Lample and Charton (2019) describe a system that uses deep learning
technology to compute symbolic, indefinite integrals, and to find symbolic
solutions to first- and second-order ordinary differential equations, when the
solutions are elementary functions. They found that, over a particular test
set, the system could find solutions more successfully than sophisticated
packages for symbolic mathematics such as Mathematica run with a long time-out.
This is an impressive accomplishment, as far as it goes. However, the system
can handle only a quite limited subset of the problems that Mathematica deals
with, and the test set has significant built-in biases. Therefore the claim
that this outperforms Mathematica on symbolic integration needs to be very much
qualified.

该论文介绍了使用深度学习技术计算符号无限积分，以及找到解析一阶和二阶常微分方程的符号解的系统，该系统的表现在某些测试集上优于 Mathematica。

使用深度学习进行符号积分：（Lample 和 Charton，2019）综述

The Use of Deep Learning for Symbolic Integration: A Review of (Lample  and Charton, 2019)

In this work we discuss the problem of selecting suitable approximators from
families of parameterized elementary functions that are known to be dense in a
Hilbert space of functions. We consider and analyze published procedures, both
randomized and deterministic, for selecting elements from these families that
have been shown to ensure the rate of convergence in $L_2$ norm of order
$O(1/N)$, where $N$ is the number of elements. We show that both randomized and
deterministic procedures are successful if additional information about the
families of functions to be approximated is provided. In the absence of such
additional information one may observe exponential growth of the number of
terms needed to approximate the function and/or extreme sensitivity of the
outcome of the approximation to parameters. Implications of our analysis for
applications of neural networks in modeling and control are illustrated with
examples.

探讨如何从函数族中选定合适的拟合函数，分析了已有的随机和确定性选取函数的方法，在提供额外信息的情况下可以成功的实现 $L_2$ 范数收敛，进一步阐述了这对于神经网络在建模和控制中的应用的含义。

随机基函数逼近：正反两面观点

Approximation with Random Bases: Pro et Contra

As is well known, the common elementary functions defined over the real
numbers can be generalized to act not only over the complex number field but
also over the skew (non-commuting) field of the quaternions. In this paper, we
detail a number of elementary functions extended to act over the skew field of
Clifford multivectors, in both two and three dimensions. Complex numbers,
quaternions and Cartesian vectors can be described by the various components
within a Clifford multivector and from our results we are able to demonstrate
new inter-relationships between these algebraic systems. One key relationship
that we discover is that a complex number raised to a vector power produces a
quaternion thus combining these systems within a single equation. We also find
a single formula that produces the square root, amplitude and inverse of a
multivector over one, two and three dimensions. Finally, comparing the
functions over different dimension we observe that $ C\ell \left (\Re^3 \right)
$ provides a particularly versatile algebraic framework.

介绍定义在 Clifford 多向量上的一些初等函数，探寻其中向量、复数、四元数之间的相互关系，并为多向量提供平方根、振幅和反函数的公式。