Symbolic Integration Algorithm Selection with Machine Learning: LSTMs vs Tree LSTMs
(2404.14973)Abstract
Computer Algebra Systems (e.g. Maple) are used in research, education, and industrial settings. One of their key functionalities is symbolic integration, where there are many sub-algorithms to choose from that can affect the form of the output integral, and the runtime. Choosing the right sub-algorithm for a given problem is challenging: we hypothesise that Machine Learning can guide this sub-algorithm choice. A key consideration of our methodology is how to represent the mathematics to the ML model: we hypothesise that a representation which encodes the tree structure of mathematical expressions would be well suited. We trained both an LSTM and a TreeLSTM model for sub-algorithm prediction and compared them to Maple's existing approach. Our TreeLSTM performs much better than the LSTM, highlighting the benefit of using an informed representation of mathematical expressions. It is able to produce better outputs than Maple's current state-of-the-art meta-algorithm, giving a strong basis for further research.
Overview
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The paper analyzes the effectiveness of LSTM and TreeLSTM models in selecting sub-algorithms for symbolic integration in Maple CAS, focusing on the structural data representation of mathematical expressions.
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TreeLSTM models, which integrate the tree structure of mathematical expressions, demonstrated superior performance in predicting sub-algorithms that reduce the integration result length more effectively than LSTMs and Maple's current algorithms.
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TreeLSTM showed a significantly higher optimal output rate and better generalization capabilities but required longer training times and has room for improvement in handling edge cases and diverse data.
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Future research directions include expanding training data, tuning hyperparameters, and comparing TreeLSTMs with other structured data machine learning architectures like graph neural networks.
Improved Sub-algorithm Selection in Symbolic Integration using TreeLSTMs
Overview
The paper presents a detailed study comparing the performance of Long Short-Term Memory (LSTM) networks and Tree-structured LSTM (TreeLSTM) models for the task of selecting sub-algorithms in symbolic integration within a Computer Algebra System (CAS), specifically Maple. The focus lies on evaluating whether embedding data that preserves the hierarchical tree structure of mathematical expressions improves the selection process over traditional sequence-based LSTM representations. Both models were trained to predict the optimal sub-algorithm that produces the shortest possible integration result in terms of output expression length, a critical metric for CAS users who benefit from simpler, more manageable results.
Approach to Machine Learning Models
Model Implementations
- LSTM: Employs a traditional sequence processing approach, where the input is a linear sequence of tokens representing a mathematical expression. This model acts as a baseline for evaluating the effectiveness of sequence-based learning in this context.
- TreeLSTM: Incorporates the natural tree structure of mathematical expressions into the learning process. The model processes nodes depending on their positions within the expression's tree, allowing it to potentially leverage structural information in making predictions.
Objective and Dataset
- Objective: Minimize the DAG (Directed Acyclic Graph) representation size of the output expression.
- Dataset: Utilized a variety of data generation methods (FWD, BWD, IBP, RISCH, SUB) to produce 100,000 samples, balancing diversity and complexity of expressions.
Results and Analysis
Performance Metrics
- Prediction Accuracy: TreeLSTM outperformed both the baseline LSTM and Maple's existing meta-algorithm in identifying the optimal sub-algorithm that minimizes the length of the output expression.
- Processing Time: TreeLSTM took significantly longer to train than LSTM (312s vs. 178s on average per model), which is expected given its more complex architecture.
Unique Contributions
- TreeLSTM not only yielded a higher percentage of optimal outputs (84.6%) compared to LSTM (56.8%) and the existing meta-algorithm (60.5%), but it also demonstrated superior capability in generalizing from training data to unseen data from the Maple test suite.
- Analysis of edge cases where TreeLSTM underperformed revealed potential areas for further optimization, suggesting that alterations in training data diversity or model hyperparameters might yield further improvements.
Future Directions
- Expansion of Training Data: Incorporating a broader array of expressions and integrating more complex structures could help in enhancing the model's predictive capacity and generalizability.
- Hyperparameter Optimization: More comprehensive tuning of model parameters, particularly for the TreeLSTM, might improve performance metrics.
- Comparative Studies: Future work could explore comparisons with other machine learning architectures tailored to structured data, such as graph neural networks, to evaluate their efficacy in a symbolic integration context.
Implications
The successful application of TreeLSTMs represents a significant step forward in the application of ML to symbolic integration, suggesting that the structural representation of mathematical data plays a crucial role in optimizing algorithm selection. Practical implications of this research extend to improved user experience in CAS environments through faster and more accurate symbolic integration capabilities, potentially transforming workflows in academic, research, and industrial settings where symbolic computation is prevalent. Further, the theoretical implications of using tree-embedded data in learning frameworks can catalyze more explorations into the application areas of TreeLSTMs and similar architectures.
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