TY - JOUR
T1 - Sparse Deep Neural Network for Nonlinear Partial Differential Equations
AU - Xu , Yuesheng
AU - Zeng , Taishan
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 1
SP - 58
EP - 78
PY - 2023
DA - 2023/01
SN - 16
DO - http://doi.org/10.4208/nmtma.OA-2022-0104
UR - https://global-sci.org/intro/article_detail/nmtma/21343.html
KW - Sparse approximation, deep learning, nonlinear partial differential equations, sparse regularization, adaptive approximation.
AB - <p style="text-align: justify;">More competent learning models are demanded for data processing due
to increasingly greater amounts of data available in applications. Data that we encounter often have certain embedded sparsity structures. That is, if they are represented in an appropriate basis, their energies can concentrate on a small number
of basis functions. This paper is devoted to a numerical study of adaptive approximation of solutions of nonlinear partial differential equations whose solutions may
have singularities, by deep neural networks (DNNs) with a sparse regularization
with multiple parameters. Noting that DNNs have an intrinsic multi-scale structure
which is favorable for adaptive representation of functions, by employing a penalty
with multiple parameters, we develop DNNs with a multi-scale sparse regularization
(SDNN) for effectively representing functions having certain singularities. We then
apply the proposed SDNN to numerical solutions of the Burgers equation and the
Schrödinger equation. Numerical examples confirm that solutions generated by the
proposed SDNN are sparse and accurate.</p>