Operator Learning for Scientific Computing

This thesis develops operator learning theory and methods for use in scientific computing. Operator learning uses data to approximate maps between infinite dimensional function spaces. As such, operator learning provides a natural framework for using machine learning in applications with partial differential equations (PDEs). While operator learning architectures have successfully modeled a variety of physical phenomena in practice, the theoretical foundations underpinning these successes remain in early stages of development. The present work takes a step towards a complete understanding of operator learning and its potential use in scientific applications. The thesis begins by studying multiscale constitutive modeling, where operator learning models can serve as surrogates to accelerate simulation and aid in model discovery of physical laws. The work proposes, and theoretically and numerically analyzes, an operator learning architecture for modeling history dependence in homogenized constitutive equations. The thesis then addresses learning solutions to an elliptic PDE in the presence of discontinuities and corner interfaces in two-dimensional materials. By proving a key continuity result for the underlying PDE, a universal approximation result is obtained. In its second half, the thesis moves on from the setting of homogenized constitutive laws and gives insight to operator learning from a broader perspective. First, error analysis bounds a form of discretization error that arises in implementations of the Fourier Neural Operator (FNO). Next, a modified form of the FNO, the Fourier Neural Mapping, accommodates finite-dimensional data while retaining the underlying function space structure. This modification allows applications where the map of interest is governed by an infinite-dimensional operator with data, such as parameters or summary statistics, in the form of finite vectors. Finally, the thesis extends a theory-to-practice gap result in finite dimensions to the infinite-dimensional operator learning setting, asserting that even for classes of architectures whose model expressivity scales well with model size, their error convergence with respect to data size scales poorly. In summary, this thesis builds understanding of operator learning from several perspectives and contributes both theoretical advancements and practical methodologies that improve the applicability of operator learning models to scientific problems.