Even if we know the physical laws that govern a dynamical system, we may seek to infer a simplified set of dynamics from measurements. Here we build a nonlinear dynamical system by identifying the coefficients attached to each term in a sum of nonlinear func- tions of the states. Many standard approaches like Sparse Identification of Nonlinear Dynamics (SINDy) identify these coefficients by minimizing the least squares mismatch in this differential equation while promoting coefficient sparsity. Unfortunately, stan- dard SINDy amplifies the effect of noise emerging from both derivative estimation and the nonlinear basis functions yielding suboptimal estimates of the dynamical system. Here we present novel approaches that perform simultaneous state variable denoising and equation recovery that lead to considerably more robust recovery at relatively large noise levels. At the core of these approaches are constraint implementation of the equation recovery as well as a reformulation of the optimization problem where the state time derivatives are solved for as the primary variables.