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.

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