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Learning time-dependent partial differential equations (PDEs) that govern evolutionary observations is one of the core challenges for data-driven inference in many fields. In this work, we propose to capture the essential dynamics of numerically challenging PDEs arising in multiscale modeling and simulation -- kinetic equations. These equations are usually nonlocal and contain scales/parameters that vary by several orders of magnitude. We introduce an efficient framework, Densely Connected Recurrent Neural Networks (DC-RNNs), by incorporating a multiscale ansatz and high-order implicit-explicit (IMEX) schemes into RNN structure design to identify analytic representations of multiscale and nonlocal PDEs from discrete-time observations generated from heterogeneous experiments. If present in the observed data, our DC-RNN can capture transport operators, nonlocal projection or collision operators, macroscopic diffusion limit, and other dynamics. We provide numerical results to demonstrate the advantage of our proposed framework and compare it with existing methods.
In this paper, we demonstrate the construction of generalized Rough Polyhamronic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the ra
There is an intimate connection between numerical upscaling of multiscale PDEs and scattered data approximation of heterogeneous functions: the coarse variables selected for deriving an upscaled equation (in the former) correspond to the sampled info
In this work, we review the connection between the subjects of homogenization and nonlocal modeling and discuss the relevant computational issues. By further exploring this connection, we hope to promote the cross fertilization of ideas from the diff
In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove the proposed multiscale method achieves nearly expo
We introduce a simple, rigorous, and unified framework for solving nonlinear partial differential equations (PDEs), and for solving inverse problems (IPs) involving the identification of parameters in PDEs, using the framework of Gaussian processes.