring{A}}$, a mean barrier-height error of $1.019\ \mathrm{kcal/mol}$, and $0.06\ \mathrm{s}$ GPU inference time per prediction, with more than $87\%$ of generated transition states satisfying chemical-accuracy criteria $(<1.58\ \mathrm{kcal/mol})$ (Tuo et al., 14 Jul 2025).

1. Chemical role and computational motivation

Transition states define the critical geometries and energy barriers underlying chemical reactivity. In kinetic and mechanistic analysis, the transition-state geometry determines the activation energy,

$$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$

and barriers govern rate constants through transition-state theory,

$$k \propto \exp\left(-\frac{\Delta G^\ddagger}{RT}\right).$$

Knowledge of the transition-state structure therefore provides direct mechanistic information, including which bonds are breaking or forming and whether the elementary step is concerted or stepwise.

The difficulty is that transition states are first-order saddle points on the potential energy surface rather than minima. Conventional quantum-chemical workflows such as DFT combined with nudged elastic band, string, dimer, or eigenvector-following methods require multiple geometry optimizations and many electronic-structure force evaluations, and they can fail or require substantial human intervention when the reaction is complex, multi-step, involves many atoms, or begins from a poor path guess. The paper positions TS-GEN as a response to this bottleneck: a model intended to replace or dramatically accelerate classical transition-state searches while maintaining or exceeding their accuracy (Tuo et al., 14 Jul 2025).

A central contrast is with prior machine-learning approaches to transition-state prediction. The paper groups these into regression models from reactant-product features to transition-state geometry, learned corrections to NEB or string methods, and score-based or diffusion generative models that still require iterative sampling. The stated limitations are Ångström-scale geometric errors, continued computational cost from iterative refinement or many diffusion steps, and difficulty in conditioning robustly on both reactant and product while preserving rotational and translational equivariance.

2. Conditional flow-matching formulation

TS-GEN is built as a continuous normalizing-flow–type generative model using conditional flow matching. Its core object is a time-dependent vector field that transports a simple prior distribution to the conditional distribution of transition-state geometries. The model evolves coordinates $\boldsymbol{x}_t$ by the deterministic ordinary differential equation

$$\frac{d\boldsymbol{x}_t}{dt} = \boldsymbol{v}^{(\theta)}(t,\boldsymbol{x}_t \mid \boldsymbol{R},\boldsymbol{P}),$$

where $\boldsymbol{R}$ and $1.019\ \mathrm{kcal/mol}$0 denote reactant and product geometries, $1.019\ \mathrm{kcal/mol}$1 is drawn from a Gaussian prior, and $1.019\ \mathrm{kcal/mol}$2 is the predicted transition-state geometry. This defines a deterministic map from latent noise to a saddle-point structure.

The model is explicitly conditional on both endpoints of the reaction. Conceptually, it learns how to deform an initial random configuration into a transition state that is consistent with the supplied reactant-product pair. Training uses a probability path between the base distribution $1.019\ \mathrm{kcal/mol}$3 and the target transition-state distribution $1.019\ \mathrm{kcal/mol}$4, and the neural vector field is fitted to the corresponding target velocity field by a flow-matching regression objective,

$1.019\ \mathrm{kcal/mol}$5

The supplementary discussion relates this formulation to optimal transport. Under that interpretation, the learned flow approximates an OT-like deformation from Gaussian noise to the empirical transition-state distribution for each conditioned reaction. The same supplementary material contrasts ODE-based flow matching with an SDE-based diffusion formulation through a Schrödinger bridge analysis. In that comparison, only an SDE with an increasing-then-decreasing diffusion schedule tending to zero at $1.019\ \mathrm{kcal/mol}$6 reaches ODE-level accuracy, which motivates the ODE-only design: fast, stable, single-pass generation without stochastic sampling noise or long diffusion trajectories (Tuo et al., 14 Jul 2025).

3. Representation, architecture, and training

The representation is graph-based and equivariant. Reactant, product, and transition-state structures are treated as 3D atomic systems with Cartesian coordinates, atom types encoded as one-hot vectors of dimension 5, and edges defined within a cutoff radius $1.019\ \mathrm{kcal/mol}$7. The architecture is an encoder–processor–decoder stack tailored to rotationally and translationally equivariant prediction of coordinate updates or velocities. This equivariance is essential because transition-state geometries are defined only up to global rigid motions.

The supplementary Table S1 specifies the main parameterization. The encoder includes $1.019\ \mathrm{kcal/mol}$8, $1.019\ \mathrm{kcal/mol}$9, $0.06\ \mathrm{s}$0, $0.06\ \mathrm{s}$1, and $0.06\ \mathrm{s}$2. The processor uses 5 transformer layers with 4 attention heads, head size 64, and hidden dimension 64. The decoder uses $0.06\ \mathrm{s}$3. In functional terms, this allows the model to encode the two endpoint conformations and propagate information through the molecular graph before emitting the velocity field required by flow matching.

Training proceeds on reaction triplets $0.06\ \mathrm{s}$4. For each triplet, a time $0.06\ \mathrm{s}$5 is sampled, an interpolated state $0.06\ \mathrm{s}$6 is constructed between a Gaussian prior sample and the reference transition state, and the target velocity $0.06\ \mathrm{s}$7 implied by the chosen probability path is computed. The training objective is

$0.06\ \mathrm{s}$8

The prior is an isotropic Gaussian,

$0.06\ \mathrm{s}$9

and the ODE solution operator $87\%$0 yields the final predicted transition-state coordinates. The supplementary text references a “Transition1x test set,” which indicates the benchmark context in which the model was evaluated (Tuo et al., 14 Jul 2025).

4. Inference, determinism, and robustness

At inference time, TS-GEN takes as input a reactant geometry $87\%$1 and a product geometry $87\%$2, draws a prior sample $87\%$3, integrates the conditional ODE from $87\%$4 to $87\%$5, and outputs

$87\%$6

The paper describes this as single-pass, deterministic generation. Because the vector field evaluation is a forward pass through the equivariant network, the integration is comparatively cheap. The same framework also permits fixing $87\%$7 or drawing multiple prior samples to generate an ensemble of candidate transition states.

Optional post-processing steps remain chemically standard: single-point DFT calculations for energies and barrier heights, brief local saddle-point refinement if desired, and frequency analysis to verify a single imaginary mode together with connectivity to the intended reactant and product minima. The model is therefore compatible both with direct prediction and with hybrid workflows in which a learned transition state initializes a conventional quantum-chemical confirmation step.

The supplementary ensemble-generation experiment reports 100 independently generated transition states for the same reactant-product pair. Their RMSD and $87\%$8 distributions are described as tight, indicating that different prior noise samples yield very similar structures and barrier predictions. This suggests that the conditional flow is strongly constrained by the reaction endpoints and tends to lock onto a consistent saddle geometry rather than producing a broad family of unrelated structures (Tuo et al., 14 Jul 2025).

5. Quantitative performance and error structure

The reported headline results are summarized below.

Quantity	TS-GEN	Prior state-of-the-art
TS geometry RMSD	$87\%$9	$(<1.58\ \mathrm{kcal/mol})$0
Mean barrier-height error	$(<1.58\ \mathrm{kcal/mol})$1	$(<1.58\ \mathrm{kcal/mol})$2
GPU inference time	$(<1.58\ \mathrm{kcal/mol})$3	—

In addition to these averages, more than $(<1.58\ \mathrm{kcal/mol})$4 of generated transition states satisfy the chemical-accuracy criterion of barrier error below $(<1.58\ \mathrm{kcal/mol})$5. The abstract also states that TS-GEN exhibits strong transferability to out-of-distribution reactions from a larger database (Tuo et al., 14 Jul 2025).

The paper highlights several outliers from the Transition1x test set. Three examples have RMSDs of $(<1.58\ \mathrm{kcal/mol})$6, $(<1.58\ \mathrm{kcal/mol})$7, and $(<1.58\ \mathrm{kcal/mol})$8, with barrier-height errors of $(<1.58\ \mathrm{kcal/mol})$9, $$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$0, and $$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$1. The accompanying inspection attributes these failures to unphysical inputs in the dataset rather than to ordinary prediction error. The cited anomalies include unrealistic $$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$2 bond angles of $$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$3 and $$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$4 where $$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$5–$$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$6 would be expected, reactant conformations lacking expected hydrogen bonds, and violations of the octet rule in carbon coordination. The paper therefore argues that major outliers are associated with chemically implausible training or test structures.

This error analysis is important for interpretation. TS-GEN’s geometric errors are not presented as uniformly small for every case; rather, the model attains extremely small mean RMSD while still admitting rare large barrier errors, and those large-error cases correlate with defective inputs. A plausible implication is that data curation remains a first-order determinant of performance even when the generative model itself is highly accurate.

6. Methodological position and practical implications

Relative to classical transition-state search methods, TS-GEN learns a direct map from conditioned structures to a near-saddle geometry. NEB, string, dimer, and related methods operate directly on the potential energy surface and provide rigorous transition-state confirmation, but they require many iterations, repeated force evaluations, and careful initialization. TS-GEN, by contrast, needs only a forward ODE integration through the learned vector field. The supplementary text explicitly presents two practical uses: as a stand-alone transition-state predictor and as a smart initializer for subsequent classical refinement. Given the reported RMSD scale of order $$\Delta E_{\mathrm{TS}} = E_{\mathrm{TS}} - E_{\mathrm{reactant}},$$7, only minimal quantum refinement would ordinarily be needed.

Relative to earlier ML-based transition-state predictors, the claimed advantages are deterministic ODE-based generation, elimination of iterative stochastic sampling, explicit conditioning on both reactant and product, and SE(3)-equivariant treatment of 3D structure. The paper also notes corresponding constraints: TS-GEN requires labeled transition-state data for training, and out-of-distribution chemistry that is very different from the training distribution may require retraining or fine-tuning. Reactions involving fundamentally different physics, such as transition-metal catalysis, strong solvent effects, or surface chemistry, are identified in the discussion as likely extension targets rather than solved cases.

The practical significance follows from the reported combination of sub-angstrom precision, sub-second inference, and robustness. The paper places TS-GEN in high-throughput reaction-network exploration, catalyst screening, and mechanistic hypothesis testing, where rapid generation of candidate transition states can be followed by targeted quantum-chemical validation. More broadly, the method exemplifies a shift from iterative path optimization to learned conditional transport in chemical structure generation: the transition state is not searched for image-by-image along a reaction path, but generated by transporting a simple prior to a saddle-point geometry conditioned on the reaction endpoints (Tuo et al., 14 Jul 2025).