  Explaining deep learning models operating on time series data is crucial in
various applications of interest which require interpretable and transparent
insights from time series signals. In this work, we investigate this problem
from an information theoretic perspective and show that most existing measures
of explainability may suffer from trivial solutions and distributional shift
issues. To address these issues, we introduce a simple yet practical objective
function for time series explainable learning. The design of the objective
function builds upon the principle of information bottleneck (IB), and modifies
the IB objective function to avoid trivial solutions and distributional shift
issues. We further present TimeX++, a novel explanation framework that
leverages a parametric network to produce explanation-embedded instances that
are both in-distributed and label-preserving. We evaluate TimeX++ on both
synthetic and real-world datasets comparing its performance against leading
baselines, and validate its practical efficacy through case studies in a
real-world environmental application. Quantitative and qualitative evaluations
show that TimeX++ outperforms baselines across all datasets, demonstrating a
substantial improvement in explanation quality for time series data. The source
code is available at \url{https://github.com/zichuan-liu/TimeXplusplus}.
