A Robust In-Context Model for Conservation Laws: Injecting Context into Flux Neural Operators via Recurrent Vision Transformers

PDE foundation models adapt to new systems from context but do so with generic backbones that ignore the numerical structure of the equations they solve. We study the opposite design for one class: families of hyperbolic conservation laws, whose finite-volume structure guarantees discrete conservation. HFluxNO infers a numerical flux operator from a short observed trajectory, with no access to the governing equation or its coefficients: a temporally recurrent vision transformer encodes the trajectory, a hypernetwork maps the encoding to the parameters of a Flux Neural Operator, and the generated flux advances the state through the conservative finite-volume update, so that conservation holds exactly for every generated solver. Across one- and two-dimensional scalar and vector-valued conservation laws, including the compressible Euler equations, HFluxNO improves rollout accuracy over recent in-context solvers, remains the most accurate model under zero-shot shifts to a flux family never seen in training and, in all but one setting, to shock-dominated initial conditions, and conserves the state exactly in every case. Code: https://anonymous.4open.science/r/in_context_flux_no/.