Solving the inverse problem in baseball trajectory

Abstract

Inferring the release conditions required to produce a desired baseball trajectory is an ill-posed inverse problem because different combinations of release velocity, spin rate, spin efficiency, spin axis, and release direction can generate similar ball flights. We trained multilayer perceptron (MLP), convolutional neural network (CNN), and gated recurrent neural network (GRU-RNN) models on MLB Statcast data from the 2022–2024 seasons to infer plausible release conditions from scalar, complete-trajectory, and partial-trajectory information. An ensemble of these architectures achieved ~ one-inch forward trajectory error across eight pitch types. We then characterized physical non-uniqueness by sampling 10,000 plausible release vectors around representative four-seam fastball, curveball, sweeper, and splitter conditions and identifying those reproducing target trajectories within multiple tolerances. Admissible solution spaces differed substantially among the four representative pitches, with the splitter exhibiting the broadest region, indicating that release-condition identifiability may be pitch-dependent. Identifiability was also parameter-dependent: full-trajectory information substantially narrowed admissible release directions, whereas release velocity, spin rate, and spin efficiency remained compensatory, including a negative spin-rate–spin-efficiency association (r = -0.44 to -0.74). The pitch-wise ordering of physical non-identifiability was associated with ensemble uncertainty, suggesting that model disagreement may reflect underlying inverse ambiguity, although this relationship was evaluated in only four representative cases. These findings recast non-uniqueness from solely a limitation into a potential pitch-design space in which alternative release strategies may produce a desired trajectory. Thus, the study distinguishes model-dependent predictive accuracy from the inherent physical identifiability of a pitch’s release conditions, providing a basis for individualized coaching, pitch development, and repertoire evaluation.