  Hardness is a materials' property with implications in several industrial
fields, including oil and gas, manufacturing, and others. However, the
relationship between this macroscale property and atomic (i.e., microscale)
properties is unknown and in the last decade several models have unsuccessfully
tried to correlate them in a wide range of chemical space. The understanding of
such relationship is of fundamental importance for discovery of harder
materials with specific characteristics to be employed in a wide range of
fields. In this work, we have found a physical descriptor for Vickers hardness
using a symbolic-regression artificial-intelligence approach based on
compressed sensing. SISSO (Sure Independence Screening plus Sparsifying
Operator) is an artificial-intelligence algorithm used for discovering simple
and interpretable predictive models. It performs feature selection from up to
billions of candidates obtained from several primary features by applying a set
of mathematical operators. The resulting sparse SISSO model accurately
describes the target property (i.e., Vickers hardness) with minimal complexity.
We have considered the experimental values of hardness for binary, ternary, and
quaternary transition-metal borides, carbides, nitrides, carbonitrides,
carboborides, and boronitrides of 61 materials, on which the fitting was
performed. The found descriptor is a non-linear function of the microscopic
properties, with the most significant contribution being from a combination of
Voigt-averaged bulk modulus, Poisson's ratio, and Reuss-averaged shear modulus.
Results of high-throughput screening of 635 candidate materials using the found
descriptor suggest the enhancement of material's hardness through mixing with
harder yet metastable structures (e.g., metastable VN, TaN, ReN$_2$,
Cr$_3$N$_4$, and ZrB$_6$ all exhibit high hardness).
