Optimal sampling and natural gradient for fast online learning in non-linear reduced order models
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In linear reduced-order models (ROMs), the online stage of reconstructing a system state from $m$ measurements is straightforward, typically involving a linear least-squares problem. For non-linear models, however, this task requires solving a non-linear counterpart, requiring in general the use of gradient-based iterative methods. By linearizing the model at each iteration, one essentially solves linear least-squares problems within the tangent space of the non-linear manifold a technique known as the Natural Gradient method \cite{NGD}. To ensure efficiency, these methods can be integrated with optimal sampling strategies \cite{OptSamp,OptNGD}, significantly reducing the computational overhead of the online stage. We will show through numerical experiments that indeed this methodology improves the online learning of non-linear compressive reduced basis (RB) \cite{NLCRB,CPN} which are a special class of non-linear ROMs that apply non-linear corrections to traditional linear methods.
