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Automated mathematical reasoning is a challenging problem that requires anagent to learn algebraic patterns that contain long-range dependencies. Twoparticular tasks that test this type of reasoning are (1) mathematical equationverification, which requires determining whether trigonometric and linearalgebraic statements are valid identities or not, and (2) equation completion,which entails filling in a blank within an expression to make it true. Solvingthese tasks with deep learning requires that the neural model learn how tomanipulate and compose various algebraic symbols, carrying this ability over topreviously unseen expressions. Artificial neural networks, including recurrentnetworks and transformers, struggle to generalize on these kinds of difficultcompositional problems, often exhibiting poor extrapolation performance. Incontrast, recursive neural networks (recursive-NNs) are, theoretically, capableof achieving better extrapolation due to their tree-like design but aredifficult to optimize as the depth of their underlying tree structureincreases. To overcome this issue, we extend recursive-NNs to utilizemultiplicative, higher-order synaptic connections and, furthermore, to learn todynamically control and manipulate an external memory. We argue that this keymodification gives the neural system the ability to capture powerful transitionfunctions for each possible input. We demonstrate the effectiveness of ourproposed higher-order, memory-augmented recursive-NN models on two challengingmathematical equation tasks, showing improved extrapolation, stableperformance, and faster convergence. Our models achieve a 1.53% averageimprovement over current state-of-the-art methods in equation verification andachieve a 2.22% Top-1 average accuracy and 2.96% Top-5 average accuracy forequation completion.

Recognizing and Verifying Mathematical Equations using Multiplicative Differential Neural Units