# Machine Learning Accelerated PDE Backstepping Observers

arXiv: https://arxiv.org/abs/2211.15044 · HTML: https://mlsanity.com/paper/2211.15044/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 24 references that are in the map.
- First posted on arXiv 28 Nov 2022.

## Builds on (most cited first, 8 of 8)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Professor Forcing: A New Algorithm for Training Recurrent Networks](https://mlsanity.com/paper/1610.09038/) (2016)
- [Control and State Estimation of the One-Phase Stefan Problem via Backstepping Design](https://mlsanity.com/paper/1703.05814/) (2017)
- [Output-Feedback Boundary Control of a Heat PDE Sandwiched Between Two ODEs](https://mlsanity.com/paper/1902.04480/) (2019)
- [PDE Traffic Observer Validated on Freeway Data](https://mlsanity.com/paper/1906.04012/) (2019)
- [Arctic Sea Ice State Estimation From Thermodynamic PDE Model](https://mlsanity.com/paper/1901.10678/) (2019)

## Built on by (most cited first, 7 of 7)

- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs](https://mlsanity.com/paper/2303.10506/) (2023)
- [PDE Control Gym: A Benchmark for Data-Driven Boundary Control of Partial Differential Equations](https://mlsanity.com/paper/2405.11401/) (2024)
- [Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems](https://mlsanity.com/paper/2411.18964/) (2024)
- [Neural Operators for Boundary Stabilization of Stop-and-go Traffic](https://mlsanity.com/paper/2312.10374/) (2023)
- [Robust Neural IDA-PBC: passivity-based stabilization under approximations](https://mlsanity.com/paper/2409.16008/) (2024)
- [Moving-Horizon Estimators for Hyperbolic and Parabolic PDEs in 1-D](https://mlsanity.com/paper/2401.02516/) (2024)

## Abstract (arXiv)

State estimation is important for a variety of tasks, from forecasting to substituting for unmeasured states in feedback controllers. Performing real-time state estimation for PDEs using provably and rapidly converging observers, such as those based on PDE backstepping, is computationally expensive and in many cases prohibitive. We propose a framework for accelerating PDE observer computations using learning-based approaches that are much faster while maintaining accuracy. In particular, we employ the recently-developed Fourier Neural Operator (FNO) to learn the functional mapping from the initial observer state and boundary measurements to the state estimate. By employing backstepping observer gains for previously-designed observers with particular convergence rate guarantees, we provide numerical experiments that evaluate the increased computational efficiency gained with FNO. We consider the state estimation for three benchmark PDE examples motivated by applications: first, for a reaction-diffusion (parabolic) PDE whose state is estimated with an exponential rate of convergence; second, for a parabolic PDE with exact prescribed-time estimation; and, third, for a pair of coupled first-order hyperbolic PDEs that modeling traffic flow density and velocity. The ML-accelerated observers trained on simulation data sets for these PDEs achieves up to three orders of magnitude improvement in computational speed compared to classical methods. This demonstrates the attractiveness of the ML-accelerated observers for real-time state estimation and control.