Geometry-aware Incremental Neural Operator for Long-Horizon PDE prediction
This work provides a more stable and accurate method for long-horizon PDE prediction, which is crucial for researchers and engineers working with complex dynamical systems.
This paper addresses the challenge of long-horizon autoregressive prediction in neural operators for PDEs, where local errors accumulate. The proposed GeoIncNO method significantly improves prediction accuracy, rollout stability, and spectral fidelity across six PDE benchmarks (1D, 2D, 3D) compared to existing neural operator baselines.
Neural operators have shown strong potential for learning solution operators of partial differential equations (PDEs). However, long-horizon autoregressive prediction remains challenging: local errors accumulate as spectral inconsistency, phase misalignment, or mean drift. Existing methods mainly improve state representations and operator backbones, while leaving the repeatedly applied latent transition increment weakly structured, allowing spectral errors and unstable channel couplings to accumulate during rollout. To address these issues, we propose a geometry-aware incremental neural operator (GeoIncNO) for stable long-horizon PDE prediction. GeoIncNO predicts latent increments for residual advancement and uses lightweight low-rank projectors to regulate channel coupling within active frequency bands derived from the increment spectral energy distribution. To reduce physical-space reconstruction errors, GeoIncNO further introduces a mean--fluctuation decoupled reconstruction mechanism, where stable mean structures and dynamic fluctuations are fused separately, and phase correction is applied only to the zero-mean fluctuation component. Extensive experiments on six PDE benchmarks, covering 1D, 2D, and 3D dynamical systems, show that GeoIncNO achieves consistently strong prediction accuracy, improved rollout stability, and better spectral fidelity compared with competitive neural-operator baselines.