On sparse regression, L_p‐regularization, and automated model discovery

Abstract

Sparse regression and feature extraction are the cornerstones of knowledge discovery from massive data. Their goal is to discover interpretable and predictive models that provide simple relationships among scientific variables. While the statistical tools for model discovery are well established in the context of linear regression, their generalization to nonlinear regression in material modeling is highly problem‐specific and insufficiently understood. Here we explore the potential of neural networks for automatic model discovery and induce sparsity by a hybrid approach that combines two strategies: regularization and physical constraints. We integrate the concept of Lp regularization for subset selection with constitutive neural networks that leverage our domain knowledge in kinematics and thermodynamics. We train our networks with both, synthetic and real data, and perform several thousand discovery runs to infer common guidelines and trends: L2 regularization or ridge regression is unsuitable for model discovery; L1 regularization or lasso promotes sparsity, but induces strong bias that may aggressively change the results; only L0 regularization allows us to transparently fine‐tune the trade‐off between interpretability and predictability, simplicity and accuracy, and bias and variance. With these insights, we demonstrate that Lp regularized constitutive neural networks can simultaneously discover both, interpretable models and physically meaningful parameters. We anticipate that our findings will generalize to alternative discovery techniques such as sparse and symbolic regression, and to other domains such as biology, chemistry, or medicine. Our ability to automatically discover material models from data could have tremendous applications in generative material design and open new opportunities to manipulate matter, alter properties of existing materials, and discover new materials with user‐defined properties.

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“…During training, our constitutive neural network learns the network weights 𝒘 = {𝑤 𝐼,1 , ..., 𝑤 𝐼,3𝑝+2 , 𝑤 𝐼𝐼,1 , ..., 𝑤 𝐼𝐼,𝑝 }, by minimizing a loss function 𝐿 [22]. This loss function is the mean-squared error, or the 𝐿 2 -norm of the difference between the model, ċ(𝒄 img ), and the training data, ̂̇𝒄(𝒄 img ), across all regions divided by the number of training points 𝑛 train ,…”