Heteroscedastic Prototype Learning for Probabilistic Second-Life Battery Degradation Prediction

Abstract

Second-life battery (SLB) cells retired from electric vehicles exhibit heterogeneous degradation trajectories shaped by diverse first-life histories, posing a challenge for fleet-scale prediction systems that must simultaneously de- liver accurate capacity forecasts, calibrated uncertainty estimates, and real-time inference. Existing methods ei- ther address population heterogeneity without uncertainty quantification, or provide uncertainty via Monte Carlo (MC) dropout at prohibitive inference cost. This paper proposes ProtoSLB-H, a heteroscedastic prototype learn- ing framework maintaining a bank of K=4 learnable prototype vectors that represent electrochemical degrada- tion archetypes discovered from data. Cosine-similarity routing assigns each cell a soft membership vector, and per-prototype heteroscedastic prediction heads output paired mean and log-standard-deviation capacity trajecto- ries. The law of total variance then yields a closed-form two-source decomposition that separates intra-archetype aleatoric uncertainty from inter-archetype routing uncertainty, with no MC sampling required at any stage. On the 39-cell Lithium-ion Second-life Battery (LSD) dataset, ProtoSLB-H achieves RMSE of 0.0533, CRPS of 0.0354, and a post-calibration 90% prediction interval coverage of 91.2%, improving probabilistic accuracy by 9.7% over the MC dropout baseline while achieving 50× faster per-sample inference (0.0024 ms vs. 0.12 ms). The two-source decomposition provides fleet operators with a predict–attribute–act loop for maintenance prioritisation.