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We present a statistical learning framework for robust identification ofpartial differential equations from noisy spatiotemporal data. Extendingprevious sparse regression approaches for inferring PDE models from simulateddata, we address key issues that have thus far limited the application of thesemethods to noisy experimental data, namely their robustness against noise andthe need for manual parameter tuning. We address both points by proposing astability-based model selection scheme to determine the level of regularizationrequired for reproducible recovery of the underlying PDE. This avoids manualparameter tuning and provides a principled way to improve the method'srobustness against noise in the data. Our stability selection approach, termedPDE-STRIDE, can be combined with any sparsity-promoting penalized regressionmodel and provides an interpretable criterion for model component importance.We show that in particular the combination of stability selection with theiterative hard-thresholding algorithm from compressed sensing provides a fast,parameter-free, and robust computational framework for PDE inference thatoutperforms previous algorithmic approaches with respect to recovery accuracy,amount of data required, and robustness to noise. We illustrate the performanceof our approach on a wide range of noise-corrupted simulated benchmarkproblems, including 1D Burgers, 2D vorticity-transport, and 3Dreaction-diffusion problems. We demonstrate the practical applicability of ourmethod on real-world data by considering a purely data-driven re-evaluation ofthe advective triggering hypothesis for an embryonic polarization system inC.~elegans. Using fluorescence microscopy images of C.~elegans zygotes as inputdata, our framework is able to recover the PDE model for the regulatoryreaction-diffusion-flow network of the associated proteins.

Stability selection enables robust learning of partial differential equations from limited noisy data