Backstepping Neural Operators for $2\times 2$ Hyperbolic PDEs

• URL: http://arxiv.org/abs/2312.16762v3
• Date: Wed, 3 Jul 2024 16:04:07 GMT
• Title: Backstepping Neural Operators for $2\times 2$ Hyperbolic PDEs
• Authors: Shanshan Wang, Mamadou Diagne, Miroslav Krstić,
• Abstract summary: We study the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants. Engineering applications include oil drilling, the Saint-Venant model of shallow water waves, and the Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow.
• Score: 2.034806188092437
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more gain kernels-a PDE structure unaddressed thus far with DeepONet. In this paper, we explore the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants by considering a simple counter-convecting $2\times 2$ coupled system in whose control a $2\times 2$ kernel PDE system in Goursat form arises. Engineering applications include oil drilling, the Saint-Venant model of shallow water waves, and the Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. We establish the continuity of the mapping from a total of five plant PDE functional coefficients to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and ensure that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our $L^2$-Globally-exponentially stabilizing (GES) approximate gain kernel-based output feedback design implies the deep learning of both the controller's and the observer's gains. Moreover, the encoding of the output-feedback law into DeepONet ensures semi-global practical exponential stability (SG-PES). The DeepONet operator speeds up the computation of the controller gains by multiple orders of magnitude. Its theoretically proven stabilizing capability is demonstrated through simulations.

Related papers

• Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernels [3.3044728148521623]
  Neural operator approximations of the gain kernels in PDE backstepping have emerged as a viable method for implementing controllers in real time. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE. We prove global stability and regulation of the plant state for a Lyapunov design of parameter adaptation.
  arXiv  Detail & Related papers  (2024-07-01T19:24:36Z)
• Unisolver: PDE-Conditional Transformers Are Universal PDE Solvers [55.0876373185983]
  We present the Universal PDE solver (Unisolver) capable of solving a wide scope of PDEs. Our key finding is that a PDE solution is fundamentally under the control of a series of PDE components. Unisolver achieves consistent state-of-the-art results on three challenging large-scale benchmarks.
  arXiv  Detail & Related papers  (2024-05-27T15:34:35Z)
• Adaptive Neural-Operator Backstepping Control of a Benchmark Hyperbolic PDE [3.3044728148521623]
  We present the first result on applying NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with recirculation. We also present numerical simulations demonstrating stability and observe speedups up to three orders of magnitude.
  arXiv  Detail & Related papers  (2024-01-15T17:52:15Z)
• Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculation [1.124958340749622]
  Gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. Recent introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation.
  arXiv  Detail & Related papers  (2024-01-04T19:45:27Z)
• Deep Equilibrium Based Neural Operators for Steady-State PDEs [100.88355782126098]
  We study the benefits of weight-tied neural network architectures for steady-state PDEs. We propose FNO-DEQ, a deep equilibrium variant of the FNO architecture that directly solves for the solution of a steady-state PDE.
  arXiv  Detail & Related papers  (2023-11-30T22:34:57Z)
• Deep Learning of Delay-Compensated Backstepping for Reaction-Diffusion PDEs [2.2869182375774613]
  Multiple operators arise in the control of PDE systems from distinct PDE classes. DeepONet-approximated nonlinear operator is a cascade/composition of the operators defined by one hyperbolic PDE of the Goursat form and one parabolic PDE on a rectangle. For the delay-compensated PDE backstepping controller, we guarantee exponential stability in the $L2$ norm of the plant state and the $H1$ norm of the input delay state.
  arXiv  Detail & Related papers  (2023-08-21T06:42:33Z)
• Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay [9.155455179145473]
  We extend the recently introduced DeepONet operator-learning framework for PDE control to an advanced hyperbolic class. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight.
  arXiv  Detail & Related papers  (2023-07-21T08:57:16Z)
• Solving High-Dimensional PDEs with Latent Spectral Models [74.1011309005488]
  We present Latent Spectral Models (LSM) toward an efficient and precise solver for high-dimensional PDEs. Inspired by classical spectral methods in numerical analysis, we design a neural spectral block to solve PDEs in the latent space. LSM achieves consistent state-of-the-art and yields a relative gain of 11.5% averaged on seven benchmarks.
  arXiv  Detail & Related papers  (2023-01-30T04:58:40Z)
• Physics-Informed Neural Operator for Learning Partial Differential Equations [55.406540167010014]
  PINO is the first hybrid approach incorporating data and PDE constraints at different resolutions to learn the operator. The resulting PINO model can accurately approximate the ground-truth solution operator for many popular PDE families.
  arXiv  Detail & Related papers  (2021-11-06T03:41:34Z)
• Kernel-Based Reinforcement Learning: A Finite-Time Analysis [53.47210316424326]
  We introduce Kernel-UCBVI, a model-based optimistic algorithm that leverages the smoothness of the MDP and a non-parametric kernel estimator of the rewards. We empirically validate our approach in continuous MDPs with sparse rewards.
  arXiv  Detail & Related papers  (2020-04-12T12:23:46Z)
• Provably Efficient Safe Exploration via Primal-Dual Policy Optimization [105.7510838453122]
  We study the Safe Reinforcement Learning (SRL) problem using the Constrained Markov Decision Process (CMDP) formulation. We present an provably efficient online policy optimization algorithm for CMDP with safe exploration in the function approximation setting.
  arXiv  Detail & Related papers  (2020-03-01T17:47:03Z)

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.