Machine Learning Accelerated PDE Backstepping Observers

TL;DR

This paper introduces a machine learning framework using Fourier Neural Operators to significantly accelerate PDE backstepping observers, enabling real-time state estimation for complex PDE systems with high accuracy.

Contribution

It presents a novel approach combining PDE backstepping with FNO to drastically reduce computational costs while maintaining convergence guarantees.

Findings

Achieves up to 1000x speedup over classical methods.

Maintains accuracy and convergence guarantees with ML acceleration.

Successfully applied to reaction-diffusion and traffic flow PDEs.

Abstract

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.