A general optimization framework for mapping local transition-state networks

TL;DR

The paper introduces MOTO, a general three-layer optimization framework to map local transition-state networks around a minimum by jointly exploring diverse initial guesses, locating index-1 saddles via a bilayer minimum-mode approach, and certifying connectivity with a deterministic bidirectional descent. The method leverages Hessian-vector products to estimate the lowest-curvature subspace with Krylov solvers, dramatically reducing memory and compute relative to explicit-Hessian approaches while maintaining accuracy, and combines this with a multi-objective NSGA-II explorer to maximize diversity and separability of the minimum-mode directions. Applied to a DFT-parameterized Pd/Fe/Ir(111) spin model and a Cartesian Ni(111) heptamer benchmark, MOTO reconstructs rich local networks including meron/antimeron–mediated mechanisms, boundary-defect–driven charge changes, and canonical rearrangements, demonstrating up to 32 pathways between topological states and successful transfer across domains. The framework’s generality, robustness, and compatibility with existing TS-search paradigms offer a scalable path to mechanism-aware landscape maps for complex materials and can support downstream kinetic modeling, catalyst design, and spintronics applications with reduced resource requirements.

Abstract

Understanding how complex systems transition between states requires mapping the energy landscape that governs these changes. Local transition-state networks reveal the barrier architecture that explains observed behaviour and enables mechanism-based prediction across computational chemistry, biology, and physics, yet current practice either prescribes endpoints or randomly samples only a few saddles around an initial guess. We present a general optimization framework that systematically expands local coverage by coupling a multi-objective explorer with a bilayer minimum-mode kernel. The inner layer uses Hessian-vector products to recover the lowest-curvature subspace (smallest k eigenpairs), the outer layer optimizes on a reflected force to reach index-1 saddles, then a two-sided descent certifies connectivity. The GPU-based pipeline is portable across autodiff backends and eigensolvers and, on large atomistic-spin tests, matches explicit-Hessian accuracy while cutting peak memory and wall time by orders of magnitude. Applied to a DFT-parameterized Néel-type skyrmionic model, it recovers known routes and reveals previously unreported mechanisms, including meron-antimeron-mediated Néel-type skyrmionic duplication, annihilation, and chiral-droplet formation, enabling up to 32 pathways between biskyrmion (Q=2) and biantiskyrmion (Q=-2). The same core transfers to Cartesian atoms, automatically mapping canonical rearrangements of a Ni(111) heptamer, underscoring the framework's generality.

Figures (4)

• Figure 1: Overview of the MOTO framework and optimization objectives. Top: three-layer workflow: First layer, a multi-objective explorer proposes diverse, feasible initial guess. Second layer, a bilater minimum-mode kernel (direction optimization + ascent) locates index-1 saddles. Third layer, deterministic ±min-mode saddle descent certifies the transition states. Representative isolated transition states found around a Néel skyrmion and an antiskyrmion in DFT-parameterized Pd/Fe/Ir(111) are shown. Bottom: objectives and strategy used during optimization, which guide automated discovery of the local saddle–minima network.
• Figure 2: Performance of the MOTO framework on artificial atomistic-spin system. a–c Inner kernel (bilatet MMF) benchmarks. a, Cosine similarity of the minimum-mode eigenvector obtained with HVP–LOBPCG and HVP–Lanczos, measured against the reference eigenvector from the explicit Hessian, for different system sizes. b, Wall-time per minimum-mode update versus system size for the explicit-Hessian method, HVP–LOBPCG, and HVP–Lanczos. c, Peak memory versus system size for the same three solvers. Each point cluster in a–c aggregates 100 runs on random spin configurations. d–f Multi-objective explorer (population size 24). Objective values over generations using eigenvalues from Lanczos: d, spectral gap between the lowest and second-lowest tangent-space eigenvalues, box-and-whisker overlays show median, interquartile range, and 1.5× inter-quartile range whiskers. e and f, Evolution of the diversity objectives: total pairwise L1 distance (L1) and Maximin (minimum-distance) diversity (L1). g and h, Outer kernel (bilayer MMF) benchmarks. Convergence of the minimum-mode force (g) and the two lowest eigenvalues (h) under L-BFGS and conjugate-gradient (CG), showing approach to an index-1 saddle. For readability, traces in (g) are smoothed with an exponential moving average (EMA)Holt2004IJF, the same smoothing convention widely used for training-loss curves in ML dashboards (e.g., TensorBoard). i and j, Initial spin configuration and the masked perturbation used in g–h. k and l, Transition states obtained from the bilayer MMF optmization with L-BFGS (k) and CG (l), respectively. All calculations are performed on Nvidia GH200 platfrom
• Figure 3: Local transition networks of a Néel skyrmion and an antiskyrmion in a DFT-parameterized Pd/Fe/Ir(111) system system size 100X100 spins with external mangetic filed B=2.7T. a, Transition-state network starting from a Néel skyrmion (left) and an antiskyrmion (right). Bottom: thumbnails of the transition states referenced in the network. Final target local minima highlighted at the top of panel a, a chiral droplet, a bi-antiskyrmion, and two skyrmions, correspond to the right-hand endpoints of panels b, c, and d, respectively. b, Bidirectional saddle descent from TS2 (antiskyrmion–meron transition state): ±min-mode downhill trajectory snapshots. c, Bidirectional saddle descent from TS4 (antiskyrmion–antimeron transition state). d, Bidirectional saddle descent from TS9 (skyrmion–meron transition state). All bidirectional saddle descent from TS1 to Ts12 are shown in support movie 1 to 12.
• Figure 4: Local transition-state landscape of a Ni heptamer on Ni(111). a, Transition-state network mapped from a seven-atom Ni island placed on a four-layer fcc(111) slab (100 Ni atoms per layer). Bottom of a: top and side views of the initial heptamer configuration. b, Top views of low-barrier saddles ($\Delta E < 1.6\,\text{eV}$) and their corresponding final states.