The transition toward 2nm nodes and complex Gate-All-Around (GAA) architectures exposes the limitations of traditional mesh-based solvers, such as Finite Element Analysis (FEA), which struggle with the” curse of dimensionality” and high computational overhead. This paper introduces a paradigm shift in semiconductor modeling by re-imagining the device as a continuous, differentiable manifold rather than a discrete lattice. We present a Physics-Informed Neural Network (PINN) framework that serves as a meshless solver, embedding the semi-classical Boltzmann Transport and Drift-Diffusion Poisson systems directly into the neural objective function. By utilizing automatic differentiation to satisfy physical residuals, our approach bypasses manual mesh generation and enables efficient inverse design. The framework achieved a Mean Squared Error (MSE) of < 10−4 when validated against industry-standard TCAD Sentaurus numerical solvers, representing a significant reduction in computational overhead for sub-2nm nodes. Our results demonstrate that this continuous solver offers a scalable, resolution-independent alternative to traditional TCAD tools, significantly accelerating the characterization and design of next-generation solid-state electronics.