This paper introduces a data-driven method for discovering the governing partial differential equations (PDEs) of a system from time series measurements in the spatial domain. The method uses sparse regression to identify the most relevant terms in the PDE that accurately represent the data, avoiding an exhaustive search through all possible candidate models. It balances model complexity and accuracy by selecting a parsimonious model via Pareto analysis. The method can be applied in either Eulerian or Lagrangian frameworks, where sensors are fixed or move with the dynamics. It is computationally efficient, robust, and has been demonstrated on various canonical problems in mathematical physics, including Navier-Stokes, quantum harmonic oscillator, and diffusion equations. The method can distinguish between potentially non-unique dynamical terms by using multiple time series with different initial data. For example, it can differentiate between a linear wave equation and the Korteweg-de Vries equation for a traveling wave. The method provides a promising technique for discovering governing equations and physical laws in parametrized spatio-temporal systems where first-principles derivations are intractable. The method, called PDE-FIND, uses sparse regression to identify the terms of the governing PDE that most accurately represent the data from a large library of potential candidate functions. It begins by collecting spatial and time series data into a single column vector and constructing a library of candidate linear and nonlinear terms and partial derivatives for the PDE. Each column of the library contains the values of a candidate term across all gridpoints. The PDE is then identified by sparse regression, which selects the most informative terms. The method uses polynomial interpolation for differentiating noisy data and applies sequential threshold ridge regression (STRidge) to find a sparse approximation to the solution vector. The method has been successfully applied to various canonical models in mathematical physics, including the diffusion equation, KdV equation, and quantum harmonic oscillator. The method is able to discover each physical system even if significantly subsampled spatially, and it has broad applicability in modern applications where first-principles derivations may be intractable. The ability to discover physical laws instead of approximate, low-dimensional subspaces enables significantly improved future state predictions and the discovery of parametric dependencies. The method represents a significant paradigm shift compared to most data-driven, machine learning architectures where accurate predictions can only be made near parameter regimes where the data was sampled.This paper introduces a data-driven method for discovering the governing partial differential equations (PDEs) of a system from time series measurements in the spatial domain. The method uses sparse regression to identify the most relevant terms in the PDE that accurately represent the data, avoiding an exhaustive search through all possible candidate models. It balances model complexity and accuracy by selecting a parsimonious model via Pareto analysis. The method can be applied in either Eulerian or Lagrangian frameworks, where sensors are fixed or move with the dynamics. It is computationally efficient, robust, and has been demonstrated on various canonical problems in mathematical physics, including Navier-Stokes, quantum harmonic oscillator, and diffusion equations. The method can distinguish between potentially non-unique dynamical terms by using multiple time series with different initial data. For example, it can differentiate between a linear wave equation and the Korteweg-de Vries equation for a traveling wave. The method provides a promising technique for discovering governing equations and physical laws in parametrized spatio-temporal systems where first-principles derivations are intractable. The method, called PDE-FIND, uses sparse regression to identify the terms of the governing PDE that most accurately represent the data from a large library of potential candidate functions. It begins by collecting spatial and time series data into a single column vector and constructing a library of candidate linear and nonlinear terms and partial derivatives for the PDE. Each column of the library contains the values of a candidate term across all gridpoints. The PDE is then identified by sparse regression, which selects the most informative terms. The method uses polynomial interpolation for differentiating noisy data and applies sequential threshold ridge regression (STRidge) to find a sparse approximation to the solution vector. The method has been successfully applied to various canonical models in mathematical physics, including the diffusion equation, KdV equation, and quantum harmonic oscillator. The method is able to discover each physical system even if significantly subsampled spatially, and it has broad applicability in modern applications where first-principles derivations may be intractable. The ability to discover physical laws instead of approximate, low-dimensional subspaces enables significantly improved future state predictions and the discovery of parametric dependencies. The method represents a significant paradigm shift compared to most data-driven, machine learning architectures where accurate predictions can only be made near parameter regimes where the data was sampled.