PaperlayerView exact revision on arXiv

Source-linked AI summary

ReCurveflow: A Flow Matching Framework that Learns Curved Reaction Trajectories to Predict Transition State Geometries

Seungheun Baek, Mogan Gim, Jaewoo Kang

arXiv:2608.20869v1cs.AIcs.LG

TL;DR

Transition-state prediction seeks molecular geometries that clarify reaction mechanisms, but existing flow-matching methods supervise straight paths unlike actual reaction trajectories. ReCurveflow instead learns from curved NEB-derived paths with off-path correction, ranking first on 11 and second on 6 evaluations.

  • Problem

    Transition-state prediction aims to infer the transition-state geometry from known reactant and product geometries.

  • Method

    ReCurveflow supervises flow matching on curved splines through full NEB image bands and adds dual off-path correction during training.

  • Results

    RECURVEFLOW ranked first on 11 evaluations and second on 6, achieving the best Dihedral MAE across all data splits.

  • Takeaways & Limitations

    ReCurveflow generates reaction trajectories with physically plausible energy profiles and provides NEB initializations that recover reference reaction channels more often and at lower cost.

  • Takeaways & Limitations

    ReCurveflow does not rank first on Angle MAE or Steric Clashes, where competing methods lead across all three splits.

Abstract

from arXiv · show

Predicting transition states (TS) in chemical reactions is crucial, as they provide insights into reaction mechanisms. Recent work on TS prediction have focused on flow matching supervised on straight linear paths that do not align with actual reaction trajectories. We propose a novel flow matching-based framework ReCurveflow that learns to predict TS geometries supervised on continuously curved reference paths interpolated from a full NEB-derived band of molecular geometries. We also introduce off-path correction, which grants ReCurveflow with the ability to produce corrective velocity fields when engaged off-path geometry states during inference rollout, leading to better resistance against exposure bias and accuracy in TS prediction. Across three data splits and six evaluation metrics, ReCurveflow achieves the best result on the majority of split-metric combinations against seven baselines. Qualitative analyses further show that ReCurveflow generates reaction trajectories with energy profiles that closely track the reference NEB path, provides initializations that ease the NEB optimization bottleneck, and exhibits the intended corrective behavior in its learned velocity fields. The ReCurveflow codebase is publicly available at https://github.com/dmis-lab/ReCurveflow.

1 Introduction

ReCurveflow reformulates transition-state prediction as flow matching along a curved reference path fitted to full NEB image bands, rather than a straight interpolation. Off-path correction addresses exposure bias, while the framework improves TS accuracy across all three splits and supports physically plausible trajectories and NEB initialization.

  • Motivation: Transition states are first-order saddle points and maximum-energy configurations along minimum-energy paths connecting reactant and product states.Chemical reactions can therefore be viewed as trajectories across a potential energy surface.
  • Motivation: NEB generates interpolated geometry images between reactant and product endpoints, then optimizes them using energy calculations to characterize minimum-energy paths.These image bands provide the trajectory information underlying ReCurveflow’s curved-path supervision.
  • Method: ReCurveflow reformulates flow matching around a continuous curved reference path spline-fit to the full NEB image band, generating transition-state geometries and reaction trajectories.This uses intermediary Transition1X images rather than only endpoint and saddle-point geometries.
  • Method: Dual off-path correction trains the model to pull perturbed states and its own rollout states back toward the reference path, countering exposure bias.For perturbed off-path states, supervision directs the learned vector field toward the reference path.
  • Results: ReCurveflow improves TS accuracy over prior baselines on all three splits, generates physically plausible energy profiles, and provides more effective, lower-cost NEB initializations.Its NEB initializations recover the reference reaction channel more often and at lower cost.

2 Related Works

Recent learning-based methods predict transition-state geometries directly using diffusion, optimal transport, or flow matching, while newer approaches model entire reaction pathways. The latter treat reaction paths as generation targets rather than focusing only on isolated transition states.

  • Direct transition-state prediction: TSDiff predicts transition-state geometries by modeling TS generation as diffusion conditioned on 2D molecular graphs.OA-ReactDiff instead incorporates reactant and product 3D structures through an object-aware equivariant diffusion model.
  • Direct transition-state prediction: React-OT and GoFlow apply optimal transport and flow matching, respectively, to transition-state prediction.
  • Reaction-pathway modeling: MEPIN and MolGen extend transition-state prediction to entire reaction pathways, learning continuous minimum energy paths or jointly generating pathways, TSs, and products.These methods model reaction dynamics beyond isolated transition states, but treat reaction paths as generation targets.

3 Methods

ReCurveflow formulates transition-state prediction as conditional flow matching from reactant and product geometries, using a curved NEB-derived reference path rather than a straight interpolant. It adds off-path correction and self-correction objectives so the learned velocity field can return deviated rollout states toward the reference trajectory.

  • Problem formulation: The model learns a conditional velocity field vθ(x, t | R, P) to generate a reaction trajectory from reactant geometry R and predict the transition state at t = 0.5.The product geometry P conditions the field alongside R, and integrating the ODE produces the ordered geometry sequence.
  • Curved reference path: ReCurveflow supervises velocity prediction on a C2-continuous curved reference path constructed by interpolating NEB-derived geometry images between R, T, and P.The path uses ten ordered geometry images, aligns them with Kabsch preprocessing and normalized distances, coerces the TS to t = 0.5, and fits a natural cubic spline.
  • Off-path correction: The framework trains on perturbed reference states and augments the target velocity with a displacement-based correction that pulls off-path states back toward the reference path.The correction strength is controlled by k, and perturbations are sampled from a mean-free Gaussian tube around the reference path.
  • Off-path correction: Rollout-based off-path self-correction exposes the model to its own intermediate predictions with added noise and applies the same path-anchored correction target.Predicted velocities are successively integrated over a sub-interval with stop-gradients before additional supervision is imposed at a sampled timestep.
  • Velocity-field parameterization: An E(3)-equivariant graph neural network rebuilds a radius graph from the current geometry and predicts per-atom velocities conditioned on atom types, time, and endpoint distances.A learned anchor term a(R − x̂t) + b(P − x̂t) is added using per-node invariant gains.

4 Experiments

ReCurveflow is evaluated across three data splits using structural-accuracy and chemical-validity metrics, ranking first in most split-metric combinations. Ablations and trajectory analyses show that dual off-path correction and curved reference paths improve robustness, energy-profile fidelity, and NEB initialization efficiency.

  • Quantitative results: ReCurveflow ranked first in 11 of 18 split-metric combinations and second in 6, while achieving the best Dihedral MAE across all three splits.It did not rank first on Angle MAE or Steric Clashes; MolGEN led Angle MAE and MEPIN led Steric Clashes across all splits.
  • Off-path correction ablation: Skipping rollout correction degraded performance more than removing perturbation correction, indicating that both off-path components are vital for mitigating exposure bias.The model trained without both components performed worst across all evaluation metrics.
  • Energy-profile analysis: ReCurveflow’s generated energy profiles closely matched reference NEB profiles, with comparable energetic trends and peaks near the transition state.The comparison used UMA energy profiles along Native-split test trajectories and reference Transition1x images.
  • Curved reference paths: Using all geometry images for curved-path interpolation produced more consistent energy peaks and local curvature than the three-image ablation.The ablated ReCurveflow (w/o OS) showed substantially higher variability across test reactions.
  • NEB initialization: ReCurveflow improved NEB optimization efficiency by approximately 1.3× over Linear and IDPP, requiring significantly fewer single-point evaluations to converge.The comparison used equal-length-resampled paths and identical NEB settings; the paired Wilcoxon signed-rank test gave p < 0.001.

5 Conclusion

RECURVEFLOW is introduced as a flow matching framework for transition-state geometry and reaction-path prediction using curved reference paths derived from NEB minimum-energy paths instead of linear interpolants. Its expected generalization is limited by reaction-path resources that mostly cover small organic molecules and few atomic species.

  • RECURVEFLOW predicts transition-state geometries and reaction paths with flow matching.
  • The framework replaces linear interpolants with curved reference paths derived from NEB minimum-energy paths.
  • Existing reaction-path resources mostly cover small organic molecules across a narrow range of atomic species, limiting generalization to larger, more realistic chemical systems.

A Analysis of Exposure Bias under Curved Reference Path Supervision … C Visualization of Energy Profiles

The analysis shows that curved reference paths add curvature-dependent supervision mismatch under rollout time shifts, motivating explicit correction toward the reference path. ReCurveflow uses corrective off-path training alongside specified architecture, optimization, and computational settings.

  • A Analysis of Exposure Bias under Curved Reference Path Supervision: Exposure bias arises because training supervises reference-state distributions, whereas inference recursively follows model-induced rollout states that accumulate velocity errors.This train–inference distribution mismatch causes rollout states to deviate from the reference trajectory.
  • A.1 Curvature-dependent Sensitivity induced by Reference Path Time-Shift: A time-shifted rollout state can remain on a curved reference path while receiving a supervision target whose tangent differs from its locally appropriate tangent.The offset δ measures the discrepancy between actual path progress and assigned flow time.
  • A.1 Curvature-dependent Sensitivity induced by Reference Path Time-Shift: Proposition 1 establishes that this tangent mismatch is curvature-dependent for twice continuously differentiable curved reference paths.The mismatch follows from a first-order Taylor expansion of the tangent field for sufficiently small δ.
  • Then the supervision mismatch induced by a time offset satisfies: Affine linear reference paths have no corresponding small-time-shift tangent mismatch, so curved supervision introduces an additional sensitivity beyond the standard distribution mismatch.This observation motivates correcting rollout states back toward the reference path during training.
  • B.1 Details on RECURVEFLOW’s Hyperparameters: ReCurveflow’s velocity field uses 6 layers and hidden dimension 512, with a 6 Å interaction cutoff and at most 32 neighbors per atom.These settings define the EGNN velocity field and bound interaction-graph memory for larger systems.
  • B.1 Details on RECURVEFLOW’s Hyperparameters: The corrective term uses gain k = 10, while the rollout term has weight λ = 0.5 and off-path states are generated with K = 25 Euler steps.The corrective gain corresponds to a flow-time constant 1/k = 0.1.
  • B.1 Details on RECURVEFLOW’s Hyperparameters: Inference uses 50 Euler steps, with the transition state at flow time t = 0.5 aligned to a discretization-step boundary.The even step count ensures this midpoint alignment.
  • B.2 Details on RECURVEFLOW’s Computational Specs: The model has 11.6M parameters, trains for approximately 8–10 hours on 4 GPUs, and infers one reaction in 15.7 ms using 0.23 GB peak memory.Experiments used NVIDIA RTX 3090 GPUs, with inference measured at batch size 128.

C.1 Ablation Results related to Inclusion of OS Images in Curved Reference Path Construction · D Reaction Path Initialization for NEB

Including On-Trajectory States (OS) in curved reference-path construction consistently improves all four generated-trajectory metrics, while the NEB initialization study uses 287 Transition1x Schreiner et al. [2022] test-split reactions with DFT reference geometries.

  • C.1 Ablation Results related to Inclusion of OS Images in Curved Reference Path Construction: The OS ablation evaluates generated reaction trajectories with four path-level metrics, and lower is better for all metrics.The metrics are Path RMSD, Endpoint RMSD, Barrier Overshoot, and Endpoint Energy Error.
  • C.1 Ablation Results related to Inclusion of OS Images in Curved Reference Path Construction: Including OS reduces Path RMSD from 0.129 to 0.101, approximately 1.3× lower than the model without OS.Path RMSD measures average geometric discrepancy after arc-length alignment.
  • C.1 Ablation Results related to Inclusion of OS Images in Curved Reference Path Construction: Including OS reduces Endpoint RMSD from 0.058 to 0.013, approximately 4.5× lower than the model without OS.Endpoint RMSD evaluates whether the generated trajectory terminates at the correct product geometry.
  • C.1 Ablation Results related to Inclusion of OS Images in Curved Reference Path Construction: Including OS reduces Barrier Overshoot from 9.26 to 2.38 kcal/mol, approximately 3.9× lower than the model without OS.Barrier Overshoot measures excess over the maximum predicted reaction-path barrier.
  • C.1 Ablation Results related to Inclusion of OS Images in Curved Reference Path Construction: Including OS reduces Endpoint Energy Error from 1.90 to 0.055 kcal/mol, yielding over 30× lower error than the model without OS.The full model’s improvements indicate better capture of geometric evolution and underlying reaction energetics.
  • D Reaction Path Initialization for NEB: The NEB initialization study is performed on the Transition1x Schreiner et al. [2022] test split containing 287 reactions.This passage provides the dataset and split used for the initialization experiments.
  • D Reaction Path Initialization for NEB: The NEB initialization study uses DFT (ωB97X/6-31G(d)) reactant, transition-state, and product geometries as references.The reference geometries cover the reactant, TS, and product states.

D.1 Experimental Setup

The experiments compare four initialization strategies and optimize all methods under identical nine-image NEB settings. Computational cost is measured hardware-independently by counting single-point energy/force evaluations during NEB optimization.

  • Initializations: Four initialization strategies are compared: Linear, IDPP, MEPIN, and Ours.MEPIN and Ours trajectories are resampled by arc-length interpolation, while MEPIN uses one sample with seed 0.
  • Common NEB settings: All methods use identical NEB settings with 9 images, including two fixed endpoints and seven interior images.Optimization uses the UMA interatomic potential through ASE and the FIRE optimizer, with a maximum of 500 steps; DFT geometries are evaluation-only.
  • Optimization cost: Optimization cost is measured by the number of single-point energy/force evaluations during NEB optimization.This provides a hardware-independent measure of computational cost.

D.2 Evaluation Protocol · E Analysis on Off-Path Self-Correction Dynamics

The evaluation uses a clean set of 249/287 reactions and prioritizes path validity over raw NEB convergence. It also compares optimization efficiency on a same-destination intersection and analyzes reaction difficulty using barrier height or reactant–product RMSD.

  • D.2 Evaluation Protocol: 249/287 reactions (87%) form the clean evaluation set after relaxing each DFT TS on UMA and retaining structures within 0.1 of the DFT TS.All subsequent analyses use this clean set.
  • D.2 Evaluation Protocol: A recovered DFT reaction requires NEB convergence, a physical single-barrier energy profile, and a permutation-aware TS RMSD below 0.2.The protocol additionally requires a barrier height above 0.05 eV.
  • D.2 Evaluation Protocol: Optimization cost is compared only on the same-destination intersection where all methods recover the DFT reaction.The reported measures are median force evaluations and optimizer steps.
  • D.2 Evaluation Protocol: Paired Wilcoxon signed-rank tests evaluate optimization cost, while McNemar tests evaluate paired success rates.These tests are applied within the efficiency comparison protocol.
  • D.2 Evaluation Protocol: The hardest 30% of reactions are analyzed using either DFT barrier height or reactant–product Kabsch RMSD.Heavy-atom count is excluded because larger molecules can yield artificially lower RMSD.
  • D.2 Evaluation Protocol: Path validity is the primary criterion because raw NEB convergence rates are similar across methods and convergence alone does not ensure the correct reaction channel.This avoids treating optimization convergence as sufficient evidence of reaction recovery.

E.1 Case Studies of Off-Path Self-Correction Dynamics

Case studies show that off-path correction generally keeps ReCurveflow rollouts closer to reference reaction paths and improves recovery toward transition states, while highly curved segments remain challenging. The learned velocity fields exhibit restoring behavior but can retain localized path-repelling regions.

  • Reaction rxn0347: For rxn0347, both models predict the transition state accurately, but the corrected rollout follows the reference path more closely and has smaller path deviation.The TS accuracy is similar, while the trajectory alignment improves throughout integration.
  • Reaction rxn2550: For rxn2550, the baseline loses guidance and deviates, whereas the corrected velocity field supplies stronger restoring force and reaches the TS more accurately.The correction keeps the rollout closer to the reference path during inference.
  • Reaction rxn4520: For rxn4520, off-path correction reduces deviation but still struggles with the sharp, highly curved segment immediately before the TS.The case indicates that paths with large local curvature remain challenging.
  • Reaction rxn5687: For rxn5687, the baseline diverges completely, while the corrected model recovers the trajectory toward the TS despite a residual deviation from a path-repelling region.The localized repelling region persists in the learned velocity field before the TS.

F Kinetics-guided Reaction Design

ReCurveflow supports kinetics-guided reaction design by producing valid transition-state geometries that enable activation-barrier ranking. On 31 reaction groups, it ranked 19 groups and correctly identified the lower-barrier pathway in 18.

  • Evaluation setup: UMA reproduced the DFT barrier ranking for 94% of reference transition states, supporting its use as a surrogate evaluator.Reference transition states were rescored before evaluating predicted geometries.
  • Transition-state validity: ReCurveflow reached the reference DFT transition state in 78% of reactions within 0.5 Å and 100% within 1.0 Å.Validity required convergence to the reference DFT transition state after Sella/UMA saddle optimization.
  • Activation-barrier ranking: ReCurveflow ranked 19 of 31 reaction groups and correctly identified the lower-barrier pathway in 18 groups, achieving 95% accuracy among successful rankings.Each group contained reactions sharing the same product but differing reactants, and both candidate transition states had to be valid for ranking.
  • Activation-barrier ranking: React-OT correctly ranked 13 of 15 feasible reaction groups, corresponding to 87% accuracy.Ranking feasibility required valid transition-state geometries for both candidate reactions.
Loading 2608.20869v1…