ConeNO: Enforcing Physical Information Flow in Neural Operators

For finite-speed PDEs, the solution at a point depends only on data inside its backward cone. Standard neural operators need not respect this constraint: a global Fourier operator trained to low error changes its prediction when data outside the cone is edited while the reference stays fixed, and a mask applied after global mixing leaves the dependence in place. We introduce CaNO, a neural operator in which every read, including the inputs that determine its attention weights, stays inside a prescribed domain of dependence. We prove that this property is closed under composition and characterize what a Jacobian audit can and cannot certify. Under matched capacity and supervision on variable-coefficient transport and a finite-volume Burgers flow map, CaNO lowers error by 26–75% relative to convolutional and local-window Fourier operators that satisfy the same constraint. The constraint costs no in-distribution accuracy: CaNO matches an unconstrained global FNO on transport and reduces its error by 45–80% on Burgers, while having zero response to exterior interventions under which the error of global FNO grows nearly tenfold.