Previous neural solvers of math word problems (MWPs) are learned with full supervision and fail to generate diverse solutions. In this paper, we address this issue by introducing a \textit{weakly-supervised} paradigm for learning MWPs. Our method only requires the annotations of the final answers and can generate various solutions for a single problem. To boost weakly-supervised learning, we propose a novel \textit{learning-by-fixing} (LBF) framework, which corrects the misperceptions of the neural network via symbolic reasoning. Specifically, for an incorrect solution tree generated by the neural network, the \textit{fixing} mechanism propagates the error from the root node to the leaf nodes and infers the most probable fix that can be executed to get the desired answer. To generate more diverse solutions, \textit{tree regularization} is applied to guide the efficient shrinkage and exploration of the solution space, and a \textit{memory buffer} is designed to track and save the discovered various fixes for each problem. Experimental results on the Math23K dataset show the proposed LBF framework significantly outperforms reinforcement learning baselines in weakly-supervised learning. Furthermore, it achieves comparable top-1 and much better top-3/5 answer accuracies than fully-supervised methods, demonstrating its strength in producing diverse solutions.

通过引入	extit{弱监督}学习范式并提出一种新的	extit{修正学习}框架（LBF），如果深度神经网络生成的'错误'解决方案树，通过符号推理实现从上到下探索并推断出最可能的修正措施，进而使其能够生成多样性的数学问题解法，相较于完全监督方法，所提出的LBF框架在弱监督学习上表现出色并且取得了可媲美的准确率。

通过修正学习:利用弱监督解决数学应用题