We propose a novel, machine-learning-based framework for constructing lattice fermions using Physics-Informed Neural Networks (PINNs). Our approach treats the formulation of the Dirac operator as an optimization problem guided by physical requirements, such as symmetries, locality and doubler-decoupling conditions. We first demonstrate that, when trained to satisfy the Ginsparg-Wilson (GW) relation as a soft constraint, a neural network reproduces the overlap fermion operator to high numerical accuracy and learns an effective sign-function mapping without explicitly using a prescribed polynomial or rational approximation. Secondly, we extend the framework from operator construction to machine-assisted algebraic discovery. Within a generalized polynomial ansatz, the network autonomously drives higher-order terms to zero and recovers the standard Ginsparg-Wilson relation. Remarkably, by changing the initial search bias, the same framework also finds a distinct solution corresponding to a Fujikawa-type generalized GW relation.

Studying phase transitions in interacting quantum field theories generally requires the numerical study of the dynamical system on a large lattice, which is, in most cases, computationally very challenging. In this work an alternative method is proposed to solve Euclidean path integrals in quantum field theories, using radial basis function-type neural networks. The method allows us to approximate observables in a very efficient manner, taking only seconds to do calculations that would otherwise take hours or even days with other existing methods. The model is used to describe phase transitions in the scalar $\phi^4$ theory for a wide range of coupling strength. The obtained phase transition line is compared to previous lattice results, giving very good agreement between them.

A fundamental issue in the renormalization-group (RG) theory of critical phenomena concerns the allowed values of critical exponents that are consistent with the continuous nature of a phase transition. Here we conjecture a lower bound for the length-scale exponent $\nu$, which should hold for the large class of continuous transitions associated with $d$-dimensional Landau-Ginzburg-Wilson (LGW) $\Phi^4$ theories with a multicomponent scalar field ${\varphi}$ and a unique ${\varphi}\cdot {\varphi}$ quadratic term (including some extensions with fermionic and gauge fields), describing many universality classes of critical phenomena. If $\Delta_\varphi=(d-2+\eta)/2$ is the dimension of the order-parameter field ${\varphi}$, and $\Delta_\varepsilon=d-1/\nu$ is the RG dimension of the energy operator $\varepsilon$, which can be identified with $[{\varphi}\cdot {\varphi}]$ (the squared field with a proper subtraction of the mixing with the identity), we conjecture the inequality $\Delta_\varepsilon \ge 2 \Delta_\varphi$, which implies $\nu \ge (2-\eta)^{-1}$ and $\gamma = (2-\eta)\nu\ge 1$. These inequalities are supported by general arguments for ferromagnetic lattice models, by $\epsilon$-expansion results for generic LGW $\Phi^4$ theories close to four dimensions, exact relations for two-dimensional minimal conformal field theories, and are consistent with all further known (numerical, perturbative, and exact) results for LGW $\Phi^4$ theories. In particular, since unitarity requires $\eta\ge 0$, the above inequality implies $\nu\ge 1/2$ for unitary theories. This lower bound is more restrictive than $\nu > 1/d$, derived by noting that $\nu=1/d$ characterizes the singular finite-size behavior at first-order transitions.

The profile of the pion valence quark distribution function (DF) remains controversial. Working from the concepts of QCD effective charges and generalised parton distributions, we show that since the pion elastic electromagnetic form factor is well approximated by a monopole, then, at large light-front momentum fraction, the pion valence quark DF is a convex function described by a large-$x$ power law that is practically consistent with expectations based on quantum chromodynamics.