We introduce DeepMoD, a deep learning based model discovery algorithmwhich seeks the partial differential equation underlying a spatio-temporal data set . DeepMoD employs sparse regression on a library of basis functions and their corresponding spatial derivatives. A feed-forward neural network approximates the data set and automatic differentiation is used to construct this function libraryand perform regression within the neural network . This construction makes it extremely robust to noise and applicable to small data sets  and, contrary to other deep learning methods, does not require a training set and is impervious to overfitting. We illustrate this approach on several physical problems , such as the Burgers', Korteweg-de Vries , advection-diffusion and Keller-Segel equations, and find that it requires as few as O (10^2) samples and works at noise levels up to 75 \%. This resilience to noise and high performance at very few samples highlights the potential of this method to be applied on experimental data. Code and examples available at URL%DIFDELCMD < \end{abstract} %DIFDELCMD < %%% \\%DIF > . Motivated by these results, we apply DeepMoD directly on noisy experimental time-series data from a gel electrophoresis experiment and find that it discovers the advection-diffusion equation describing this system.
We introduce DeepMoD, a Deep learning based Model Discovery algorithm. DeepMoD discovers the partial differential equation underlying a spatio-temporal data set using sparse regression on a library of possible functions and their derivatives. A neural network approximates the data and constructs the function library, but it also performs the sparse regression . This construction makes it extremely robust to noise , applicable to small data sets , and, contrary to other deep learning methods, does not require a training set . We benchmark our approach on several physical problems such as the Burgers', Korteweg-de Vries and Keller-Segel equations, and find that it requires as few as \mathcal{O (10^2) samples and works at noise levels up to 75 %DIFDELCMD < \end{abstract} %DIFDELCMD < %%% \\%DIF > . Motivated by these results, we apply DeepMoD directly on noisy experimental time-series data from a gel electrophoresis experiment and find that it discovers the advection-diffusion equation describing this system.
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