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

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

cs.LGOct 7, 2026

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

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

Organizations: University of Tennessee, Knoxville · McGill University

Abstract

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.

Figures & tables

Figure 1: ASPIRE: Atomistic Saddle-Point Inference with Refinement for Events. (a) Overall workflow. Ev-Quiformer generates $M$ saddle-point proposals from a single initial configuration, followed by physical refinement using Dimer searches on the original interatomic potential. Panels (b–d) detail its three neural modules. (b) Equivariant encoding. A six-layer Equiformer extracts invariant scalar and equivariant vector features for each atom. (c) Geometry-conditioned event slots. Scalar self-attention and repeated slot–atom attention update slots initialized from learned scalar seeds and pooled atom vectors. (d) Multi-source displacement decoding. Invariant coefficients combine atom, slot, and anchor-relative vectors into a full atomic displacement field through a gated term and an ungated residual. Adding this field to the initial configuration yields a candidate for subsequent physical refinement.

Stage type	$H$	$\eta$	$\mathrm{N}$	$\gamma$	$F$	$W_{U}$	Updated
	Scalar attn.			Vector attn.	Update
Initialization	1	0 or 1	Id	$0$	$0$	$I$	$s_{m}^{(t+1)}=s_{m}^{(t)}$ , $Q_{m}^{(t+1)}=U_{m}^{(t)}$
Vector–scalar	1	0	Id	learned	MLP	learned	$s_{m}^{(t+1)}$ , $Q_{m}^{(t+1)}$
Scalar-only	8	0	LN	$0$	$W_{o}z$ +FF	–	$s_{m}^{(t+1)}$ , $Q_{m}^{(t+1)}=Q_{m}^{(t)}$
Readout	1	0	Id	$1$	–	–	none; outputs $A^{(T)}_{ma},z^{(T)}_{m},s^{(T)}_{m},Q^{(T)}_{m}$

Table 1: Slot stage types. Settings of Equations ( 4 )–( 8 ). “MLP” is a two-layer perceptron on

[s,z,\|U\|]

; “

W_{o}z

+FF” is Equation ( 8 ); “–” means the map is absent;

\gamma

of the vector–scalar stage is learned from initial value 1. Detailed stage counts and parameterization are given in Appendix C.2 .

Figure 2: Energy background, saddles, and trajectories. (a) Sammon-projected view of the energy function and initial, saddle, and product configurations. (b) MDS-coordinate views of ASPIRE and random-start Dimer searches on one configuration. ASPIRE proposal jumps effectively predict approximate saddles, followed by light Dimer refinement, whereas random Dimer goes long paths; Each trajectory energy point denotes every five translation steps, and color denotes the energy value above initial at the point. (Appendix I ). a.u.: arbitrary units.

Figure 3: Reference-event recovery versus search cost. Coverage (left) and RWC (right) versus the mean recorded Dimer search force calls per configuration, evaluated on 200 test configurations. Curves vary the judge threshold for ASPIRE (up to 96 proposals) or the number of random-start Dimer searches (up to 80).

Figure 4: Proposal geometry and refinement outcomes. Each point represents a proposal and its subsequent Dimer search. Initial error is measured by moving-union RMS or Euclidean distance to the nearest reference saddle, and refinement effort by the number of translation steps. Colors distinguish nearest-reference recovery (fastest or slower), recovery of another reference, valid unmatched outputs, invalid outputs, and searches reaching the cap. The dashed line marks the 300-step cap.

Variant	Slots	Anchor	Self-attn.	O2M	Loss	$\beta$ (Å)	Cov. (%) $\uparrow$	RWC (%) $\uparrow$	$\bar{F}_{\mathrm{search}}$ $\downarrow$
	Architecture			Supervision			Results
Full Ev-Quiformer	96	$\checkmark$	$\checkmark$	4	Uniform	0.03	76.13	87.34	75,768
Cross-attention only	96	$\checkmark$	$\times$	4	Uniform	0.03	75.60	84.29	71,033
Single matching round	96	$\checkmark$	$\checkmark$	1	Uniform	0.03	74.09	87.28	87,688
Active-aware loss	96	$\checkmark$	$\checkmark$	4	Active-aware	0.03	23.29	27.18	138,302
Fewer slots	32	$\checkmark$	$\checkmark$	4	Uniform	0.03	62.10	78.78	26,461

Table 2: Ablation of Ev-Quiformer. Judge selection is disabled for all variants. O2M denotes the number of sequential matching rounds; Uniform denotes all-atom SmoothL1 supervision. The active-aware objective is specified in Appendix D.2 .

\bar{F}_{\mathrm{search}}

is the mean recorded Dimer search force-call count per configuration (Appendix L.5 ). Evaluated on 50 test configurations selected with topology-layered sampling covering all five vacancy-connectivity classes (Appendix L.1 ).

Figure 5: Recovery and wall time compared with event reuse. (a) Reference-event coverage and rate-weighted coverage. After Random Dimer, methods are ordered by increasing RWC. (b) Measured wall time per configuration and per recovered reference event. Random Dimer and the reuse baselines use 16 CPUs; ASPIRE uses 16 CPUs and one A100 GPU. ASPIRE uses judge-based proposal selection with threshold

\tau=10^{-5}

, refining only proposals with

\widehat{p}_{m}\geq\tau

. Evaluated on 50 test configurations with topology-layered sampling

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	AV states	Events	Vacancies	Use
BccFe-4vacAV-4000	4,000	65,450	4	Reported evaluation
BccFe-1to4Vac	5,372	83,372	1–4	Future trajectory studies

Appendix

Table 3: Dataset records. Each state is one retained local AV. The two collections use their respective AV-selection protocols and have complementary roles in evaluation and future trajectory studies.

Setting	Value
Point-defect-reduction cutoff, DCut4PDR	$4.0\,\text{\AA}$
DActive , DBuffer , DFixed	$11,1,11\,\text{\AA}$
Dimer starts, NSearch	80 per searched AV
Buffer search, SearchBuffer	False
Maximum translation iterations, NMax4Trans	900
Trial / maximum step size	$0.015/0.05\,\text{\AA}$

Appendix

Table 4: Reference-catalogue generation settings. Dataset collection uses the following protocol; evaluation settings are specified in Appendix F .

PDR fragmentation signature	Selected parents
$4$	35
$3+1$	583
$2+2$	246
$2+1+1$	1,254
$1+1+1+1$	1,882
Total	4,000

Appendix

Table 5: Four-vacancy parent sampling. Counts are for the recorded PDR fragmentation signatures used in dataset construction.

Vacancies	Released AVs	Events
1	1	12
2	38	492
3	1,121	16,073
4	4,212	66,795
Total	5,372	83,372

Appendix

Table 6: One-to-four-vacancy release. Counts summarize the final AV collection after construction and filtering.

Quantity / native key	Value	Scope
DimerSep	$0.005\,\text{\AA}$	supplied setting/default
Ordinary trial / maximal step	$0.015/0.05\,\text{\AA}$	Random defaults
Fine trial / maximal step	$0.003/0.01\,\text{\AA}$	preload defaults
FConv	$10^{-6}$	native force-magnitude threshold
FMin4Rot / FThres4Rot	$0.001/0.1$	native rotation tests
NMax4Rot	$3$	native rotation counter setting

Appendix

Table 7: Native numerical controls and defaults. Defaults identify the implementation’s parameter families. Each experiment uses its resolved settings and the reported translation cap.

Component-size signature	Existing 30	Added 20	Final 50
$4$	0	2	2
$3+1$	4	4	8
$2+2$	0	5	5
$2+1+1$	11	5	16
$1+1+1+1$	15	4	19
Total	30	20	50

Appendix

Table 8: Vacancy-connectivity coverage of the fixed evaluation subset. The existing 30 cases are augmented by 20 cases to obtain the 50-case set used for both ablation and wall-time comparisons. Counts report the configuration selection in each connectivity class.

Operation	Criterion / threshold	Target-reference use
Active-aware training	Reference displacement $\geq 0.05\,\text{\AA}$ ; largest-reference-atom fallback	Training only
Judge movers	Proposal/reference union uses $\geq 0.3\,\text{\AA}$	Offline labels only
Judge positive label	Nearest-union RMS $<0.35\,\text{\AA}$ and union maximum atom error $<0.5\,\text{\AA}$	Offline labels only
Judge selection	$\widehat{p}_{m}\geq\tau$ after full-pool features	None
KDB / KeyBank retrieval	Maximum aligned atom error $\leq 0.3\,\text{\AA}$	None
Reuse candidate deduplication	Both saddle and product maximum field differences $\leq 0.3\,\text{\AA}$	None

Appendix

Table 9: Thresholds in the implementation. “Reference” denotes the target configuration’s reference catalogue. Event-reuse methods retrieve stored mechanisms from their training libraries.

Experiment	States	Proposal pool and selection	Translation cap
Recovery curves	200	ASPIRE: up to 96 proposals, varying judge threshold; Random: up to 80 starts	300 / 450
Ablations	50	Judge disabled for every variant; slot counts and objectives as in Table 2	300
Wall time	50	ASPIRE: $\widehat{p}_{m}\geq 10^{-5}$ ; reuse library sizes as in Section 5.4	300; Random 450

Appendix

Table 10: Settings of the main-text evaluation populations. The 50-state ablation and wall-time studies use the same held-out subset, but different judge settings. Proposal count, number of submitted searches, and translation cap are distinct quantities.

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