  Recent advances in the field of data-driven dynamics allow for the discovery
of ODE systems using state measurements. One approach, known as Sparse
Identification of Nonlinear Dynamics (SINDy), assumes the dynamics are sparse
within a predetermined basis in the states and finds the expansion coefficients
through linear regression with sparsity constraints. This approach requires an
accurate estimation of the state time derivatives, which is not necessarily
possible in the high-noise regime without additional constraints. We present an
approach called Derivative-based SINDy (DSINDy) that combines two novel methods
to improve ODE recovery at high noise levels. First, we denoise the state
variables by applying a projection operator that leverages the assumed basis
for the system dynamics. Second, we use a second order cone program (SOCP) to
find the derivative and governing equations simultaneously. We derive
theoretical results for the projection-based denoising step, which allow us to
estimate the values of hyperparameters used in the SOCP formulation. This
underlying theory helps limit the number of required user-specified parameters.
We present results demonstrating that our approach leads to improved system
recovery for the Van der Pol oscillator, the Duffing oscillator, and the
R\"{o}ssler attractor.
