Symbolic Integration by Integrating Learning Models With Different Strengths and Weaknesses

Abstract

Integration is indispensable, not only in mathematics, but also in a wide range of other fields. A deep learning method has recently been developed and shown to be capable of integrating mathematical functions that could not previously be integrated on a computer. However, that method treats integration as equivalent to natural language translation and does not reflect mathematical information. In this study, we adjusted the learning model to take mathematical information into account and developed a wide range of learning models that learn the order of numerical operations more robustly. In this way, we achieved a 98.80% correct answer rate with symbolic integration, a higher rate than that of any existing method. We judged the correctness of the integration based on whether the derivative of the primitive function was consistent with the integrand. By building an integrated model based on this strategy, we achieved a 99.79% rate of correct answers with symbolic integration. In summary, we have developed a more accurate method of selecting the correct model than the existing method by judging the result of symbolic integration based on whether the output of the model equals the input formula when the output is differentiated.

Keywords

• Deep learning
• encoder-decoder
• supervised learning
• symbolic integration
• translation model