Learning the Geometry of Collider Events with Metric-Aware Deep SetsmanuscriptsmanuscriptsOptimal transport gives structured data a geometry, but exact evaluation is costly in large pairwise analyses that exploit relationships among distances. Learned surrogates are faster, but need not preserve this metric structure. We develop a Deep Sets surrogate for OT between variable-size weighted point clouds that enforces non-negativity, exchange symmetry, and zero self-distance, leaving the triangle inequality unconstrained. Applied to the Energy Mover’s Distance between collider events in a particle physics application, the Metric-Aware Particle Flow Network achieves percent-level mean absolute percentage error while significantly improving inference throughput over other exact and approximate methods surveyed. The architectural constraints are found to improve properties that are not explicitly enforced: across 10^6 held-out event triplets, triangle-inequality violations fall from 199 for a matched unconstrained network to 2, and the maximum from 149.5 to 5.8 GeV. These results demonstrate that targeted inductive biases can yield fast neural surrogates with substantially improved geometric fidelity.Lauren HayAuthorRishabh JainAuthorMatt LeBlancAuthorJennifer RoloffAuthor2026-09-09Optimal TransportLarge Hadron ColliderParticle PhysicsAI/MLmetric spacesdeep setsneural surrogatesAttribution 4.0 International (CC BY 4.0)In Copyright10.26300/1f5r-bs32