Data-driven Einstein-dilaton model for pure Yang-Mills thermodynamics and glueball spectrum

We develop a machine learning assisted holographic model that consistently describes both the equation of state and glueball spectrum of pure Yang-Mills theory, achieved through neural network reconstruction of Einstein-dilaton gravity. Our framework incorporates key nonperturbative constraints of lattice QCD data: The ground ( <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mn>0</a:mn> <a:mrow> <a:mo>+</a:mo> <a:mo>+</a:mo> </a:mrow> </a:msup> </a:math> ) and first-excited ( <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mn>0</c:mn> <c:mrow> <c:mo>+</c:mo> <c:mo>+</c:mo> <c:mo>*</c:mo> </c:mrow> </c:msup> </c:math> ) scalar glueball masses pins down the infrared (IR) geometry, while entropy density data anchors the ultraviolet (UV) behavior of the metric. A multistage neural network optimization then yields the full gravitational dual—warp factor <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>A</e:mi> <e:mo stretchy="false">(</e:mo> <e:mi>z</e:mi> <e:mo stretchy="false">)</e:mo> </e:math> and dilaton field <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi mathvariant="normal">Φ</i:mi> <i:mo stretchy="false">(</i:mo> <i:mi>z</i:mi> <i:mo stretchy="false">)</i:mo> </i:math> —that satisfies both spectroscopic and thermodynamic constraints. The resulting model accurately reproduces the deconfinement phase transition thermodynamics (pressure, energy density, trace anomaly) and predicts higher glueball excitations ( <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:msup> <n:mn>0</n:mn> <n:mrow> <n:mo>+</n:mo> <n:mo>+</n:mo> <n:mo>*</n:mo> <n:mo>*</n:mo> </n:mrow> </n:msup> </n:math> , <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:msup> <p:mn>0</p:mn> <p:mrow> <p:mo>+</p:mo> <p:mo>+</p:mo> <p:mo>*</p:mo> <p:mo>*</p:mo> <p:mo>*</p:mo> </p:mrow> </p:msup> </p:math> ) consistent with available lattice calculations. This work establishes a new paradigm for data-driven holographic reconstruction, solving the long-standing challenge of unified description of confinement thermodynamics and spectroscopy.