Data-Driven, ML-assisted Approaches to Problem Well-Posedness
This work addresses a gap in applying differential equations to realistic data acquisition scenarios, offering a practical method for researchers in computational mathematics and physics.
The paper tackles the problem of inferring well-posedness features for differential equations when data is available as patches or multiple solutions, rather than precise boundary conditions, using machine and manifold learning tools. It demonstrates a data-driven approach to assess existence and uniqueness without relying on rigorous theorems.
Classically, to solve differential equation problems, it is necessary to specify sufficient initial and/or boundary conditions so as to allow the existence of a unique solution. Well-posedness of differential equation problems thus involves studying the existence and uniqueness of solutions, and their dependence to such pre-specified conditions. However, in part due to mathematical necessity, these conditions are usually specified "to arbitrary precision" only on (appropriate portions of) the boundary of the space-time domain. This does not mirror how data acquisition is performed in realistic situations, where one may observe entire "patches" of solution data at arbitrary space-time locations; alternatively one might have access to more than one solutions stemming from the same differential operator. In our short work, we demonstrate how standard tools from machine and manifold learning can be used to infer, in a data driven manner, certain well-posedness features of differential equation problems, for initial/boundary condition combinations under which rigorous existence/uniqueness theorems are not known. Our study naturally combines a data assimilation perspective with an operator-learning one.