Active delta-learning for surface hopping: efficient fitting of potential energy surfaces

Active learning has become central to modern machine-learning workflows for trajectory surface hopping, enabling automatic construction of training sets for excited-state potentials. However, currently used protocols can struggle with convergence, requiring large datasets and extensive human intervention. Here, we combine active learning with Δ-learning - where the ML model predicts the difference between a target quantum chemical method and a computationally efficient baseline - into an end-to-end protocol for active delta-learning (ADL) of potential energy surfaces for trajectory surface hopping dynamics. The protocol is tested on a benchmark system, fulvene, where a twofold data efficiency is achieved with respect to fine-tuning from a pre-trained foundational model, OMNI-P2x. Next, we analyze the case of PSB4, a protonated Schiff base with multiple competing torsional deactivation channels. Here, pure ML active learning fails to converge despite extensive sampling of over 14 000 points, while ADL converges within just two iterations and 900 sampled points, yielding good agreement with the reference.