  Topological phases of matter are conventionally characterized by the
bulk-boundary correspondence in Hermitian systems: The topological invariant of
the bulk in $d$ dimensions corresponds to the number of $(d-1)$-dimensional
boundary states. By extension, higher-order topological insulators reveal a
bulk-edge-corner correspondence, such that $n$-th order topological phases
feature $(d-n)$-dimensional boundary states. The advent of non-Hermitian
topological systems sheds new light on the emergence of the non-Hermitian skin
effect (NHSE) with an extensive number of boundary modes under open boundary
conditions. Still, the higher-order NHSE remains largely unexplored,
particularly in the experiment. We introduce an unsupervised approach --
physics-graph-informed machine learning (PGIML) -- to enhance the data mining
ability of machine learning with limited domain knowledge. Through PGIML, we
experimentally demonstrate the second-order NHSE in a two-dimensional
non-Hermitian topolectrical circuit. The admittance spectra of the circuit
exhibit an extensive number of corner skin modes and extreme sensitivity of the
spectral flow to the boundary conditions. The violation of the conventional
bulk-boundary correspondence in the second-order NHSE implies that modification
of the topological band theory is inevitable in higher dimensional
non-Hermitian systems.
