Processing symbolic mathematics algorithmically is an important field of research. It has applications in computer algebra systems and supports researchers as well as applied mathematicians in their daily work. Recently, exploring the ability of neural networks to grasp mathematical concepts has received special attention. One complex task for neural networks is to understand the relation of two mathematical expressions to each other. Despite the advances in learning mathematical relationships, previous studies are limited by small-scale datasets, relatively simple formula construction by few axiomatic rules and even artifacts in the data. With this work, we aim at overcoming these limitations and provide a deeper insight into the representation power of neural networks for classifying mathematical relations. We introduce a novel data generation algorithm to allow for more complex formula compositions and fully include mathematical fields up to high-school level. We research several tree-based and sequential neural architectures for classifying mathematical relations and conduct a systematic analysis of the models against rule-based as well as neural baselines with a focus on varying dataset complexity, generalization abilities, and understanding of syntactical patterns. Our findings show the potential of deep learning models to distinguish high-school level mathematical concepts.

Recent studies have shown that many deep metric learning loss functions perform very similarly under the same experimental conditions. One potential reason for this unexpected result is that all losses let the network focus on similar image regions or properties. In this paper, we investigate this by conducting a two-step analysis to extract and compare the learned visual features of the same model architecture trained with different loss functions: First, we compare the learned features on the pixel level by correlating saliency maps of the same input images. Second, we compare the clustering of embeddings for several image properties, e.g. object color or illumination. To provide independent control over these properties, photo-realistic 3D car renders similar to images in the Cars196 dataset are generated. In our analysis, we compare 14 pretrained models from a recent study and find that, even though all models perform similarly, different loss functions can guide the model to learn different features. We especially find differences between classification and ranking based losses. Our analysis also shows that some seemingly irrelevant properties can have significant influence on the resulting embedding. We encourage researchers from the deep metric learning community to use our methods to get insights into the features learned by their proposed methods.

To the present day, computer algebra systems used for calculating mathematical relations on a symbolic level are mainly rule-based systems. Only recently, deep learning has been applied to the task of symbolic mathematics. Researchers succeeded in developing neural networks which can do symbolic calculations of the equivalence of multiple pairing of equations. However, the generator used to create mathematical terms in previous research has several drawbacks and as with all deep learning tasks, the quality of the data used for training the models has a significant impact on the quality of the results. In this work, we propose a new generator for polynomials. We use it to train several recursive neural networks to recognize the equivalence, derivative, or variable substitution between pairs of polynomials for the first time. Our results indicate that these mathematical relations are identifiable with the help of deep learning.