4 accepted papers
It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of the problem. A practical way to apply such a scheme is to co…
We propose GMsFEM-NO, a novel hybrid framework that combines the robustness of the Generalized Multiscale Finite Element Method (GMsFEM) with the computational speed of neural operators (NOs) to create an efficient method for solving heterogeneous partial differential equations (PDEs). GMsFEM build…
In ``Large Associative Memory Problem in Neurobiology and Machine Learning,'' Dmitry Krotov and John Hopfield introduced a general technique for the systematic construction of neural ordinary differential equations with non-increasing energy or Lyapunov function. We study this energy function and id…
Deep learning solvers for partial differential equations typically have limited accuracy. We propose to overcome this problem by using them as preconditioners. More specifically, we apply discretization-invariant neural operators to learn preconditioners for the flexible conjugate gradient method (F…