Neural Operators for Approximating Gain Kernels in PDE Control
Luke Bhan, University of California, San Diego
Boundary control of PDEs is an important subset of control theory with applications to biological modeling, combustion processes, chemical reactions, and materials manufacturing. The typical approach for boundary control of PDEs - backstepping - requires the solution to a separate, Goursat PDE (called the gain kernel) in the control feedback law. As such, it is not possible to implement real-time control as the computation for numerically solving PDEs is expensive. In this work, we introduce a recent mathematical breakthrough, neural operators, for approximating the solution to the PDE at every timestep. We prove the existence of such a neural network approximation as well as stability guarantees on the control feedback law under the neural operator approximated gain kernel. We then showcase numerical simulations exhibiting speedups up to three orders of magnitude faster than a traditional finite difference approach.