  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.

Artificial Intelligence;Computation and Language;Machine Learning