Neuro-Symbolic Learning for Galois Groups: Unveiling Probabilistic Trends in Polynomials

We review, for a general audience, a variety of recent experiments on extracting structure from machine-learning mathematical data that have been compiled over the years. Focusing on supervised machine-learning on labeled data from different fields ranging from geometry to representation theory, from combinatorics to number theory, we present a comparative study of the accuracies on different problems. The paradigm should be useful for conjecture formulation, finding more eff...

Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated tasks in mathematics, such as symbolic integration and solving differential equations. We propose a syntax for representing mathematical problems, and methods for generating large datasets that can be used to train sequence-to-sequence models...

Quantum entanglement, a cornerstone of quantum mechanics, remains challenging to classify, particularly in multipartite systems. Here, we present a new interpretation of entanglement classification by revealing a profound connection to Galois groups, the algebraic structures governing polynomial symmetries. This approach not only uncovers hidden geometric relationships between entangled quantum states and polynomial roots but also introduces a method for quantifying entanglem...

Group theory has been used in machine learning to provide a theoretically grounded approach for incorporating known symmetry transformations in tasks from robotics to protein modeling. In these applications, equivariant neural networks use known symmetry groups with predefined representations to learn over geometric input data. We propose MatrixNet, a neural network architecture that learns matrix representations of group element inputs instead of using predefined representat...

Current advances in Artificial Intelligence (AI) and Machine Learning (ML) have achieved unprecedented impact across research communities and industry. Nevertheless, concerns about trust, safety, interpretability and accountability of AI were raised by influential thinkers. Many have identified the need for well-founded knowledge representation and reasoning to be integrated with deep learning and for sound explainability. Neural-symbolic computing has been an active area of ...

Neural networks adapt very well to distributed and continuous representations, but struggle to generalize from small amounts of data. Symbolic systems commonly achieve data efficient generalization by exploiting modularity to benefit from local and discrete features of a representation. These features allow symbolic programs to be improved one module at a time and to experience combinatorial growth in the values they can successfully process. However, it is difficult to desig...

We introduce the Deep Symbolic Network (DSN) model, which aims at becoming the white-box version of Deep Neural Networks (DNN). The DSN model provides a simple, universal yet powerful structure, similar to DNN, to represent any knowledge of the world, which is transparent to humans. The conjecture behind the DSN model is that any type of real world objects sharing enough common features are mapped into human brains as a symbol. Those symbols are connected by links, representi...

We introduce a method to design a computationally efficient $G$-invariant neural network that approximates functions invariant to the action of a given permutation subgroup $G \leq S_n$ of the symmetric group on input data. The key element of the proposed network architecture is a new $G$-invariant transformation module, which produces a $G$-invariant latent representation of the input data. This latent representation is then processed with a multi-layer perceptron in the net...

State-of-the-art neurosymbolic learning systems use probabilistic reasoning to guide neural networks towards predictions that conform to logical constraints over symbols. Many such systems assume that the probabilities of the considered symbols are conditionally independent given the input to simplify learning and reasoning. We study and criticise this assumption, highlighting how it can hinder optimisation and prevent uncertainty quantification. We prove that loss functions ...

Neurosymbolic AI aims to integrate deep learning with symbolic AI. This integration has many promises, such as decreasing the amount of data required to train a neural network, improving the explainability and interpretability of answers given by models and verifying the correctness of trained systems. We study neurosymbolic learning, where we have both data and background knowledge expressed using symbolic languages. How do we connect the symbolic and neural components to co...