# Symbolic integration by integrating learning models with different strengths and weaknesses

arXiv: https://arxiv.org/abs/2103.05497 · HTML: https://mlsanity.com/paper/2103.05497/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 27 references that are in the map.
- First posted on arXiv 9 Mar 2021.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [What Does BERT Look At? An Analysis of BERT's Attention](https://mlsanity.com/paper/1906.04341/) (2019)
- [Analysing Mathematical Reasoning Abilities of Neural Models](https://mlsanity.com/paper/1904.01557/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [On the Turing Completeness of Modern Neural Network Architectures](https://mlsanity.com/paper/1901.03429/) (2019)
- [Neural Symbolic Machines: Learning Semantic Parsers on Freebase with Weak Supervision](https://mlsanity.com/paper/1611.00020/) (2016)
- [Improve Transformer Models with Better Relative Position Embeddings](https://mlsanity.com/paper/2009.13658/) (2020)
- [Adversarial Neural Machine Translation](https://mlsanity.com/paper/1704.06933/) (2017)

## Built on by (most cited first, 1 of 1)

- [SC-Square: Future Progress with Machine Learning?](https://mlsanity.com/paper/2209.04361/) (2022)

## Abstract (arXiv)

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.