  Variational methods have been widely used in soft matter physics for both
static and dynamic problems. These methods are mostly based on two variational
principles: the variational principle of minimum free energy (MFEVP) and
Onsager's variational principle (OVP). Our interests lie in the applications of
these variational methods to active matter physics. In our former work [Soft
Matter, 2021, 17, 3634], we have explored the applications of OVP-based
variational methods for the modeling of active matter dynamics. In the present
work, we explore variational (or energy) methods that are based on MFEVP for
static problems in active elastic solids. We show that MFEVP can be used not
only to derive equilibrium equations, but also to develop approximate solution
methods, such as Ritz method, for active solid statics. Moreover, the power of
Ritz-type method can be further enhanced using deep learning methods if we use
deep neural networks to construct the trial solutions of the variational
problems. We then apply these variational methods and the deep Ritz method to
study the spontaneous bending and contraction of a thin active circular plate
that is induced by internal asymmetric active contraction. The circular plate
is found to be bent towards its contracting side. The study of such a simple
toy system gives implications for understanding the morphogenesis of solid-like
confluent cell monolayers. In addition, we introduce a so-called activogravity
length to characterize the importance of gravitational forces relative to
internal active contraction in driving the bending of the active plate. When
the lateral plate dimension is larger than the activogravity length (about 100
micron), gravitational forces become important. Such gravitaxis behaviors at
multicellular scales may play significant roles in the morphogenesis and in the
up-down symmetry broken during tissue development.
