Time series classification remains a fundamental challenge in machine learning, requiring models that capture complex temporal dynamics while maintaining interpretability. Current deep learning approaches, while effective, often lack theoretical grounding and provide limited mechanistic insights. This paper introduces Koopman Operator-Based Deep Neural Networks (KODNet), a novel architecture that integrates dynamical systems theory with deep learning for time series classification. Our approach leverages Koopman operator theory to transform nonlinear temporal dynamics into linear representations in an eigenfunction space, enabling efficient learning with improved interpretability. The dual-network architecture consists of a Koopman autoencoder that learns coordinate transformations linearizing the dynamics, followed by a classification network operating on these representations with temporal attention. We evaluate KODNet on five diverse datasets from the UCR Time Series Archive spanning medical, automotive, manufacturing, and motion capture domains. Experimental results demonstrate consistent improvements over standard LSTM and BiLSTM baselines, with average accuracy gains of 2.34% (96.28% vs. 93.94% for BiLSTM). Ablation studies validate the contribution of physics-informed constraints, while analysis of learned Koopman modes reveals interpretable dynamical patterns aligned with domain knowledge. Our work demonstrates that incorporating structure from dynamical systems theory provides meaningful advantages for time series classification.