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Building Hybrid B-Spline And Neural Network Operators

Raffaele Romagnoli, Jasmine Ratchford, Mark H. Klein

TL;DR

The paper introduces a hybrid B-spline neural operator that learns to predict future states of nonlinear autonomous systems in real time by mapping initial conditions to B-spline control points. Building on universal approximation theory for nonlinear operators and the DeepONet paradigm, the authors prove approximation and error bounds and validate the approach on a $12$-D quadrotor state with a $3$rd-order B-spline basis. They compare FCNN and GRU-based architectures, showing trade-offs between speed and accuracy, and highlight the convex hull property of B-splines as a path toward real-time safety checks. The work has practical implications for safety-critical CPS by enabling verifiable, real-time trajectory prediction and by providing a framework for future non-autonomous extensions and equivariant neural designs.

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.

Building Hybrid B-Spline And Neural Network Operators

TL;DR

The paper introduces a hybrid B-spline neural operator that learns to predict future states of nonlinear autonomous systems in real time by mapping initial conditions to B-spline control points. Building on universal approximation theory for nonlinear operators and the DeepONet paradigm, the authors prove approximation and error bounds and validate the approach on a -D quadrotor state with a rd-order B-spline basis. They compare FCNN and GRU-based architectures, showing trade-offs between speed and accuracy, and highlight the convex hull property of B-splines as a path toward real-time safety checks. The work has practical implications for safety-critical CPS by enabling verifiable, real-time trajectory prediction and by providing a framework for future non-autonomous extensions and equivariant neural designs.

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.
Paper Structure (14 sections, 7 theorems, 31 equations, 3 figures, 1 table)

This paper contains 14 sections, 7 theorems, 31 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nonlinear Systems
  4. Neural Networks
  5. The Universal Approximation Theorem for Nonlinear Operators
  6. B-splines
  7. Problem Statement
  8. B-spline-Based Deep Neural Operator
  9. Initial condition and control points mapping
  10. Multidimensional Solution and Error Bounds
  11. Experiments
  12. Results
  13. Conclusions
  14. acknowledgements

Key Result

Theorem 1

For any $\epsilon > 0$, there exists a positive integer $N$, $\theta_0^i \in \mathbb{R}$, $\theta_i \in \mathbb{R}^n$, $i=1,...,N$ independent of $g \in C(K;\mathbb{R})$, and constants $c_i(g)$, $i=1,...,N$ depending on $g$ such that holds for all $z \in K$ and $g \in \mathcal{U}$. Moreover, each $c_i(g)$ is a linear continuous functional defined on $\mathcal{U}$.

Figures (3)

  • Figure 1: (top) FCNN and (bottom) RNN root mean squared error versus initial condition radius. The error is positively correlated with distance.The overall error can be seen in the profile histogram right of the scatter plot.
  • Figure 2: (left) FCNN and (right) RNN predicted and least-squares fitted trajectories for sets of initial condition, varied about a central value as indicated on the legend. Only four dimensions are shown to conserve space.
  • Figure 3: The initial condition is rotated around the Z axis to 85 different angles from $\pi/8$ to $15\pi/8$. The (top) FCNN or (bottom) RNN is applied. The root mean squared difference is compared to the radius of the initial condition in the 12-D ball.

Theorems & Definitions (11)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Lemma 1
  • proof
  • Proposition 1
  • Lemma 2
  • ...and 1 more