In this paper, we propose a systematic approach for accelerating finite
element-type methods by machine learning for the numerical solution of partial
differential equations (PDEs). The main idea is to use a neural network to
learn the solution map of the PDEs and to do so in an element-wise fashion.
This map takes input of the element geometry and the PDEs' parameters on that
element, and gives output of two operators -- (1) the in2out operator for
inter-element communication, and (2) the in2sol operator (Green's function) for
element-wise solution recovery. A significant advantage of this approach is
that, once trained, this network can be used for the numerical solution of the
PDE for any domain geometry and any parameter distribution without retraining.
Also, the training is significantly simpler since it is done on the element
level instead on the entire domain. We call this approach element learning.
This method is closely related to hybridizbale discontinuous Galerkin (HDG)
methods in the sense that the local solvers of HDG are replaced by machine
learning approaches. Numerical tests are presented for an example PDE, the
radiative transfer equation, in a variety of scenarios with idealized or
realistic cloud fields, with smooth or sharp gradient in the cloud boundary
transition. Under a fixed accuracy level of 10β3 in the relative L2
error, and polynomial degree p=6 in each element, we observe an approximately
5 to 10 times speed-up by element learning compared to a classical finite
element-type method