  We develop a novel deep learning technique, termed Deep Orthogonal
Decomposition (DOD), for dimensionality reduction and reduced order modeling of
parameter dependent partial differential equations. The approach consists in
the construction of a deep neural network model that approximates the solution
manifold through a continuously adaptive local basis. In contrast to global
methods, such as Principal Orthogonal Decomposition (POD), the adaptivity
allows the DOD to overcome the Kolmogorov barrier, making the approach
applicable to a wide spectrum of parametric problems. Furthermore, due to its
hybrid linear-nonlinear nature, the DOD can accommodate both intrusive and
nonintrusive techniques, providing highly interpretable latent representations
and tighter control on error propagation. For this reason, the proposed
approach stands out as a valuable alternative to other nonlinear techniques,
such as deep autoencoders. The methodology is discussed both theoretically and
practically, evaluating its performances on problems featuring nonlinear PDEs,
singularities, and parametrized geometries.
