Approximate solutions to partial differential equations on unstructured grids can be represented as graphs, according to a new arXiv preprint. The paper, listed as 2610.27888, considers numerical optimization problems involving stiff evolutionary and static PDEs, which often require solving large systems of equations.
The work centers on automatic differentiation with constraints, with applications to both forward and inverse problems. Representing PDE solutions on graphs appears to be a way to make such constrained differentiation more practical when working with unstructured grids.
The abstract is short, so details of the method and validation are not yet available. The preprint signals a growing interest in combining graph-based representations of PDEs with constrained optimization and automatic differentiation.