An adaptive Galerkin neural-network method for minimum-norm control of wave equations

Abstract

We develop an adaptive Galerkin neural-network method for computing minimum-$L^2$-norm interior controls for wave equations with spatially varying coefficients. Within the Hilbert Uniqueness Method, adjoint waves are approximated directly in space-time by least-squares Galerkin projections onto trial spaces enriched by neural-network basis functions selected through residual-based training. Linear Galerkin solves determine the adjoint approximations, and a separate linear system reconstructs the control. The least-squares objective provides an indicator for adaptive enrichment and stopping; the experiments use a Ritz-based terminal-data version whose continuous-error interpretation additionally requires an eigenpair consistency estimate. The main difficulty is that adjoint errors perturb this reconstruction system. Assuming observability of the continuous equation, we derive sufficient stability conditions involving a worst-case factor $\sqrt{n}$ multiplying the maximal adjoint error, where $n$ is the number of retained terminal modes. A fully discrete error estimate separates terminal-space truncation, adjoint approximation, and assembly errors, yielding convergence to the minimum-norm control when the truncation error, the $\sqrt{n}$-weighted adjoint error, and the assembly errors tend to zero as $n$ grows. For spectral terminal bases, additional regularity and uniform contraction assumptions yield quantitative convergence rates and show that $O(\log n)$ accepted enrichment steps per mode suffice to balance the neural and truncation errors. Numerical experiments on a nonconvex Cassini oval, a square containing a low-speed lens, and a three-dimensional ball with a graded coefficient illustrate the method.