New Neural Operator Improves Long-Horizon PDE Prediction Stability

Neural operators are powerful tools for learning solutions to partial differential equations (PDEs), but their application to long-horizon autoregressive prediction often suffers from accumulating local errors, leading to spectral inconsistencies and instability. Existing efforts have focused on improving state representations and operator backbones, yet the incremental latent transitions, which are repeatedly applied, often lack sufficient structure, allowing errors to compound. The proposed GeoIncNO (Geometry-aware Incremental Neural Operator) tackles these issues by predicting latent increments for residual advancement. It employs lightweight low-rank projectors to control channel coupling within active frequency bands, which are derived from the increment's spectral energy distribution. Furthermore, GeoIncNO introduces a mean-fluctuation decoupled reconstruction mechanism, separating stable mean structures from dynamic fluctuations and applying phase correction only to the zero-mean fluctuation component, thereby reducing physical-space reconstruction errors. Extensive experiments across six PDE benchmarks, including 1D, 2D, and 3D systems, demonstrate that GeoIncNO consistently achieves superior prediction accuracy, enhanced rollout stability, and better spectral fidelity compared to other neural operator baselines. This advancement offers a more robust method for simulating complex physical systems over extended periods.