Building Hybrid B-Spline And Neural Network Operators

This paper introduces a hybrid B-spline neural network operator for real-time prediction of cyber-physical system behavior, leveraging the inductive bias of B-splines and the learning capabilities of neural networks to enhance safety and reliability in complex systems.

Abstract

Control systems are indispensable for ensuring the safety of cyber-physical systems (CPS), spanning various domains such as automobiles, airplanes, and missiles. Safeguarding CPS necessitates runtime methodologies that continuously monitor safety-critical conditions and respond in a verifiably safe manner. A fundamental aspect of many safety approaches involves predicting the future behavior of systems. However, achieving this requires accurate models that can operate in real time. Motivated by DeepONets, we propose a novel strategy that combines the inductive bias of B-splines with data-driven neural networks to facilitate real-time predictions of CPS behavior. We introduce our hybrid B-spline neural operator, establishing its capability as a universal approximator and providing rigorous bounds on the approximation error. These findings are applicable to a broad class of nonlinear autonomous systems and are validated through experimentation on a controlled 6-degree-of-freedom (DOF) quadrotor with a 12 dimensional state space. Furthermore, we conduct a comparative analysis of different network architectures, specifically fully connected networks (FCNN) and recurrent neural networks (RNN), to elucidate the practical utility and trade-offs associated with each architecture in real-world scenarios.

Method

The authors used a hybrid methodology that combines B-spline functions with neural networks, specifically inspired by the DeepONets architecture, to create a hybrid B-spline neural operator. This operator is designed to predict the behavior of cyber-physical systems in real time by learning the mapping from initial conditions to control points of a B-spline approximation. The approach involves using a neural network to approximate the solution of nonlinear autonomous systems, with the B-spline functions serving as the basis for the approximation. The authors also conducted experiments on a 6-degree-of-freedom quadrotor to validate their method and compared the performance of fully connected neural networks (FCNNs) and recurrent neural networks (RNNs) to evaluate the practical utility of different network architectures.

Main Finding

The authors discovered that their hybrid B-spline neural operator could effectively serve as a universal approximator for predicting the behavior of nonlinear autonomous systems. They provided theoretical bounds on the approximation error, which is applicable to a broad class of systems. Through experimentation on a controlled 6-degree-of-freedom quadrotor, they demonstrated the practicality of their approach. Additionally, they found that recurrent neural networks (RNNs) could achieve lower validation loss with fewer parameters compared to fully connected neural networks (FCNNs), although FCNNs showed better computational efficiency.

Conclusion

The conclusion of the paper is that the proposed hybrid B-spline neural operator is a promising approach for real-time prediction of cyber-physical system behavior, offering a balance between accuracy and computational efficiency. The authors suggest that future work could focus on improving the neural network architecture, leveraging the B-spline convex hull property for real-time safety assessments, and extending the framework to non-autonomous systems. This research contributes to enhancing the safety and reliability of cyber-physical systems through improved predictive modeling.

Keywords

B-spline, neural network, DeepONets, cyber-physical systems, control systems, universal approximator, approximation error, quadrotor, fully connected neural networks, recurrent neural networks, real-time prediction, safety-critical systems, scientific machine learning, system identification, model predictive control, digital twins, isogeometric analysis, finite element methods, nonlinear systems, autonomous systems, runtime safety monitoring, trajectory prediction, convex hull property, rotational equivariance, least squares fitting, ODE solver, residual blocks, long-short-term memory cell, equivariant network structures, SympNets, group equivariant convolutional neural networks.