Average time complexity halving for Gaussian Process Saddle searches

Gaussian Process (GP) surrogates provide a powerful framework for accelerating molecular saddle point searches, yet the O(N3) scaling of hyperparameter optimization severely hampers their practical utility. We introduce a robust, scalable algorithm that resolves this bottleneck through key innovations in data subsampling and optimization stability. To achieve scalability, our method employs a Farthest Point Sampling (FPS) algorithm to actively curate a fixed-size subset of training configurations from the search history. The novelty of our approach resides in the distance metric: an intensive, permutation-invariant variant of the Wasserstein-1 (Earth Mover's) distance. By averaging minimum atomic displacements on a per-element basis, this metric yields a size-independent measure of geometric dissimilarity, enabling the selection of a truly representative and diverse training set.

This geometry-aware subsampling facilitates near-constant-time GP model updates, fundamentally improving the algorithm's time complexity. We further enhance the optimization process with a Hyperparameter Oscillation Detection (HOD) heuristic that adaptively expands the training subset to stabilize learning and an adaptive logarithmic barrier that regularizes the signal variance. The same intensive Wasserstein metric affords a physically meaningful trust radius for early stopping, preventing model exploitation in unexplored regions. On a diverse benchmark of 238 molecular systems, our Optimal Transport GP (OT-GP) method demonstrates state-of-the-art performance. It reduces the mean wall-time by more than half compared to previous GP-accelerated approaches and cuts the number of expensive quantum chemistry calculations by nearly an order of magnitude relative to the standard Dimer method. This work presents a complete framework for creating stable, data-efficient, and computationally tractable machine learning force fields for on-the-fly exploration of complex potential energy surfaces.