Integral-Form Refinement Suppresses Error Accumulation in Neural Operator Rollout

Neural operators (NOs) forecast PDE trajectories rapidly, but in autoregressive rollout each window becomes the next input and errors compound until the forecast is unusable. We refine each predicted window against the governing equation before feeding it forward, with the NO frozen. The residual is the integral (Duhamel) form of the equation, or the discrete balance of a numerical scheme where no suitable linear part exists. This lowers rollout error by 40–72% on Burgers, 83% on turbulent Navier–Stokes and 33–61% on the highly nonlinear degenerate Richards equation of variably saturated flow. The improvement is substantial for the tested NOs, and refinement largely equalizes them, though our primitive NO keeps an edge at small correction budgets. More importantly, the refinement stops rollout error from compounding on non-chaotic systems. The uncorrected Richards error grows 2.7-fold over a 38-day forecast, whereas the corrected error settles three to thirteen times lower. In turbulent flow, where chaos forbids any bound, the error still grows but stays 56 times lower. The integral-form residual is critical: at the coarse frame spacing of NOs, both the standard pointwise residual and a one-step scheme residual are dominated by their own time-discretization error, and refining against them worsens the rollout or diverges. On Burgers, our training-free correction also outperforms learned refiners, which need per-system training. Furthermore, the benefit can be anticipated from the residual’s own error, measured once per system. This simple screen correctly identifies when refinement helps. Lastly, the correction can be strategically enhanced: semi-Lagrangian transport lowers Navier–Stokes error from 0.050 to 0.0083 in 17 ms per trajectory, below what any residual-minimizing refiner can reach. Together, these results show NO rollout errors to be measurable, predictable and suppressible, and establish test-time refinement as a practical tool for long forecasts.