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The data-driven recovery of the unknown governing equations of dynamicalsystems has recently received an increasing interest. However, theidentification of governing equations remains challenging when dealing withnoisy and partial observations. Here, we address this challenge and investigatevariational deep learning schemes. Within the proposed framework, we jointlylearn an inference model to reconstruct the true states of the system and thegoverning laws of these states from series of noisy and partial data. In doingso, this framework bridges classical data assimilation and state-of-the-artmachine learning techniques. We also demonstrate that it generalisesstate-of-the-art methods. Importantly, both the inference model and thegoverning model embed stochastic components to account for stochasticvariabilities, model errors, and reconstruction uncertainties. Variousexperiments on chaotic and stochastic dynamical systems support the relevanceof our scheme w.r.t. state-of-the-art approaches.

Variational Deep Learning for the Identification and Reconstruction of Chaotic and Stochastic Dynamical Systems from Noisy and Partial Observations