  The partial differential equation (PDE) plays a significantly important role
in many fields of science and engineering. The conventional case of the
derivation of PDE mainly relies on first principles and empirical observation.
However, the development of machine learning technology allows us to mine
potential control equations from the massive amounts of stored data in a fresh
way. Although there has been considerable progress in the data-driven discovery
of PDE, the extant literature mostly focuses on the improvements of discovery
methods, without substantial breakthroughs in the discovery process itself,
including the principles for the construction of candidates and how to
incorporate physical priors. In this paper, through rigorous derivation of
formulas, novel physically enhanced machining learning discovery methods for
control equations: GSNN (Galileo Symbolic Neural Network) and LSNN (Lorentz
Symbolic Neural Network) are firstly proposed based on Galileo invariance and
Lorentz invariance respectively, setting forth guidelines for building the
candidates of discovering equations. The adoption of mandatory embedding of
physical constraints is fundamentally different from PINN in the form of the
loss function, thus ensuring that the designed Neural Network strictly obeys
the physical prior of invariance and enhancing the interpretability of the
network. By comparing the results with PDE-NET in numerical experiments of
Burgers equation and Sine-Gordon equation, it shows that the method presented
in this study has better accuracy, parsimony, and interpretability.
