To lift or not to lift: Operator inference for nonlinear PDEs with boundary control

Abstract

Operator inference (OpInf) provides a nonintrusive framework for learning projection-based reduced-order models directly from data, but its standard formulations do not explicitly account for the fact that boundary actuation changes the admissible solution space of a PDE. This paper develops and evaluates a boundary-consistent, lifting-based OpInf formulation for nonlinear PDEs with time-varying inhomogeneous boundary conditions. The proposed approach decomposes the state into a known lifting component that satisfies the prescribed boundary conditions and a homogeneous component represented in a POD basis. Substituting this decomposition into the governing dynamics reveals a boundary-consistent reduced model structure in which the learned dynamics includes bilinear state-input and quadratic input-input interactions for quadratically nonlinear PDEs. The resulting operators are inferred from data, preserving the nonintrusive character of OpInf while guaranteeing boundary condition satisfaction by construction. Numerical experiments on boundary-actuated heat, Burgers, and lid-driven cavity problems show that the lifted formulation improves boundary compliance and can enhance prediction accuracy, solver stability, and computational efficiency, although these gains vary across problems and reduced model dimensions. Overall, this study establishes lifting as a principled extension of OpInf for boundary-actuated PDEs, clarifies when and how this formulation improves performance, and indicates that standard OpInf can remain an effective alternative in certain regimes.