  With the recent success of representation learning methods, which includes
deep learning as a special case, there has been considerable interest in
developing techniques that incorporate known physical constraints into the
learned representation. As one example, in many applications that involve a
signal propagating through physical media (e.g., optics, acoustics, fluid
dynamics, etc), it is known that the dynamics of the signal must satisfy
constraints imposed by the wave equation. Here we propose a matrix
factorization technique that decomposes such signals into a sum of components,
where each component is regularized to ensure that it {nearly} satisfies wave
equation constraints. Although our proposed formulation is non-convex, we prove
that our model can be efficiently solved to global optimality. Through this
line of work we establish theoretical connections between wave-informed
learning and filtering theory in signal processing. We further demonstrate the
application of this work on modal analysis problems commonly arising in
structural diagnostics and prognostics.
