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Equivariant network architectures are a well-established tool for predicting invariant or equivariant quantities. However, almost all learning problems considered in this context feature a global symmetry, i.e. each point of the underlying…
Neural networks that satisfy invariance with respect to input permutations have been widely studied in machine learning literature. However, in many applications, only a subset of all input permutations is of interest. For heterogeneous…
We propose a framework for the specification of behaviour-preserving reconfigurations of systems modelled as Petri nets. The framework is based on open nets, a mild generalisation of ordinary Place/Transition nets suited to model open…
In these lectures, we discuss two approaches to studying orbit spaces of algebraic Lie groups. Due to algebraic approach orbit space, or quotient, is an algebraic manifold, while from the differential viewpoint a quotient is a differential…
Reversible computation is an unconventional form of computing that extends the standard forward-only mode of computation with the ability to execute a sequence of operations in reverse at any point during computation. As such, in this…
Neural networks embed the geometric structure of a data manifold lying in a high-dimensional space into latent representations. Ideally, the distribution of the data points in the latent space should depend only on the task, the data, the…
The crucial role played by the underlying symmetries of high energy physics and lattice field theories calls for the implementation of such symmetries in the neural network architectures that are applied to the physical system under…
We give a definition of $\mathsf{Q}$-net, a generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in…
We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant…
We present a unified framework to describe lattice gauge theories by means of tensor networks: this framework is efficient as it exploits the high amount of local symmetry content native of these systems describing only the gauge invariant…
Let G be a simple algebraic group. Labelled trivalent graphs called webs can be used to product invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine…
We present an embedding of Petri nets into B abstract systems. The embedding is achieved by translating both the static structure (modelling aspect) and the evolution semantics of Petri nets. The static structure of a Petri-net is captured…
Metanetworks are neural architectures designed to operate directly on pretrained weights to perform downstream tasks. However, the parameter space serves only as a proxy for the underlying function class, and the parameter-function mapping…
We introduce a notion of a sheaf of vector spaces on a graph, and develop the foundations of homology theories for such sheaves. One sheaf invariant, its "maximum excess," has a number of remarkable properties. It has a simple definition,…
In past few decades, tensor algebra also known as multi-linear algebra has been developed and customized as a tool to be used for various engineering applications. In particular, with the help of a special form of tensor contracted product,…
Petri nets proved useful to describe various real-world systems, but many of their properties are very hard to check. To alleviate this difficulty, subclasses are often considered. The class of weighted marked graphs with relaxed place…
Conventional machine learning algorithms have traditionally been designed under the assumption that input data follows a vector-based format, with an emphasis on vector-centric paradigms. However, as the demand for tasks involving set-based…
Linear constraint transformation is an essential step to solve the forbidden state problem in Petri nets that contain uncontrollable transitions. This work studies the equivalent transformation from a legal-marking set to its…
Loop invariants are properties of a program loop that hold before and after each iteration of the loop. They are often employed to verify programs and ensure that algorithms consistently produce correct results during execution.…
This paper presents a transformative framework for artificial neural networks over graded vector spaces, tailored to model hierarchical and structured data in fields like algebraic geometry and physics. By exploiting the algebraic…