System Identification Through Lipschitz Regularized Deep Neural Networks

TL;DR

This paper introduces a neural network approach with Lipschitz regularization to accurately learn system dynamics from data, improving smoothness and generalization over existing methods, especially under noisy conditions.

Contribution

It proposes a novel Lipschitz regularized neural network method for system identification that does not require prior knowledge and can handle high-dimensional systems.

Findings

Lipschitz regularization yields smoother, more generalizable models.

The method outperforms non-regularized neural networks in noisy scenarios.

Applicable to systems of any dimension without prior knowledge.

Abstract

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 $\dot{x}(t) = f(t, x(t))$ 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 don't 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.