  Discovering governing equations from data is critical for diverse scientific
disciplines as they can provide insights into the underlying phenomenon of
dynamic systems. This work presents a new representation for governing
equations by designing the Mathematical Operation Network (MathONet) with a
deep neural network-like hierarchical structure. Specifically, the MathONet is
stacked by several layers of unary operations (e.g., sin, cos, log) and binary
operations (e.g., +,-), respectively. An initialized MathONet is typically
regarded as a super-graph with a redundant structure, a sub-graph of which can
yield the governing equation. We develop a sparse group Bayesian learning
algorithm to extract the sub-graph by employing structurally constructed priors
over the redundant mathematical operations. By demonstrating the chaotic Lorenz
system, Lotka-Volterra system, and Kolmogorov-Petrovsky-Piskunov system, the
proposed method can discover the ordinary differential equations (ODEs) and
partial differential equations (PDEs) from the observations given limited
mathematical operations, without any prior knowledge on possible expressions of
the ODEs and PDEs.
