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Interdisciplinary Machine Learning in Science and Engineering ()

Yalchin Efendiev, Department of Mathematics
Reduced-Order Deep Learning for Flow Dynamics. The Interplay Between Deep Learning and Model Reduction


In this talk, we investigate neural networks applied to multiscale simulations of porous media flows and discuss a design of a novel deep neural network model reduction approach for multiscale problems. In practice, low-order models are derived to reduce the computational cost. We use a non-local multicontinuum (NLMC) approach, which represents the solution on a coarse grid for porous media flows. In specific, we construct a reduced dimensional space which takes into account multi-continuum information. The numerical solutions are then sought in such space. Using multi-layer learning techniques, we formulate and learn input-output maps constructed with NLMC on a coarse grid. We study the features of the coarse-grid solutions that neural networks capture via relating the input-output optimization to minimization of flow solutions. In proposed multi-layer networks, we can learn the forward operators in a reduced way. We present soft thresholding operators as activation function, which our studies show to have some advantages. With these activation functions, the neural network identifies and selects important multiscale features which are crucial in modeling the underlying flow. Using trained neural network approximation of the input-output map, we construct a reduced-order model for the solution approximation. We use multi-layer networks for the time stepping and reduced-order modeling, where at each time step the appropriate important modes are selected. For a class of nonlinear flow problems, we suggest an efficient strategy. Numerical examples are presented to examine the performance of our method.