## Scientific Computing and Numerics (SCAN) Seminar

Operator learning is a rapidly growing field of machine learning relevant to modeling continuum mechanics. It uses data from experiments or numerical simulations to learn operators between spaces of functions, often consisting of input and output spatial fields such as force and displacement on a structure. This talk will discuss how much data is needed to construct accurate approximations of operators, focusing primarily on learning finite-rank approximations of solution operators for non-self-adjoint uniformly elliptic PDEs. Low-rank approximation of large matrices using matrix-vector product queries generally requires querying the matrix transpose (adjoint), or otherwise querying the matrix as many times as columns in the matrix. In contrast, we show that finite-rank approximations of solution operators for uniformly elliptic PDEs can be learned without querying the adjoint solution operator, and we provide sample complexity estimates for learning using only forward queries. This is fortunate because the adjoint is usually inaccessible in experimental settings. The key ingredient is the additional structure imposed on the operator by elliptic regularity. These results illuminate some of the fundamental challenges in operator learning and provide a path towards truly data-efficient learning for linear and nonlinear operators by exploiting known structure.