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Unlike ODEs, whose models involve system matrices and whose controllersinvolve vector or matrix gains, PDE models involve functions in those rolesfunctional coefficients, dependent on the spatial variables, and gain functionsdependent on space as well. The designs of gains for controllers and observersfor PDEs, such as PDE backstepping, are mappings of system model functions intogain functions. These infinite dimensional nonlinear operators are given in animplicit form through PDEs, in spatial variables, which need to be solved todetermine the gain function for each new functional coefficient of the PDE. Theneed for solving such PDEs can be eliminated by learning and approximating thesaid design mapping in the form of a neural operator. Learning the neuraloperator requires a sufficient number of prior solutions for the design PDEs,offline, as well as the training of the operator. In recent work, we developedthe neural operators for PDE backstepping designs for first order hyperbolicPDEs. Here we extend this framework to the more complex class of parabolicPDEs. The key theoretical question is whether the controllers are stillstabilizing, and whether the observers are still convergent, if they employ theapproximate functional gains generated by the neural operator. We provideaffirmative answers to these questions, namely, we prove stability in closedloop under gains produced by neural operators. We illustrate the theoreticalresults with numerical tests and publish our code on github. The neuraloperators are three orders of magnitude faster in generating gain functionsthan PDE solvers for such gain functions. This opens up the opportunity for theuse of this neural operator methodology in adaptive control and in gainscheduling control for nonlinear PDEs.

Neural Operators of Backstepping Controller and Observer Gain Functions for Reaction-Diffusion PDEs