Adaptive Pruning for Increased Robustness and Reduced Computational Overhead in Gaussian Process Accelerated Saddle Point Searches

TL;DR

This work tackles the computational bottleneck of Gaussian Process–accelerated saddle point searches on high-dimensional potential energy surfaces. It introduces OT-GP, which couples a Dimer-based saddle point search with a permutation-invariant, geometry-aware Gaussian Process learned on a chemically diverse, Farther Point Sampling subset guided by Earth Mover’s Distance. The framework integrates an adaptive variance barrier, hyperparameter-oscillation detection, and a data-driven trust radius with rotation removal to achieve robust, wall-time-efficient performance. Benchmarking on 238 molecular systems shows OT-GP substantially reduces wall-time and the number of expensive energy evaluations while maintaining high saddle-point accuracy and reliability. The approach promises to enable routine, scalable exploration of complex chemical landscapes and can be extended to other domains requiring efficient, symmetry-aware surrogate modeling of high-dimensional energy surfaces.

Abstract

Gaussian process (GP) regression provides a strategy for accelerating saddle point searches on high-dimensional energy surfaces by reducing the number of times the energy and its derivatives with respect to atomic coordinates need to be evaluated. The computational overhead in the hyperparameter optimization can, however, be large and make the approach inefficient. Failures can also occur if the search ventures too far into regions that are not represented well enough by the GP model. Here, these challenges are resolved by using geometry-aware optimal transport measures and an active pruning strategy using a summation over Wasserstein-1 distances for each atom-type in farthest-point sampling, selecting a fixed-size subset of geometrically diverse configurations to avoid rapidly increasing cost of GP updates as more observations are made. Stability is enhanced by permutation-invariant metric that provides a reliable trust radius for early-stopping and a logarithmic barrier penalty for the growth of the signal variance. These physically motivated algorithmic changes prove their efficacy by reducing to less than a half the mean computational time on a set of 238 challenging configurations from a previously published data set of chemical reactions. With these improvements, the GP approach is established as, a robust and scalable algorithm for accelerating saddle point searches when the evaluation of the energy and atomic forces requires significant computational effort.

Figures (10)

• Figure 1: The previously developed GPDimer method for accelerating saddle point searches using Gaussian process regression. The GP generated surrogate surface is considered reliable if the guardrails of Eq. \ref{['eq:mindistatm']} and \ref{['eq:max1dlog']} are satisfied.
• Figure 2: Overview of the OT-GP method. Red arrows indicate rejected proposals (outside the trust radius); blue dashed arrows denote the transition from a freshly-updated GP model back to the relaxation loop; dark-red bold arrows highlight the final verification step on the true PES. The flowchart therefore encapsulates the hierarchical control strategy of OT-GP: (i) cheap GP-driven exploration, (ii) data-driven trust-radius and variance regularisation, and (iii) intermittent high-fidelity validation that guarantees convergence to the first-order saddle point on the PES.
• Figure 3: Comparison of the performance of the OTGPD saddle point search method against GPDimer and regular dimer without GP acceleration for 238 molecular systems. A calculation exceeding 4 hours or raising an error in the electronic structure calculation counts as a failure. (A, B) Success outcomes for each system, ordered along the horizontal axis by mean root-mean-square deviation (RMSD) between the saddle point and the initial atomic structure. Red dots denote systems where only OTGPD succeeded, blue dots where only the alternate method succeeded, and orange dots where both failed. (C) Bar chart summarizing success rate of each method for initial structures that represent a single fragment and those representing two fragments. The single-fragment cases represent typical saddle point searches, while the two-fragment cases test performance on more complex, often arbitrary dissociation pathways. The OTGPD method demonstrates a clear advantage. In addition to a mutual success rate exceeding 93% in both comparisons, OTGPD uniquely finds the saddle point for an additional 11 systems (4.6%) compared to GPDimer (1.3%) and additional 9 systems (3.8%) compared to Dimer (1.7%).
• Figure 4: Evolution of the values for the hyperparameters in a representative failure case, system D110. (Left) The GPDimer search fails after 30 iterations as the signal variance (Var) explodes, leading to an error after 38 minutes. (Right) The OTGPD framework maintains a stable signal variance, which saturates at the adaptive barrier limit. This stability allows the search to converge successfully.
• Figure 5: Comparison of saddle search outcomes for system D150, highlighting the stability of the OTGPD method. Bond coloring visualizes the Wiberg bond order, with dots used when the order is below 0.5, representing non-bonded interaction. (A) The initial geometry for the saddle point searches. (B) The GPDimer search fails and ends up fragmenting the molecule. (C) Both the standard dimer and the OTGPD method successfully converge to the same saddle point.