The mechanisms for enhancing learning and human learning are quite similar, and DeepMind has successfully applied enhanced learning in scenarios like AlphaGo and Atari games. The collaborative research by Afan Research Institute, University of Electronic Science and Technology, and Peking University introduced an automatic solver for arithmetic application problems based on DQN (Deep Q-Network). This approach transforms the problem-solving process into a Markov decision process, leveraging the BP neural network's strong generalization, storage, and approximation capabilities to enhance the Q-value of state-action pairs during learning. Experimental results show that this algorithm performs well on standard test sets, increasing average accuracy by nearly 15%.
Research Background
The study of automatic solving mathematical application problems (MWP) dates back to the 1960s and continues to attract attention in recent years. The automatic solution of applied mathematics involves mapping human-readable sentences into machine-understandable logical forms and then making inferences. This process cannot be solved simply through pattern matching or end-to-end classification technology. Therefore, designing an automatic problem solver with semantic understanding and reasoning ability has become an essential step in the development of general artificial intelligence.
For a math application solver, given a text-based math problem, it is not sufficient to train it end-to-end using text question and answer pairs to directly get the answer. Instead, the solver must use text processing and numerical reasoning to derive the expression and compute the result. Thus, this task requires a deep understanding of the text and strong logical reasoning ability, which makes it a challenging focus in natural language understanding research.
In recent years, researchers have designed algorithms from various angles and developed solving systems to automatically tackle math problems. These include template-based methods, statistical-based methods, expression tree-based methods, and deep learning-based approaches. However, there are still challenges such as limited training datasets, weak robustness, low efficiency, and poor performance in solving math application questions. Because math problems require a solid understanding of natural language, strong numerical reasoning, and common sense, most existing methods rely heavily on manual intervention, limiting their versatility. As data complexity increases, many algorithms experience a sharp decline in performance, making the design of an effective and efficient automatic solver a difficult but crucial goal.
Related Work
Arithmetic Application Solver:
Early attempts at solving arithmetic problems used verb classification and state transition reasoning, which could only handle addition and subtraction. Later, label-based methods were introduced, using extensive mapping rules to convert variables and numbers into logical expressions for reasoning. However, these methods required too much manual input and were hard to scale.
Expression Tree-Based Methods aim to identify relevant numbers and classify operators between them, building an expression tree from the bottom up. Some methods also consider ratio units to ensure correctness. A more brute-force approach involves enumerating all possible equation trees using integer linear programming. However, as the number of digits increases, the number of trees grows exponentially, making this method computationally intensive.
Equations Application Solver:
For solving equation-based application problems, current methods mainly rely on templates. These methods classify text into predefined systems of equations and infer the placement of unknowns using manually crafted features. While this approach can work for simple cases, its performance drops significantly when the number of questions per template decreases or the template becomes more complex.
Main Contributions:
This paper presents the first attempt to use deep reinforcement learning to design a general framework for solving mathematical problems. It introduces a deep Q-network tailored for application problems, with specific designs for states, actions, rewards, and network structure. The proposed method was validated on major arithmetic application datasets, achieving good results in both solution efficiency and accuracy.
Introduction
An enhanced learning framework for solving mathematical problems is proposed, achieving an accuracy rate of 15%. The figure above illustrates the framework.
Mathematical Application Solver Based on Deep Q Network
The framework presented in this paper is shown in the figure. Given a math application problem, the system first extracts relevant numbers using a digital pattern, then reorders them according to rules. For example, in "3+4*5," the system calculates 4*5 first. The number 5 has the unit "yuan/hour," while 4 has "hour" and 3 has "yuan." In such cases, the system adjusts 4 and 5 to the front of the sequence before constructing the expression tree from the bottom up. Features related to each number and their relationships with the problem are extracted as the state in the reinforcement learning component.
These features are input into the forward neural network of the deep Q network, which outputs Q-values for six actions: "+", "-", reverse "-", "*", "/", and reverse "/". Using epsilon-greedy, the appropriate operator is selected, and the expression tree is built accordingly. The process repeats for other number pairs until no more relevant numbers remain.
State:
For each pair of numbers, the system extracts single-number, two-number, and problem-related features, along with whether the numbers have been used in the expression tree. These features help the network choose the correct operator and determine the hierarchical position of the numbers in the tree.
Action:
Since this paper focuses on simple arithmetic problems, the actions considered are addition, subtraction, multiplication, division, inverse subtraction, and inverse division. Different orders in subtraction and division lead to different results, so inverse operations are included.
Reward Function:
During training, the deep Q network selects the correct action based on the current numbers. If correct, it receives a positive reward; otherwise, a negative penalty is given.
Parameter Learning:
A two-layer forward neural network is used to calculate expected Q-values. The network parameters are updated based on environmental feedback. Experience replay memory stores transitions, and batch sampling is used to update the network. The loss function is defined as follows:
[Image: Loss function formula]
Gradient descent is used to update parameters, minimizing the difference between predicted and target Q-values:
[Image: Gradient update formula]
Algorithm Flow:
[Image: Algorithm flow diagram]
Experiments
Three arithmetic application datasets—AI2, IL, and CC—are used in the experiments. AI2 contains 395 questions with irrelevant numbers involving only addition and subtraction. IL has 562 questions with irrelevant numbers covering all four operations. CC includes 600 questions without irrelevant numbers, involving multi-step operations.
Accuracy Results:
[Image: Accuracy chart]
The proposed method achieves the best results on AI2 and CC datasets. ALGES performs well on IL but poorly on AI2 and CC, indicating better versatility. UnitDep shows little effect on AI2, while Context features improve performance on CC but not as much on AI2. This highlights the limitations of manual features. On the CC dataset, reordering improves performance, while on AI2 and IL, which involve only single-step operations, the effect remains stable.
Additionally, the paper conducted single-step and multi-step breakpoint analysis, showing that the proposed method performs well in multi-step tasks.
Running Time:
[Image: Time comparison]
CC, being a multi-step dataset, takes longer to solve. ALGES spends the most time enumerating candidate trees, while the proposed method ranks second after SVM and ExpTree.
Average Reward and Accuracy Trends:
[Images: Reward and accuracy trends]
Conclusion
This paper introduces the first enhanced learning framework for solving mathematical problems, demonstrating good performance on benchmark datasets. Future work will focus on improving depth and reinforcement learning to develop an automatic solver that reduces reliance on manual features. Additionally, the approach will be tested on larger and more diverse datasets to solve equation-based application problems.
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