Seminar Details
2019-11-26 (14:00) : Some geometric and machine learning methods for model order reduction
At Euler building (room A.002)
Organized by Mathematical Engineering
Speakers :
Rolando Mosquera (LaSIE, Université de la Rochelle, France
)
,
and
Antoine Falaize (LaSIE, Université de la Rochelle, France
)
Abstract :
Geometric methods have proved to be effective for data mining (simulations, measurements) in the context of nonlinear model order reduction problems in mechanics (e.g. nonlinear methods for dimensionality reduction such as Local Principal Component Analysis, LPCA or Locally Linear Embedding, LLE). In addition, deep learning methods by so-called neural network (NN) architectures proved to be effective for a class of problems involving data classification, detection and compression.
We first present a set of geometric tools for the interpolation of reduced bases obtained by orthogonal decomposition (POD) of simulation results. More precisely, by exploiting the intrinsic geometry of the sub- spaces engendered by the POD bases, we give the construction of different interpolators on the Grassmann manifold.
Then, we use the architecture of neural networks to extract geometric informations about the data manifold. More precisely, a metric on the manifold can be found from the representations of coordinates learned by the network. Conversely, classical architectures of neural networks can be informed by the geometry of the data to build optimizers that respect the local geometry. These approaches are illustrated on model order reduction problems in mechanics.