All roads lead to Markov kernel networks: A representation theorem for discretization-stable learning

A central goal of operator learning is to design networks that are stable to changes in the discretization of the input function, a property sometimes called "discretization invariance". We consider a quantitative refinement of this idea called *uniform discretization stability* that provides a Hölder-stability bound controlling changes in a network's output in terms of the Wasserstein distance between input measures. Our main result is a representation theorem showing all uniformly discretization stable operators can be written exactly as a *stable Markov kernel network*—an abstract architecture that is a joint generalization of transformers, Fourier neural operators (FNO), Flowers, and other successful learned operators. Our representation theorem characterizes a design space for discretization-stable operator learning; designing new architectures becomes a question of which parts of a Markov kernel network to parameterize, and how. We further show that the read kernels of these networks can be realized as solutions of regularized variational problems, with particular choices of costs and regularizers recovering the kernels of transformers, FNO, Flowers, their hybrids, and beyond. This variational formulation provides a systematic approach to architecture design, together with sufficient conditions for uniform discretization stability.