Journal of Computational Physics
In this paper we use neural networks to learn governing equations from data. Specifically we reconstruct the right-hand side of a system of ODEs x˙(t)=f(t,x(t)) directly from observed uniformly time-sampled data using a neural network. In contrast with other neural network-based approaches to this problem, we add a Lipschitz regularization term to our loss function. In the synthetic examples we observed empirically that this regularization results in a smoother approximating function and better generalization properties when compared with non-regularized models, both on trajectory and non-trajectory data, especially in presence of noise. In contrast with sparse regression approaches, since neural networks are universal approximators, we do not need any prior knowledge on the ODE system. Since the model is applied component wise, it can handle systems of any dimension, making it usable for real-world data.
Deep learning, Generalization gap, Machine learning, Ordinary differential equations, Regularized network, System identification
Negrini, Elisa; Citti, Giovanna; and Capogna, Luca, "System Identification Through Lipschitz Regularized Deep Neural Networks" (2021). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.