  Using symbolic regression to discover physical laws from observed data is an
emerging field. In previous work, we combined genetic algorithm (GA) and
machine learning to present a data-driven method for discovering a wave
equation. Although it managed to utilize the data to discover the
two-dimensional (x,z) acoustic constant-density wave equation
u_tt=v^2(u_xx+u_zz) (subscripts of the wavefield, u, are second derivatives in
time and space) in a homogeneous medium, it did not provide the complete
equation form, where the velocity term is represented by a coefficient rather
than directly given by v^2. In this work, we redesign the framework, encoding
both velocity information and candidate functional terms simultaneously. Thus,
we use GA to simultaneously evolve the candidate functional and coefficient
terms in the library. Also, we consider here the physics rationality and
interpretability in the randomly generated potential wave equations, by
ensuring that both-hand sides of the equation maintain balance in their
physical units. We demonstrate this redesigned framework using the acoustic
wave equation as an example, showing its ability to produce physically
reasonable expressions of wave equations from noisy and sparsely observed data
in both homogeneous and inhomogeneous media. Also, we demonstrate that our
method can effectively discover wave equations from a more realistic
observation scenario.
