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。

用于非线性偏微分方程的稀疏深度神经网络

Sparse Deep Neural Network for Nonlinear Partial Differential Equations

We describe how the couplings in an asynchronous kinetic Ising model can be
inferred. We consider two cases, one in which we know both the spin history and
the update times and one in which we only know the spin history. For the first
case, we show that one can average over all possible choices of update times to
obtain a learning rule that depends only on spin correlations and can also be
derived from the equations of motion for the correlations. For the second case,
the same rule can be derived within a further decoupling approximation. We
study all methods numerically for fully asymmetric Sherrington-Kirkpatrick
models, varying the data length, system size, temperature, and external field.
Good convergence is observed in accordance with the theoretical expectations.

描述了异步动力学 Ising 模型中的耦合如何被推测，并讨论了两种情况下的学习规则和数值方法。

异步更新伊辛模型的极大似然重构

Maximum likelihood reconstruction for Ising models with asynchronous  updates

In this letter, we investigate the detailed epidemic spreading process in
scale-free networks with links' weights that denote familiarity between two
individuals and find that spreading velocity reaches a peak quickly then decays
in a power-law form. Numerical study exhibits that the nodes with larger
strength is preferential to be infected, but the hierarchical dynamics are not
clearly found, which is different from the well-known result in unweighed
network case. In addition, also by numerical study, we demonstrate that larger
dispersion of weight of networks results in slower spreading, which indicates
that epidemic spreads more quickly on unweighted scale-free networks than on
weighted scale-free networks with the same condition.

本文研究网络中带权链接的无标度网络的传染病传播过程，并发现传播速度会迅速达到高峰，然后呈现幂律形式的衰减。数值研究表明，具有较大强度值的节点更容易受到感染，但层次动力学并不明显，这与未加权网络的结果不同。此外，数值研究还表明，网络权重的较大差异导致传播速度较慢，这表明与具有相同条件的无权无标度网络相比，流行病在带权无标度网络上传播更慢。