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The sparse identification of nonlinear dynamics (SINDy) is a regressionframework for the discovery of parsimonious dynamic models and governingequations from time-series data. As with all system identification methods,noisy measurements compromise the accuracy and robustness of the modeldiscovery procedure. In this work, we develop a variant of the SINDy algorithmthat integrates automatic differentiation and recent time-stepping constrainedmotivated by Rudy et al. for simultaneously (i) denoising the data, (ii)learning and parametrizing the noise probability distribution, and (iii)identifying the underlying parsimonious dynamical system responsible forgenerating the time-series data. Thus within an integrated optimizationframework, noise can be separated from signal, resulting in an architecturethat is approximately twice as robust to noise as state-of-the-art methods,handling as much as 40% noise on a given time-series signal and explicitlyparametrizing the noise probability distribution. We demonstrate this approachon several numerical examples, from Lotka-Volterra models to thespatio-temporal Lorenz 96 model. Further, we show the method can identify adiversity of probability distributions including Gaussian, uniform, Gamma, andRayleigh.

Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data