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.

本文通过使用深度神经网络结合稀疏规则化技术，针对非线性偏微分方程的解进行了数值研究，结果表明该方法在求解 Burgers 方程和 Schrödinger 方程时能够生成稀疏且准确的解。