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ASPIRE: Saddle-Point Discovery through Set Prediction and Physical Refinement

Yucheng Zhao, Quanyou Zhang, Shaoxiang Qin, Haixuan Xu, Xiongye Xiao

arXiv:2610.10969v1cs.LG

TL;DR

Event-driven simulation needs saddle discovery to identify rearrangement mechanisms and barriers, but one metastable state can have multiple possible events and repeated searches are costly. ASPIRE predicts a set of saddle candidates from one initial configuration and refines them with Dimer searches on the original potential. On the four-vacancy bcc Fe benchmark, it reports higher reference-event coverage than conventional Dimer with roughly half the recorded search-force calls.

  • Problem

    Saddle discovery must identify multiple possible rearrangements from one metastable state even though their number and product configurations are unknown, and renewed searches along trajectories are costly.

    Source: §1 · pp. 1–2

  • Method

    ASPIRE predicts a set of saddle candidates from one initial configuration and physically refines them using Dimer searches on the original potential.

    Source: §1 · pp. 2–3

  • Results

    77.20% reference-event coverage versus Dimer’s 75.73%, with roughly half the recorded search-force calls under the respective protocols.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The reported results support learning approximate saddle geometry as a means of accelerating physical event discovery.

    Source: §6 · p. 9

    Reference sets are finite and non-exhaustive, evaluation is restricted to four-vacancy bcc Fe under a fixed potential, and coverage and RWC do not establish full KMC trajectory accuracy.

    Source: §6 · p. 9

Abstract

from arXiv · show

Predicting thermally activated diffusion and defect evolution with event-driven models requires identifying atomic rearrangement mechanisms and their activation barriers. Discovering the associated saddle points is a major computational bottleneck: multiple rearrangements may originate from one metastable state, while costly local searches can fail or repeatedly converge to the same saddle. To address this challenge, we introduce ASPIRE (Atomistic Saddle-Point Inference with Refinement for Events), a framework that predicts a set of saddle candidates from a single initial atomic environment and refines them through Dimer searches on the original interatomic potential. The framework's equivariant set predictor, Ev-Quiformer, integrates (i) geometry-conditioned scalar-vector event slots for generating multiple saddle-point proposals and (ii) a decoder that maps each slot to a full atomic displacement field by combining atom, slot, and anchor-relative vectors with invariant coefficients. We also contribute two datasets: (i) BCCFE4VACAV-4000, comprising 4,000 four-vacancy body-centered cubic iron configurations and 65,450 reference events grouped by initial state for set supervision and post-refinement evaluation; and (ii) BCCFE-1TO4VAC, comprising 5,372 configurations with one to four vacancies each. Theoretically, we establish conditions for proposal equivariance. Experimentally, ASPIRE achieves 77.20% reference-event coverage on this benchmark, compared with 75.73% for a conventional Dimer baseline, while requiring approximately half as many Dimer force evaluations. In a timing evaluation on 50 configurations, ASPIRE reduces wall time per configuration from 478.8 s to 176.3 s under the stated hardware settings.

1 INTRODUCTION

ASPIRE frames saddle discovery as predicting multiple rearrangements from one initial configuration, then physically refining the proposals. The introduction presents a benchmark and reports greater reference-event coverage with roughly half the recorded search-force calls of conventional Dimer.

1 INTRODUCTION

  • The motivation is to resolve atomic rearrangements underlying defect migration, recombination, and interactions in irradiated iron-based materials.The introduction gives interstitial-loop interactions and vacancy diffusion during dislocation climb as examples.Source: §1 · p. 1
  • Saddle discovery must identify multiple possible events from one state, although their number and product configurations are unknown in advance.New states along a trajectory may require renewed searches, making event discovery computationally costly.Source: §1 · pp. 1–2
  • Endpoint-conditioned methods do not directly address on-the-fly discovery when product configurations are unknown.React-OT and OA-ReactDiff use reactant and product configurations as inputs.Source: §1 · p. 2
  • ASPIRE combines set prediction from a single initial configuration without supplied products with physical refinement on the original potential.Ev-Quiformer uses geometry-conditioned scalar/vector event slots and a three-source displacement decoder; the paper establishes conditions for proposal equivariance.Source: §1 · pp. 2–3, Figure 1 · p. 2
  • The bcc Fe benchmark groups 65,450 reference events by 4,000 four-vacancy configurations for set supervision and post-refinement evaluation.This grouping supports evaluation of distinct-event recovery after refinement.Source: §1 · pp. 2–3
  • 77.20% reference-event coverage versus Dimer’s 75.73%, with roughly half the recorded search-force calls under the respective protocols.Evaluated on 200 four-vacancy test configurations.Source: §1 · p. 3

2 RELATED WORK

Related work spans event-driven simulation and saddle search, equivariant atomistic representations, and attention-based set prediction. ASPIRE combines these directions to predict unordered saddle candidates from one initial configuration.

2 RELATED WORK

  • On-the-fly KMC discovers events through saddle searches, while KDB and k-ART also use stored mechanisms or event catalogues for reuse.The Dimer method searches for first-order saddles without a supplied product configuration (Henkelman & J´onsson, 1999).Source: §2 · p. 3
  • Equivariant models encode geometric transformation rules, while NequIP and MACE learn interatomic energies and forces.The cited examples include Tensor Field Networks, SE(3)-Transformers, EGNN, and Equiformer (Thomas et al., 2018; Fuchs et al., 2020; Satorras et al., 2021; Liao & Smidt, 2022).Source: §2 · p. 3
  • Set-prediction methods use learned queries or slots, attention, and matching; Ev-Quiformer adapts such designs to predict an unordered set of saddle configurations.The cited designs include Set Transformer (Lee et al., 2019), DETR (Carion et al., 2020), and Slot Attention (Locatello et al., 2020).Source: §2 · p. 3

3 PROBLEM FORMULATION

The problem is to predict and validate multiple saddle candidates from a metastable configuration without product states. The target is recovery of distinct events at low search cost.

3 PROBLEM FORMULATION

  • The input is a metastable local minimum of a fixed interatomic potential, and the number, identities, and outcomes of accessible events are unknown a priori.The formulation calls this open saddle configuration set prediction.Source: §3 · p. 3
  • A retained event requires a first-order saddle whose unstable-mode perturbations relax to the initial state and a distinct product minimum.The search-and-validation map returns a saddle only when the target saddle and connectivity conditions hold; Dimer-based search and validation instantiate the framework.Source: §3 · p. 4
  • The objective is to recover as many distinct events as possible at low search cost without using product configurations as input.The resulting event set is determined by the initial configuration and validated saddle configurations.Source: §3 · p. 4

4 METHODS

ASPIRE predicts multiple approximate saddle geometries from one metastable configuration and uses them to initialize Dimer refinement on the original potential. Ev-Quiformer encodes the environment, generates geometry-conditioned event slots, and decodes them into displacement proposals for matching and optional confidence-based selection.

4 METHODS

  • ASPIRE predicts approximate saddle geometries as alternative rearrangements, then initializes Dimer refinement on the original interatomic potential.Force-based refinement seeks nearby saddle points.Source: §4 · p. 4

4.1 EQUIVARIANT ATTENTION ENVIRONMENT ENCODING

  • Six Equiformer blocks encode each atom with an invariant scalar feature and an equivariant vector feature from a directed radius graph.The graph connects atoms within a maximum radius, subject to a per-atom neighbor limit.Source: §4.1 · p. 4

4.2 GEOMETRY-CONDITIONED EVENT SLOTS

  • Geometry-conditioned slots pair learned scalar fingerprints with geometry-pooled equivariant vectors, then attend over atoms to update slot states.The stage schedule includes initialization, vector–scalar updates, scalar-only updates, and final readout.Source: §4.2 · pp. 4–5, Table 1 · p. 5

4.3 DISPLACEMENT DECODING FROM THREE VECTOR SOURCES

  • Each slot’s atomic displacement combines atom-, slot-, and anchor-derived vectors using invariant learned coefficients, with gated participation and residual motion.The gate permits localized motion without hard sparsity, while the residual accommodates surrounding relaxations.Source: §4.3 · pp. 5–6

4.4 SET MATCHING AND TRAINING OBJECTIVE

  • Hungarian matching assigns targets to distinct slots, then repeats against the full catalogue for up to four rounds, allowing targets to supervise multiple slots across rounds.Each slot is used at most once.Source: §4.4 · p. 6

4.5 CONFIDENCE READOUT FOR PROPOSAL SELECTION

  • An optional judge predicts proposal confidence from invariant latent and geometric features without force evaluations; thresholding trades reference-event recovery against physical search cost.Only proposals meeting the selected confidence threshold undergo Dimer refinement.Source: §4.5 · p. 6

5 EXPERIMENTS AND RESULTS

ASPIRE is evaluated for distinct-event recovery and physical search cost on finite reference sets of vacancy configurations. The results compare recovery against Dimer search effort, examine proposal refinement and model ablations, and measure wall time against event-reuse baselines.

5.1 DATASETS

  • Reference sets contain 65,450 events across 4,000 four-vacancy configurations, with 2–43 events per configuration; they are finite and not exhaustive.The datasets use relaxed active volumes and reference events collected through 80 random-start Dimer searches and SEAKMC validation; BCCFE-1TO4VAC contains 5,372 configurations and 83,372 events for future KMC trajectory simulation and analysis.Source: §5.1 · p. 7

5.2 EVALUATION PROTOCOL

  • Evaluation fixes each configuration’s reference set, counts each reference at most once, and caps random-start Dimer searches at 450 translations versus 300 for ASPIRE.Events match when their maximum atom-wise displacement difference is at most 0.05 Å and their barrier difference is at most 0.02 eV; RWC is a kinetic-focused proxy, not full KMC trajectory kinetics, and reported values use 800 K.Source: §5.2 · pp. 7–8

5.3 REFERENCE-EVENT RECOVERY AND FORCE COST

  • ASPIRE achieves higher reference-event coverage and comparable rate-weighted coverage at substantially lower search cost than 80 random-start Dimer searches.The comparison is evaluated on 200 test configurations; 80 searches per active volume were chosen as a reasonable parameter for obtaining a relatively comprehensive set in SEAKMC deployment.Source: Figure 3 · p. 7, §5.3 · p. 8
  • Small moving-union RMS errors are associated with rapid convergence to the nearest reference saddle, while larger errors yield more variable refinement outcomes.The authors suggest geometric error as an offline proxy for model quality.Source: §5.3 · p. 8
  • The full Ev-Quiformer has the highest coverage among tested variants, with 76.13% coverage, 87.34% RWC, and 75,768 mean search force calls per configuration.The ablation comparison disables judge selection; the full model combines a larger proposal set, slot interaction, repeated matching, and uniform all-atom supervision.Source: Table 2 · p. 8, §5.3 · p. 9

5.4 WALL TIME AND COMPARISON WITH EVENT REUSE

  • With comparable RWC, event-reuse baselines are substantially slower than ASPIRE, while faster alternatives achieve much lower RWC under the stated hardware settings.The comparison reports measured recovery–wall-time tradeoffs at the evaluated operating points.Source: §5.4 · p. 9, Figure 5 · p. 9

6 CONCLUSION AND LIMITATION

ASPIRE predicts saddle candidates from one initial atomic configuration and refines them on the original potential. On the four-vacancy bcc Fe benchmark it improves reference-event coverage relative to the reported random Dimer endpoint at lower search cost, while the reference sets and evaluation remain limited in scope.

6 CONCLUSION AND LIMITATION

  • ASPIRE achieves higher reference-event coverage than the 80-start Random Dimer endpoint at approximately half the recorded search-force cost, with lower measured runtime than event-reuse baselines at comparable RWC.The runtime comparison is under the stated hardware settings.Source: §6 · p. 9
  • The reference sets are finite and non-exhaustive, and evaluation is restricted to four-vacancy bcc Fe under a fixed potential; reference-relative metrics do not establish full KMC trajectory accuracy.More complete reference catalogues and trajectory-level evaluation are needed to assess long-time kinetics.Source: §6 · p. 9

A.1 BCC FE, VACANCIES, AND THE FIXED INTERATOMIC POTENTIAL

  • The bcc Fe defect task uses a fixed M07 EAM potential for refinement, while the predictor targets complete displacement fields that capture vacancy-directed motion and surrounding relaxation.Multiple vacancies change local coordination and can produce rearrangements with differing saddle geometries and barriers.Source: §A.1 · p. 15

A.2 PERIODIC IMAGES AND A LOCAL CARTESIAN REPRESENTATION

  • Consistent local Cartesian coordinates preserve atom correspondence and displacement targets across periodic boundaries, supporting the predictor’s geometric operations and reference matching.The local representation places atoms near a defect in a common image; trajectories winding around the cell use path unwrapping.Source: §A.2 · p. 16, §A.6 · p. 20

A.3 ACTIVE VOLUMES, SURROUNDING ATOMS, AND FREE COORDINATES

  • The local active volume has radius 11 Å, and its physical locality model includes interactions across the boundary when specifying refinement forces and barriers.The free-coordinate representation specifies which atoms can move during refinement.Source: §A.3 · p. 17

A.4 DIMER SADDLE SEARCHES AND LEARNED INITIALIZATION

  • Dimer seeks an index-one saddle without a supplied product configuration; ASPIRE initializes it with a proposed center geometry and an orientation derived from predicted displacement.The search combines ascent along an unstable direction with transverse relaxation.Source: §A.4 · pp. 17–18

A.5 TRANSITION RATES AND KINETIC MONTE CARLO

  • The benchmark measures both distinct reference-event recovery and search work; ASPIRE targets the proposal-and-refinement stage that supplies mechanisms for downstream event-based modeling.Coverage summarizes recovered event identities, while RWC measures their reference escape-rate mass.Source: §A.5 · p. 19

B.1 PHYSICAL CONSTRUCTION AND RECORDED EXAMPLES

  • BCCFE-4VACAV-4000 supplies the evaluation data, while BCCFE-1TO4VAC was constructed for future KMC trajectory simulation and analysis.The learning examples pair relaxed 11 Å active volumes with variable-size event sets collected through 80 random-start Dimer searches and SEAKMC validation.Source: §B.1 · p. 20

B.2 FOUR-VACANCY BENCHMARK CONSTRUCTION

The appendix details construction of the four-vacancy benchmark and the complementary one-to-four-vacancy collection, then specifies symmetry-aware configuration deduplication, event matching, and data splitting.

B.2 FOUR-VACANCY BENCHMARK CONSTRUCTION

  • 65,450 connected events, with 2–43 per AV and a mean of 16.36, were collected using one WS-selected AV per parent.Four-vacancy patterns were selected from 21,297 symmetry-distinct candidates; 4,000 parents were sampled across fragmentation classes and diameter bins.Source: §B.2 · p. 21

B.3 ONE-TO-FOUR-VACANCY COLLECTION

  • 5,372 deduplicated AVs from 4,812 parent IDs comprise the one-to-four-vacancy release, with 83,372 events.Native-CN AVs are independently audited with WS detection; approximately one ninth of topology families may contribute a second AV.Source: §B.3 · pp. 21–22

B.4 REFERENCE-EVENT RECORDS AND CORPUS DISTINCTION

  • The four-vacancy corpus uses one WS-selected AV per parent, while the one-to-four-vacancy collection uses native-CN AVs and can retain a second AV.Each corpus is deduplicated under its own construction protocol; reference events are grouped by initial AV with positions, barriers, connectivity, and identifiers.Source: §B.4 · p. 22

B.5 THE 48 CUBIC OPERATIONS USED FOR CONFIGURATION DEDUPLICATION

  • Configuration deduplication uses all 48 cubic point-group operations, formed from coordinate permutations and independent sign choices.The operations comprise 24 proper rotations and 24 improper operations.Source: §B.5 · pp. 22–23

B.6 EQUIVALENCE OF VACANCY PATTERNS AND AN EXPLICIT CANONICAL KEY

  • The canonical vacancy-pattern key removes translation and identifies patterns equivalent under a common cubic operation.It sorts relative vacancy coordinates over group operations and vacancy anchors, yielding an exact equivalence criterion for ideal lattice coordinates.Source: §B.6 · p. 23

B.7 CONFIGURATION DEDUPLICATION, EVENT MATCHING, AND DATA SPLITTING

  • Event recovery matches full atomwise displacements and barriers within a fixed state, and each output can recover a reference identity only once.The four-vacancy split groups targets by initial state into 3,700 training, 100 monitoring, and 200 held-out states.Source: §B.7 · p. 24

C ARCHITECTURE AND TRAINING DETAILS

The architecture encodes atomic environments equivariantly, refines geometry-conditioned event slots, and decodes each slot into a full displacement field trained against reference events.

C.1 ENCODER

  • The encoder builds a radius-6 Å directed graph with up to 128 neighbors per atom and equivariant scalar and vector features.Its six Equiformer blocks use radially conditioned tensor products and attention; the stated equivariance result assumes compatible graph construction and neighbor truncation.Source: §C.1 · pp. 24–25

C.2 SLOT REFINEMENT AND CROSS-ATTENTION

  • Event slots acquire directional information by attention-weighted pooling of encoded atomic vectors, alongside scalar updates and scalar-only refinement stages.The full model enables slot self-attention at initialization; the cross-attention-only ablation disables it.Source: §C.2 · pp. 25–26

C.3 DECODER DETAILS

  • The decoder combines atom-derived, slot-derived, and anchor-relative vectors with invariant coefficients to synthesize each atom’s displacement.The anchor-relative offset provides a geometry-conditioned spatial reference, and a residual outside the activity gate permits small-participation atoms to express surrounding relaxation.Source: §C.3 · p. 26

D.1 DISPLACEMENT FIELDS AND THE UNIFORM OBJECTIVE

  • The predictor outputs multiple full displacement fields, supervised against reference event fields using an all-atom SmoothL1 objective.The main Uniform model uses β = 0.03 Å for both proposal assignment and matched regression, averaging over all atoms, including zero-displacement atoms.Source: §D.1 · p. 27

D.2 THE ACTIVE-AWARE ABLATION

The appendix specifies active-aware supervision and sequential slot assignment, then documents predictor optimization and the frozen-feature proposal judge.

D.2 THE ACTIVE-AWARE ABLATION

  • The active-aware ablation combines reduced-weight all-atom loss with active-atom, maximum-displacement, magnitude, and cosine terms.Its 0.05 Å training activity cutoff differs from the judge’s 0.3 Å moving-atom cutoff and score threshold.Source: §D.2 · p. 27

D.3 SEQUENTIAL ASSIGNMENT AND LOSS AGGREGATION

  • Sequential Hungarian assignment matches slots to reference events over at most four rounds, reusing the full reference set at each round.A slot is used at most once, while each target can receive one slot per round; later-round pairs provide repeated supervision.Source: §D.3 · p. 28

D.4 PREDICTOR OPTIMIZATION AND CHECKPOINT SCOPE

  • Predictor training uses AdamW with an 8 × 10^-5 learning rate and a 120-epoch cosine schedule, selecting the checkpoint by monitoring first-round displacement loss.The split comprises 3,700 training, 100 monitoring, and 200 held-out configurations.Source: §D.4 · p. 28

E.1 FROZEN PREDICTOR AND INFERENCE INPUTS

  • The judge scores the full predicted displacement pool using frozen encoder features and proposals, with reference displacements used only as offline labels.The frozen features include scalar and vector representations and do not require force evaluations for scoring.Source: §E.1 · pp. 28–29

E.2 ATOM FEATURES, POOLING, AND NETWORK WIDTHS

  • The judge’s atom features include proposal displacement amplitudes, which are pooled into a readout used to score each proposal.The amplitude feature is defined as the norm of an atom’s predicted displacement for a proposal.Source: §E.2 · p. 29

E.3 THE TEN GEOMETRIC DESCRIPTORS, IN IMPLEMENTATION ORDER

The appendix defines proposal-pool descriptors and geometric judge labels, then gives the selection procedure and the physical Dimer path used to refine proposals.

E.3 THE TEN GEOMETRIC DESCRIPTORS, IN IMPLEMENTATION ORDER

  • The ten judge descriptors summarize proposal amplitudes, moving-atom counts and clusters, pool-relative redundancy and rank, and directional coherence.Moving-atom clusters connect atoms within 3 Å; the judge computes these distances in the prepared nonperiodic local coordinates.Source: §E.3 · pp. 29–30

E.4 OFFLINE LABELS AND JUDGE TRAINING

  • Offline positive labels require the RMS-nearest reference to meet strict displacement thresholds, and judge training uses training configurations with monitoring ROC AUC for checkpoint selection.The label tests use thresholds of 0.35 Å and 0.5 Å; final event evaluation instead uses all-atom displacement and barrier criteria.Source: §E.4 · p. 30

E.5 DEPLOYMENT AND PROPOSAL IDENTIFIERS

  • Deployment computes descriptors on the complete proposal pool, then refines proposals above the score threshold while retaining their slot and search identifiers.The 50-configuration wall-time comparison uses τ = 10^-5; recovery curves vary the threshold, and ablations disable selection.Source: §E.5 · p. 31

F.1 ENABLED PATH AND COORDINATE CONVENTION

  • The physical refinement uses the native SEAKMC Dimer implementation on the fixed potential, with LocalRelax enabled and several optional search branches disabled.Search coordinates cover active atoms with fixed and buffer surroundings; LocalRefine is disabled in the main Dimer search.Source: §F.1 · p. 31

F.2 RANDOM AXES, PROPOSAL LOADING, AND ORIENTATION RESET

  • For a learned displacement, the bridge removes its mean to define the search axis while preloading the candidate center with the complete field.The orientation reset uses a configured coefficient of 0.2; the conditional analysis assumes an admissible field and configured initialization.Source: §F.2 · pp. 31–32

F.3 CURVATURE ESTIMATION, ROTATION, AND FORCE PROJECTION

These appendices specify the native Dimer search and the KDB and KeyBank retrieval baselines, including candidate refinement, acceptance, and numerical accounting. They also describe a shared embedding used to visualize search trajectories.

F.3 CURVATURE ESTIMATION, ROTATION, AND FORCE PROJECTION; F.4 TWO-STAGE TRANSLATION AND FINE-STEP CONTROLS

  • Reused-force curvature estimates and native rotation updates determine the Dimer search orientation, while two-stage translation uses trial forces and configured step limits.The ordinary trial/maximal defaults are 0.015/0.05 Å and the fine-step defaults are 0.003/0.01 Å.Source: §F.3 · p. 32, §F.4 · p. 33

F.5 SEARCH LOOP, STOPPING, AND CAP HANDLING

  • Searches alternate rotation and translation; convergence requires both force and step flags, while reaching the translation cap terminates the search and invalidates its output.The cap is checked at NTSITR >= NMax4Trans, and scoring also requires a translation count strictly below the cap.Source: §F.5 · pp. 33–34

F.6 NATIVE TWO-SIDED RELAXATION AND CONNECTIVITY

  • Two seeds from the final search segment undergo native relaxation, and the event is connected when the nearer relaxed branch falls within Tol4Connect, subject to the farther-branch validity test.Connectivity uses a maximum coordinate-component tolerance; reference matching instead uses an all-atom Euclidean displacement gate.Source: §F.6 · p. 34

F.7 RETURNED SADDLE GEOMETRY AND THE NUMERICAL MAP

  • The numerical map returns saddle coordinates, barrier, and native status flags only when configured search and acceptance checks pass.The formal connection to reference recovery assumes local success for this numerical map and the final reference matcher.Source: §F.7 · p. 34

F.8 SEARCH-FORCE ACCOUNTING AND RUN CONFIGURATION

  • The search-force ledger counts each Dimer force-routine invocation, including preload work and failed, duplicate, or capped searches, but excludes initialization and SPSRELAX forces.Reproduction also depends on each run’s resolved settings and timer boundaries.Source: §F.8 · p. 34

G.1 STORED MECHANISMS AND THE SCOPE OF THE BASELINE; G.2 MOBILE ATOMS AND THE RETAINED LOCAL TEMPLATE; G.5 GLOBAL CANDIDATE DEDUPLICATION AND DIMER SUBMISSION

  • KDB-494 contains 494 stored event templates represented by 988 directed query entries, while its retained local template includes mobile atoms and their neighbors.The mobile-atom threshold is 0.7 Å; retained candidates initialize Dimer searches capped at 300 translations, followed by common acceptance and reference-matching gates.Source: §G.1 · p. 35, §G.2 · p. 35, §G.5 · p. 36

G.3 GRAPH MATCHING AND GEOMETRIC SCREENING; G.4 ALIGNED STATE TRANSFER AND DISPLACEMENT EXTRACTION; G.5 GLOBAL CANDIDATE DEDUPLICATION AND DIMER SUBMISSION

  • KDB queries use graph mappings and geometric screening, then transfer aligned saddle and product states before globally deduplicating candidates by fit score.The alignment uses an SVD least-squares fit screened by maximum atomwise residual; transferred saddle and product predictions inform deduplication.Source: §G.3 · pp. 35–36, §G.4 · p. 36, §G.5 · p. 36

H.1 ROOTED GRAPHS AND DIRECTED LIBRARY ENTRIES; H.2 EXACT GRAPH MAPPINGS AND NECESSARY DISTANCE CHECKS; H.3 ORTHOGONAL FIT AND DIRECT DISPLACEMENT TRANSFER; H.4 MAPPING LIMIT, SORTING, AND CANDIDATE DEDUPLICATION

  • KeyBank retrieves rooted local-graph proposals by VF2 exact colored-graph matching with matching root labels and species; least-squares-fit maximum residual scores and gates acceptance, then transferred motions seed native Dimer searches.Retained fields initialize Dimer searches capped at 300 translations; recovery uses all-atom 0.05 Å displacement and 0.02 eV barrier criteria.Source: §H.1 · p. 37, §H.2 · p. 37, §H.3 · p. 38

I.1 ILLUSTRATIVE SAMPLE AND ONE SHARED COORDINATE SYSTEM

The appendix constructs a shared low-dimensional visualization of selected search trajectories and landmarks, then records the local-convergence conditions motivating the analysis.

I.1 ILLUSTRATIVE SAMPLE AND ONE SHARED COORDINATE SYSTEM

  • Figure 2(b) embeds 33 successful searches—17 ASPIRE and 16 Random trajectories—using one shared coordinate system and 4,994 frames.The displayed sample visualizes search geometry; recovery and cost experiments use separately specified evaluation populations.Source: §I.1 · p. 38

I.2 CONFIGURATION FEATURES, LANDMARKS, AND RADIAL PREPROCESSING

  • The shared map includes the initial state and 41 supplied landmarks, with saddle distances setting the reference radius for radial compression before dimensionality reduction.The feature metric is Cartesian RMS over corresponding atoms, and energies are retained separately for coloring.Source: §I.2 · p. 39

I.3 SIX CLASSES OF DISTANCE CONSTRAINTS

  • Six edge classes constrain the embedding, combining trajectory-wide, frame-to-reference, temporal, reference-to-reference, and original-space nearest-neighbor relationships.The sampled global-pair block balances the 33 selected successful searches through uniform trajectory selection.Source: §I.3 · p. 39

I.4 BLOCK NORMALIZATION AND WEIGHTED STRESS

  • The two-dimensional variables minimize weighted metric stress, with block weights balancing trajectory, temporal, local, and global geometric constraints.The implementation uses block-specific base weights and normalization to set the relative contributions.Source: §I.4 · p. 40

I.5 INITIALIZATION, OPTIMIZATION, AND REPRODUCIBILITY

  • The displayed map starts from a saved shared raw-metric embedding and is optimized in two compressed-space stages, each allowing 800 iterations.Both stages reached their iteration limits, and the saved inputs and hyperparameters support reproduction.Source: §I.5 · p. 40

I.6 COMMON DISPLAY ALIGNMENT, MARKERS, AND ENERGY COLORS

  • A common orthogonal Procrustes alignment places the display in the saved baseline orientation without fitted rescaling; axes remain in arbitrary units.Lines show all saved frames, while markers identify every fifth translation step and each trajectory’s endpoints.Source: §I.6 · p. 41

J.1 LOCAL CONVERGENCE MOTIVATION

  • The local-convergence proposition assumes a nondegenerate first-order saddle, a Lipschitz Hessian with exactly one negative eigenvalue, and a sufficiently close initial point.Under those conditions, the ideal minimum-mode iterates remain in the neighborhood and satisfy the stated convergence bound.Source: §J.1 · p. 41

J.2 PROOF OF GEOMETRIC CONSISTENCY

The appendix establishes rotation and atom-permutation equivariance for deterministic proposals under specified architectural and representation conditions. It also relates recovered-event coverage to restricted one-step kinetics and its high-temperature limit.

J.2 PROOF OF GEOMETRIC CONSISTENCY

  • Invariant attention and readouts preserve scalar features while vector channels transform covariantly, supporting the slot-wise transformation law used in the proof.The decoder’s invariant scalar coefficients combine covariant vector terms to yield covariant displacement proposals.Source: §J.2 · pp. 41–42
  • Proposal equivariance holds under deterministic inference, compatible graph construction, an equivariant encoder, invariant decoder conditioning, and the stated vector-map and normalization conditions.The result concerns rotations and atom permutations on the prepared Cartesian representation.Source: §J.2 · pp. 41–42

K.1 REFERENCE RATES AND RESTRICTED ONE-STEP MODELS

  • For a fixed state, reference rates define a single-step event distribution and waiting time; restricting to recovered events renormalizes their probabilities while retaining their rates.The recovered subset’s rate-mass coverage determines the corresponding ratio of mean waiting times.Source: §K.1 · pp. 42–43

K.2 AVERAGE RECOVERY PROBABILITY AND THE HIGH-TEMPERATURE LIMIT

  • Averaged recovery probability equals average RWC over the fixed evaluation bank, and under a common prefactor the high-temperature limit is the recovered fraction of reference events.With event-specific prefactors, the limiting normalized weights are proportional to those prefactors.Source: §K.2 · p. 43

K.3 METRIC INTERPRETATION AND CONVENTIONS

RWC summarizes recovered reference escape-rate mass across configurations, separating relative event recovery from absolute waiting-time scale.

K.3 METRIC INTERPRETATION AND CONVENTIONS

  • RWC uses fixed reference rates; with no recovered events it is zero, but the restricted exit distribution and finite mean waiting time are undefined.Aggregating event identities into product-basin labels can decrease total variation.Source: §K.3 · p. 43

L.1 FIXED 50-CONFIGURATION EVALUATION SET AND TOPOLOGY COVERAGE

The appendix specifies a shared 50-configuration evaluation subset and documents the acceptance, matching, metric, force-count, and timing conventions used in the experiments.

L.1 FIXED 50-CONFIGURATION EVALUATION SET AND TOPOLOGY COVERAGE

  • The fixed 50-case subset combines 30 existing cases with 20 selected to cover all five vacancy-connectivity classes, using topology, target-label coverage, and configuration ID.The added cases include all remaining cases in classes 4 and 2 + 2, with priority given to specified rare target-PDR combinations.Source: §L.1 · pp. 43–44, Table 8 · p. 44

L.2 NUMERICAL ACCEPTANCE AND REFERENCE-EVENT MATCHING

  • Completed searches qualify for reference matching only under stated validity, connectivity, translation, displacement, and barrier criteria; repeated matches to one reference count once.Accepted unmatched outputs receive no reference-coverage credit and remain candidates for further mechanism analysis and catalogue expansion.Source: §L.2 · pp. 44–45

L.3 DISTINCT THRESHOLDS SERVE DISTINCT OPERATIONS

  • Training activity, judge labels, retrieval, numerical connectivity, and reference matching use distinct thresholds and definitions.Dataset deduplication uses the 48 bcc operations, while event recovery uses the fixed-initial-state matching predicate.Source: §L.3 · p. 45

L.4 REFERENCE METRICS AND EXPERIMENT-SPECIFIC SETTINGS

  • RWC is evaluated at 800 K with a common prefactor that cancels, and configuration averages weight evaluation states uniformly.The reference catalogue, temperature, and prefactor convention remain fixed across judge-threshold and random-start comparisons.Source: §L.4 · p. 45

L.5 RECORDED SEARCH-FORCE COUNT

  • Mean search-force cost counts Dimer search-force invocations, including failed, capped, duplicate, and accepted-unmatched searches, but excludes initialization and other specified work.ASPIRE and reuse use a 300-translation cap; Random uses up to 80 searches with a 450-translation cap, and each translation may involve multiple force calls.Source: §L.5 · p. 46

L.6 WALL TIME, RETRIEVAL SIZE, AND EXPERIMENT RECORDS

  • Wall-time comparisons use the shared 50-state subset; ASPIRE runs with 16 CPUs and one A100 GPU, while Random and reuse use 16 CPUs.ASPIRE's judge threshold is τ = 10^-5; wall time is reported separately from search-force counts.Source: §L.6 · p. 46
  • The experiments include a 200-configuration recovery test, a 50-configuration ablation and timing subset, and 33 selected successful MDS trajectories; KDB-494 has 494 templates and 988 directed queries.KeyBank-32/128 denotes libraries built from 32 or 128 training configurations, and retained proposals undergo native Dimer refinement.Source: §L.6 · p. 46
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