mlsanityData to 7 Oct 2026

Neural Operators for PDE Backstepping Control of First-Order Hyperbolic PIDE with Recycle and Delay

arXiv 2307.11436 ↗First posted 21 Jul 2023math.OC cs.LG cs.SYcited by 11

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Abstract (arXiv)

The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output or input. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator, mapping functions on a spatial domain into functions on a spatial domain, and where this gain-generating operator's inputs are the PDE's coefficients. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight. Once we produce this approximation-theoretic result in infinite dimension, with it we establish stability in closed loop under feedback that employs approximate gains. In addition to supplying such results under full-state feedback, we also develop DeepONet-approximated observers and output-feedback laws and prove their own stabilizing properties under neural operator approximations. With numerical simulations we illustrate the theoretical results and quantify the numerical effort savings, which are of two orders of magnitude, thanks to replacing the numerical PDE solving with the DeepONet.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 19 of its 56 references that are in the map. First posted on arXiv 21 Jul 2023.

Builds on

19 of 56 references.

References · 19

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7 more
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Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 3; 2025: 4; 2026: 1.

11 papers.

  1. 2024Gain Scheduling with a Neural Operator for a Transport PDE with Nonlinear Recirculationcited by 9
  2. 2024Neural Operators for Predictor Feedback Control of Nonlinear Delay Systemscited by 9
  3. 2023Deep Learning of Delay-Compensated Backstepping for Reaction-Diffusion PDEscited by 7
  4. 2023Neural Operators for Boundary Stabilization of Stop-and-go Trafficcited by 7
  5. 2023Backstepping Neural Operators for 2× 2 Hyperbolic PDEscited by 4
  6. 2024Adaptive control of reaction-diffusion PDEs via neural operator-approximated gain kernelscited by 4
  7. 2025Stabilization of nonlinear systems with unknown delays via delay-adaptive neural operator approximate predictorscited by 2
  8. 2025Delay-adaptive Control of Nonlinear Systems with Approximate Neural Operator Predictorscited by 1
  9. 2025Neural Operator based Reinforcement Learning for Control of first-order PDEs with Spatially-Varying State Delaycited by 0
  10. 2025Employing Deep Neural Operators for PDE control by decoupling training and optimizationcited by 0
  11. 2026Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospectscited by 0