AI 与机器学习2019openreview.net
面向符号数学的深度学习
Deep Learning for Symbolic Mathematics
进阶 · 约 90 分钟读完前置知识：Transformer 基础、seq2seq 序列建模、微积分与符号积分
导读
这篇论文探索用深度学习（Transformer + seq2seq）求解符号数学问题：将表达式树编码为逆波兰表示（RPN）序列，让模型直接生成不定积分和微分方程的解。作者报告其准确率在某些任务上超过了 Mathematica 等计算机代数系统，展示了神经网络在符号推理上的潜力。
HN 关注点：神经网络能否做符号推理这一反直觉结果，讨论焦点集中在实验方法是否公平、结果是否过于乐观。
后来怎么样
该工作开创了用 Transformer 做符号数学的路线，后续衍生出大量符号推理与定理证明方向的研究，成为 AI for Math 领域的代表性早期成果。
HN 评论精选
这几条评论里：质疑 3 补充 1
质疑质疑与 Mathematica 的对比不公平：30 秒超时过短，且可能未对齐复数域等实验条件
作者展示了一个漂亮且（对我而言）反直觉的结果，但他们与 Mathematica 的对比有点不严谨：他们比较的是自己算法的答案正确率，而 Mathematica 是在 30 秒内能否找到正确解——这是很重要的区别！Mathematica 从不会悄悄返回错误答案（软件 bug 除外），而且它常常需要几分钟才算完一个看似简单的积分，30 秒超时太短了。另外还有其他细节问题：Mathematica 默认在复数域工作，使许多运算更难，但作者把含复系数的表达式当作“无效”丢弃，说明他们可能隐式地在实数域工作——有没有对 Mathematica 做同样限制？有没有先调用 FullSimplify[] 这类常用技巧？希望看到更详细的方法说明。
wwalker3查看英文原文
The authors have shown a very nice and (to me) non-intuitive result. But they're playing a little fast and loose with their comparison to Mathematica. They're comparing their algorithm's accuracy (solution correctness vs. incorrectness), with Mathematica's ability to find the correct solution in less than 30 seconds. This is a very important distinction! Mathematica will never silently return an incorrect solution (barring software bugs, of course). And Mathematica can often take minutes to evaluate what appears to be a simple integral, so a 30-second timeout is far too short, unless you're simply trying to compare the computational efficiency of the two approaches. There may be other subtleties as well. Mathematica works in the complex domain by default, which makes many operations more difficult, but the authors discard expressions which contain complex-valued coefficients as "invalid", which makes me think they're implicitly working in the real domain. Do they restrict Mathematica to the real domain when they invoke it? Perhaps, but they don't say one way or the other. And do they try common tricks like invoking FullSimplify[] on an expression/equation before attempting to operate on it? I'd like to see more details of their methodology.
质疑怀疑模型只是学到解模板匹配，并担忧其在无解析解问题上会给出错误答案
有意思，作者把表达式树编码成 RPN 格式，用普通 transformer 加 seq2seq 解码出某些积分和微分方程的解。我怀疑在如此受限的问题空间里，神经网络其实是在把输入格式匹配到解的模板，学到了“什么运算的逆是什么”的模式，权重里甚至可能埋着乘积法则的某些影子。我很好奇它在训练集格式之外的表现，比如需要技巧的积分 exp(x)*sin(x)，或嵌套复合 A/(1+exp(-k*(x-a)))，对输入长度是否敏感？1/sqrt(x^2 - y^4) 这种小而刁钻的积分会不会难倒它？最重要的是，现实中大多数不定积分没有解析解，实用系统必须知道何时该放弃——输入 exp(-x^2) 时最好别给出答案。这是我对这类方法的主要担忧：在覆盖更广数学问题的真实场景中，无法信任一个对自己能力边界不是 100% 确定的系统。
Cybiote查看英文原文
This is interesting, the authors encode expression trees in RPN format and use vanilla transformers plus seq2seq to decode solutions to certain integrals and differential equations. I suspect what's happening, given the constrained space of problems, is the NN is matching input formats to solution templates. For that space, it would have learned some patterns of what undoes to what. Perhaps there's even something that's captured aspects of the product rule buried in those weights. I would be curious to see how it does on problems outside the training set format. Or on integrals that require a trick, such as exp(x) * sin(x) or those with nested composition: A/(1+exp(-k*(x-a))). Is it sensitive to input size? Would a smaller but slightly tricky integral like 1/sqrt(x^2 - y^4) trip it? Most important, since the bulk of real world indefinite integrals are not analytically tractable, it is vital that a practical system knows when to give up. If I put in exp(-x^2), it better not return an answer. This is my main worry for methods like this. In a real world setting covering a wider range of mathematical problems, you can't trust something that's not 100% on what it knows how to do.
质疑质疑测试集生成方式可能有偏，建议用真实问题语料库评估
结果好得几乎不真实。表达式生成方式可能存在偏差，无法保证它们能代表真实问题，而论文似乎没有讨论这一点。我认为应该整理一个真实世界问题的语料库，然后在上面测性能，这样才能得到客观有用的指标，而不是基于可能有偏的样本。
andrepd查看英文原文
The results look almost too good to be true. There could be a bias in the way the expressions are generated; there is no guarantee that this is in any way representative of real problems, and I don't think this is addressed in the paper. A good thing to do, in my opinion, is compile a corpus of real-world, occurring problems, and then measure performance on that. Then you would know how it performs on some objectively useful metric, not on a potentially biased sample.
补充猜测模型可能学到了让微积分变简单的函数嵌入
有可能存在一种函数的自然向量空间嵌入，使得积分/求导成为简单操作，深度学习网络或许能找到这样的嵌入。
lapink查看英文原文
It is a possibility that there is a natural vector space embedding for functions in which integration/derivation is a simple operation. A deep learning network could find such an embedding.
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