System Identification Through Lipschitz Regularized Deep Neural Networks

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.

References in corpus (18)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
• Deep learning for universal linear embeddings of nonlinear dynamics
• Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations
• Data-driven discovery of coordinates and governing equations
• Applied Koopmanism
• Machine Learning of Linear Differential Equations using Gaussian Processes
• Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations
• Data Driven Governing Equations Approximation Using Deep Neural Networks
• Sparse Dynamics for Partial Differential Equations
• Spectrally-normalized margin bounds for neural networks
• Data-driven discovery of PDEs in complex datasets
• Training robust neural networks using Lipschitz bounds
• Nonlinear Systems Identification Using Deep Dynamic Neural Networks
• On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems
• Learning Equations for Extrapolation and Control
• Neural network augmented inverse problems for PDEs

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