Learning Topological Features of $\widehat Z$-invariants

Machine learning and data analysis techniques have recently emerged as powerful tools for identifying patterns and formulating conjectures in mathematical research, most notably in the field of low-dimensional topology. In this paper, we initiate a systematic approach to handling mathematical data structured as (truncated) infinite $q$-series, or equivalently, infinite series of integers. To apply this data analysis pipeline, we construct a comprehensive dataset of $\widehat{Z}$-invariants (homological blocks) for plumbed 3-manifolds. We demonstrate that neural networks can reliably extract essential topological information, such as homology class and underlying graph structure, directly from the $q$-series coefficients. A central feature of our methodology is a focus on interpretability; by contrasting local gradient sensitivity with global feature relevance, we reveal that the networks learn to bypass complex topological rules in favor of specific spectral and geometric proxies. Finally, we apply this pipeline to probe homology cobordism, discovering a high-accuracy predictive relationship between the $\widehat{Z}$-invariant exponents and the Heegaard Floer $d$-invariant (correction term). These results suggest that $\widehat{Z}$-invariants capture subtle geometric information regarding cobordism equivalences, warranting a new direction for the study of quantum invariants.

Topological Effects in Neural Network Field Theory

Neural network field theory formulates field theory as a statistical ensemble of fields defined by a network architecture and a density on its parameters. We extend the construction to topological settings via the inclusion of discrete parameters that label the topological quantum number. We recover the Berezinskii--Kosterlitz--Thouless transition, including the spin-wave critical line and the proliferation of vortices at high temperatures. We also verify the T-duality of the bosonic string, showing invariance under the exchange of momentum and winding on $S^1$, the transformation of the sigma model couplings according to the Buscher rules on constant toroidal backgrounds, the enhancement of the current algebra at self-dual radius, and non-geometric T-fold transition functions.

Virasoro Symmetry in Neural Network Field Theories

Neural Network Field Theories (NN-FTs) can realize global conformal symmetries via embedding space architectures. These models describe Generalized Free Fields (GFFs) in the infinite width limit. However, they typically lack a local stress-energy tensor satisfying conformal Ward identities. This presents an obstruction to realizing infinite-dimensional, local conformal symmetry typifying 2d Conformal Field Theories (CFTs). We present the first construction of an NN-FT that encodes the full Virasoro symmetry of a 2d CFT. We formulate a neural free boson theory with a local stress tensor $T(z)$ by properly choosing the architecture and prior distribution of network parameters. We verify the analytical results through numerical simulation; computing the central charge and the scaling dimensions of vertex operators. We then construct an NN realization of a Majorana Fermion and an $\mathcal{N}=(1,1)$ scalar multiplet, which then enables an extension of the formalism to include super-Virasoro symmetry. Finally, we extend the framework by constructing boundary NN-FTs that preserve (super-)conformal symmetry via the method of images.

Conformal Defects in Neural Network Field Theories

Neural Network Field Theories (NN-FTs) represent a novel construction of arbitrary field theories, including those of conformal fields, through the specification of the network architecture and prior distribution for the network parameters. In this work, we present a formalism for the construction of conformally invariant defects in these NN-FTs. We demonstrate this new formalism in two toy models of NN scalar field theories. We develop an NN interpretation of an expansion akin to the defect OPE in two-point correlation functions in these models.