  Understanding ENSO dynamics has tremendously improved over the past decades.
However, one aspect still poorly understood or represented in conceptual models
is the ENSO diversity in spatial pattern, peak intensity, and temporal
evolution. In this paper, a physics-informed auto-learning framework is
developed to derive ENSO stochastic conceptual models with varying degrees of
freedom. The framework is computationally efficient and easy to apply. Once the
state vector of the target model is set, causal inference is exploited to build
the right-hand side of the equations based on a mathematical function library.
Fundamentally different from standard nonlinear regression, the auto-learning
framework provides a parsimonious model by retaining only terms that improve
the dynamical consistency with observations. It can also identify crucial
latent variables and provide physical explanations. Exploiting a realistic
six-dimensional reference recharge oscillator-based ENSO model, a hierarchy of
three- to six-dimensional models is derived using the auto-learning framework
and is systematically validated by a unified set of validation criteria
assessing the dynamical and statistical features of the ENSO diversity. It is
shown that the minimum model characterizing ENSO diversity is four-dimensional,
with three interannual variables describing the western Pacific thermocline
depth, the eastern and central Pacific sea surface temperatures (SSTs), and one
intraseasonal variable for westerly wind events. Without the intraseasonal
variable, the resulting three-dimensional model underestimates extreme events
and is too regular. The limited number of weak nonlinearities in the model are
essential in reproducing the observed extreme El Ni\~nos and nonlinear
relationship between the eastern and western Pacific SSTs.
