Gain Scheduling with a Neural Operator for a Transport PDE with
  Nonlinear Recirculation
Maxence Lamarque Luke Bhan Rafael Vazquez Miroslav Krstic
To stabilize PDE models, control laws require space-dependent functional
gains mapped by nonlinear operators from the PDE functional coefficients. When
a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent,
a gain-scheduling (GS) nonlinear design is the simplest approach to the design
of nonlinear feedback. The GS version of PDE backstepping employs gains
obtained by solving a PDE at each value of the state. Performing such PDE
computations in real time may be prohibitive. The recently introduced neural
operators (NO) can be trained to produce the gain functions, rapidly in real
time, for each state value, without requiring a PDE solution. In this paper we
introduce NOs for GS-PDE backstepping. GS controllers act on the premise that
the state change is slow and, as a result, guarantee only local stability, even
for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear
recirculation using both a "full-kernel" approach and the "gain-only" approach
to gain operator approximation. Numerical simulations illustrate stabilization
and demonstrate speedup by three orders of magnitude over traditional PDE
gain-scheduling. Code (Github) for the numerical implementation is published to
enable exploration.