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While most classical approaches to Granger causality detection assume linear dynamics, many interactions in real-world applications, like neuroscience and genomics, are inherently nonlinear. In these cases, using linear models may lead to inconsistent estimation of Granger causal interactions. We propose a class of nonlinear methods by applying structured multilayer perceptrons (MLPs) or recurrent neural networks (RNNs) combined with sparsity-inducing penalties on the weights. By encouraging specific sets of weights to be zero--in particular, through the use of convex group-lasso penalties--we can extract the Granger causal structure. To further contrast with traditional approaches, our framework naturally enables us to efficiently capture long-range dependencies between series either via our RNNs or through an automatic lag selection in the MLP. We show that our neural Granger causality methods outperform state-of-the-art nonlinear Granger causality methods on the DREAM3 challenge data. This data consists of nonlinear gene expression and regulation time courses with only a limited number of time points. The successes we show in this challenging dataset provide a powerful example of how deep learning can be useful in cases that go beyond prediction on large datasets. We likewise illustrate our methods in detecting nonlinear interactions in a human motion capture dataset.
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