The paper introduces a sparse regression method, PDE-FIND, for discovering the governing partial differential equations (PDEs) from time series data in the spatial domain. The method leverages sparsity-promoting techniques to select the most accurate nonlinear and partial derivative terms from a large library of candidate functions, bypassing the combinatorial search through all possible models. It balances model complexity and regression accuracy using Pareto analysis. The method can handle both Eulerian and Lagrangian frameworks and has been successfully applied to various canonical problems in mathematical physics, including the Navier-Stokes equation, the quantum harmonic oscillator, and the diffusion equation. PDE-FIND is demonstrated to be robust and computationally efficient, capable of distinguishing between potentially non-unique dynamical terms using multiple time series with different initial data. This approach provides a promising tool for discovering governing equations and physical laws in complex, parametrized spatio-temporal systems where first-principles derivations are intractable.The paper introduces a sparse regression method, PDE-FIND, for discovering the governing partial differential equations (PDEs) from time series data in the spatial domain. The method leverages sparsity-promoting techniques to select the most accurate nonlinear and partial derivative terms from a large library of candidate functions, bypassing the combinatorial search through all possible models. It balances model complexity and regression accuracy using Pareto analysis. The method can handle both Eulerian and Lagrangian frameworks and has been successfully applied to various canonical problems in mathematical physics, including the Navier-Stokes equation, the quantum harmonic oscillator, and the diffusion equation. PDE-FIND is demonstrated to be robust and computationally efficient, capable of distinguishing between potentially non-unique dynamical terms using multiple time series with different initial data. This approach provides a promising tool for discovering governing equations and physical laws in complex, parametrized spatio-temporal systems where first-principles derivations are intractable.