  This paper introduces a novel learning-based methodology for Direction of
Arrival (DoA) estimation in environments featuring an Intelligent Reflecting
Surface (IRS) or Reconfigurable Intelligent Surface (RIS), particularly under
challenging Non-Line-of-Sight (NLoS) conditions. The core innovation lies in
the integration of a specialized Neural Network (NN) layer, termed the NN-based
RIS layer, within a broader network architecture.
  The NN-based RIS layer is meticulously designed to autonomously learn the
optimal phase shifts for the RIS, specifically tailored to enhance the accuracy
of DoA estimation. In this framework, pre-processed real-valued input
observations are directed into the RIS layer, which possesses a distinct and
specialized structure. Unlike conventional neural network layers, the NN-based
RIS layer imposes constraints on its weights, requiring them to be sinusoidal
functions. The phase arguments of these sinusoidal functions serve as the
tunable parameters during the training process, effectively enabling the layer
to simulate the physical behavior of an RIS.
  To accommodate the unique properties of the NN-based RIS layer, the standard
feed-forward and back-propagation algorithms are adapted, ensuring that the
learning process aligns with the specialized requirements of the RIS-based
architecture. This innovative approach allows the network to seamlessly
integrate the RIS functionality into the DoA estimation process.
  Extensive numerical simulations validate the effectiveness of the proposed
machine learning-based technique, demonstrating its superior performance
compared to traditional non-learning-based methods across a range of
RIS-assisted scenarios. The results underscore the potential of this approach
to significantly enhance DoA estimation accuracy in complex wireless
communication environments, paving the way for future advancements in
RIS-enabled systems.
