PAIR: Prediction with Additive Intermediate Readouts for Long-Horizon PDE Forecasting

Long-horizon accuracy in autoregressive PDE forecasting depends on both prediction errors and their propagation through the dynamics. Multiscale architectures construct coarse-grid features, but these representations typically contribute to the predicted physical field implicitly through the remaining decoder. In our periodic Navier–Stokes simulations, we observed that low-frequency initial perturbations produce larger deviations for long-horizon responses than equal-norm high-frequency perturbations. Motivated by this scale dependence, we investigate whether giving intermediate features an explicit coarse-scale prediction role can improve both coarse-scale learning and the full-resolution predictor. To this end, we introduce **Prediction with Additive Intermediate Readouts (PAIR)**, which maps intermediate features to a coarse physical-field contribution, upsamples it with a fixed operator, and adds it to the backbone’s full-resolution prediction. A unified prediction loss on the combined output jointly trains both paths and their shared features, providing an additional route for the prediction loss to reach intermediate representations, allowing the coarse readout and the full-resolution path to adapt together without a separate coarse-scale supervision objective. Across periodic Navier–Stokes flow, reaction–diffusion, and heterogeneous acoustic scattering, PAIR reduces mean endpoint error by 72.8%, 87.5%, and 72.6%, respectively, relative to the corresponding U-Net baselines after 99–100 autoregressive steps. Further experiments show that joint training improves the full-resolution path’s predictions in a high-frequency band outside the coarse readout’s spectral support, suggesting our paired design decomposing the dynamics offer a better control of error propagation for long-term rollout.