  Identifying differential operators from data is essential for the
mathematical modeling of complex physical and biological systems where massive
datasets are available. These operators must be stable for accurate predictions
for dynamics forecasting problems. In this article, we propose a novel
methodology for learning sparse differential operators that are theoretically
linearly stable by solving a constrained regression problem. These underlying
constraints are obtained following linear stability for dynamical systems. We
further extend this approach for learning nonlinear differential operators by
determining linear stability constraints for linearized equations around an
equilibrium point. The applicability of the proposed method is demonstrated for
both linear and nonlinear partial differential equations such as 1-D scalar
advection-diffusion equation, 1-D Burgers equation and 2-D advection equation.
The results indicated that solutions to constrained regression problems with
linear stability constraints provide accurate and linearly stable sparse
differential operators.
