Neural networks such as encoder-decoder architectures, UNet, and Transformers have achieved remarkable success in image processing and sequence modeling, yet a comprehensive mathematical understanding of their structure remains limited. In this talk, we present a unified, operator-theoretic framework that interprets these architectures through the lens of control theory, multigrid methods, and continuous modeling. We show that popular encoder-decoder networks—including UNet—can be derived as time-discretized solutions to control problems using operator-splitting and multigrid decomposition. Specifically, we introduce PottsMGNet, a network derived from the two-phase Potts model, and demonstrate how it generalizes many encoder-decoder designs. We further extend this perspective to Transformers, modeling self-attention as a non-local integral operator within a continuous integro-differential framework, and interpreting normalization as time-dependent constraints. These insights not only offer a rigorous theoretical foundation for key neural architectures but also open new paths for principled architecture design, robustness, and interpretability across tasks in vision and language.