Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for
  Solving PDEs   In this paper, we propose a a machine learning approach via
model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by
a model to solve PDEs based on operator representation with regularization from
data. For linear PDEs, we use a DNN to parameterize the Green's function and
obtain the neural operator to approximate the solution according to the Green's
method. To train the DNN, the empirical risk consists of the mean squared loss
with the least square formulation or the variational formulation of the
governing equation and boundary conditions. For complicated problems, the
empirical risk also includes a few labels, which are computed on coarse grid
points with cheap computation cost and significantly improves the model
accuracy. Intuitively, the labeled dataset works as a regularization in
addition to the model constraints. The MOD-Net solves a family of PDEs rather
than a specific one and is much more efficient than original neural operator
because few expensive labels are required. We numerically show MOD-Net is very
efficient in solving Poisson equation and one-dimensional radiative transfer
equation. For nonlinear PDEs, the nonlinear MOD-Net can be similarly used as an
ansatz for solving nonlinear PDEs, exemplified by solving several nonlinear PDE
problems, such as the Burgers equation.

20
The categories of this paper are: acc-phys
34
The categories of this paper are: chem-ph
25
The categories of this paper are: patt-sol
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
30
The categories of this paper are: q-alg
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph