Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems

• URL: http://arxiv.org/abs/2411.18964v1
• Date: Thu, 28 Nov 2024 07:30:26 GMT
• Title: Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems
• Authors: Luke Bhan, Peijia Qin, Miroslav Krstic, Yuanyuan Shi,
• Abstract summary: We introduce a new perspective on predictor designs by recasting the predictor formulation as an operator learning problem. We prove the existence of an arbitrarily accurate neural operator approximation of the predictor operator. Under the approximated-predictor, we achieve semiglobal practical stability of the closed-loop nonlinear system.
• Score: 3.0248879829045388
• License:
• Abstract: Predictor feedback designs are critical for delay-compensating controllers in nonlinear systems. However, these designs are limited in practical applications as predictors cannot be directly implemented, but require numerical approximation schemes. These numerical schemes, typically combining finite difference and successive approximations, become computationally prohibitive when the dynamics of the system are expensive to compute. To alleviate this issue, we propose approximating the predictor mapping via a neural operator. In particular, we introduce a new perspective on predictor designs by recasting the predictor formulation as an operator learning problem. We then prove the existence of an arbitrarily accurate neural operator approximation of the predictor operator. Under the approximated-predictor, we achieve semiglobal practical stability of the closed-loop nonlinear system. The estimate is semiglobal in a unique sense - namely, one can increase the set of initial states as large as desired but this will naturally increase the difficulty of training a neural operator approximation which appears practically in the stability estimate. Furthermore, we emphasize that our result holds not just for neural operators, but any black-box predictor satisfying a universal approximation error bound. From a computational perspective, the advantage of the neural operator approach is clear as it requires training once, offline and then is deployed with very little computational cost in the feedback controller. We conduct experiments controlling a 5-link robotic manipulator with different state-of-the-art neural operator architectures demonstrating speedups on the magnitude of $10^2$ compared to traditional predictor approximation schemes.

Related papers

• Linearization Turns Neural Operators into Function-Valued Gaussian Processes [23.85470417458593]
  We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators. Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator's predictions. We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief.
  arXiv  Detail & Related papers  (2024-06-07T16:43:54Z)
• Deep Neural Networks Tend To Extrapolate Predictably [51.303814412294514]
  neural network predictions tend to be unpredictable and overconfident when faced with out-of-distribution (OOD) inputs. We observe that neural network predictions often tend towards a constant value as input data becomes increasingly OOD. We show how one can leverage our insights in practice to enable risk-sensitive decision-making in the presence of OOD inputs.
  arXiv  Detail & Related papers  (2023-10-02T03:25:32Z)
• Guaranteed Approximation Bounds for Mixed-Precision Neural Operators [83.64404557466528]
  We build on intuition that neural operator learning inherently induces an approximation error. We show that our approach reduces GPU memory usage by up to 50% and improves throughput by 58% with little or no reduction in accuracy.
  arXiv  Detail & Related papers  (2023-07-27T17:42:06Z)
• Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems [0.0]
  This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. The unpredictability of the accuracy of neural operators impacts their applications in downstream problems of inference, optimization, and control.
  arXiv  Detail & Related papers  (2023-06-21T06:30:56Z)
• Approximate Bayesian Neural Operators: Uncertainty Quantification for Parametric PDEs [34.179984253109346]
  We provide a mathematically detailed Bayesian formulation of the ''shallow'' (linear) version of neural operators. We then extend this analytic treatment to general deep neural operators using approximate methods from Bayesian deep learning. As a result, our approach is able to identify cases, and provide structured uncertainty estimates, where the neural operator fails to predict well.
  arXiv  Detail & Related papers  (2022-08-02T16:10:27Z)
• Scalable computation of prediction intervals for neural networks via matrix sketching [79.44177623781043]
  Existing algorithms for uncertainty estimation require modifying the model architecture and training procedure. This work proposes a new algorithm that can be applied to a given trained neural network and produces approximate prediction intervals.
  arXiv  Detail & Related papers  (2022-05-06T13:18:31Z)
• Training Feedback Spiking Neural Networks by Implicit Differentiation on the Equilibrium State [66.2457134675891]
  Spiking neural networks (SNNs) are brain-inspired models that enable energy-efficient implementation on neuromorphic hardware. Most existing methods imitate the backpropagation framework and feedforward architectures for artificial neural networks. We propose a novel training method that does not rely on the exact reverse of the forward computation.
  arXiv  Detail & Related papers  (2021-09-29T07:46:54Z)
• Neural Operator: Learning Maps Between Function Spaces [75.93843876663128]
  We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations.
  arXiv  Detail & Related papers  (2021-08-19T03:56:49Z)
• Relaxing the Constraints on Predictive Coding Models [62.997667081978825]
  Predictive coding is an influential theory of cortical function which posits that the principal computation the brain performs is the minimization of prediction errors. Standard implementations of the algorithm still involve potentially neurally implausible features such as identical forward and backward weights, backward nonlinear derivatives, and 1-1 error unit connectivity. In this paper, we show that these features are not integral to the algorithm and can be removed either directly or through learning additional sets of parameters with Hebbian update rules without noticeable harm to learning performance.
  arXiv  Detail & Related papers  (2020-10-02T15:21:37Z)
• Bayesian Perceptron: Towards fully Bayesian Neural Networks [5.5510642465908715]
  Training and predictions of a perceptron are performed within the Bayesian inference framework in closed-form. The weights and the predictions of the perceptron are considered Gaussian random variables. This approach requires no computationally expensive gradient calculations and further allows sequential learning.
  arXiv  Detail & Related papers  (2020-09-03T15:08:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.