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.
Jiaquan Zhang, Shuxu Chen, Haifan Meng et al.· 0 citations
Neural operators provide fast surrogates for time-dependent partial differential equations (PDEs) by applying a learned evolution operator recursively to its own predictions, but this autoregressive rollout feeds every prediction error back as input, so local errors accumulate. Existing rollout-training strategies reduce the mismatch between training inputs and self-generated states, yet their supervision still measures only the absolute discrepancy from the ground-truth trajectory. Such supervision is therefore uninformative about whether the operator has overcome the long-horizon failure behaviors it exhibited earlier during optimization. We propose history-enriched rollout training (HERO), which augments conventional absolute trajectory supervision with relative supervision derived from the model's optimization history. HERO ranks detached candidate rollouts from a periodically refreshed lagged operator, the current model, and a perturbed input by rollout error, spectral discrepancy, energy drift, and error growth, and selects the strongest failure trajectory as reference. This reference enters a margin-based objective as a fixed comparison baseline, inducing a bounded, sample-dependent reweighting of the ground-truth rollout gradient rather than an independent gradient direction, which we further analyze theoretically. Experiments on nine PDE benchmarks with spectral and attention-based backbones show that HERO consistently improves long-horizon accuracy, stable rollout length, and out-of-distribution robustness at no inference-time cost. These results indicate that history-enriched relative supervision is effective for stabilizing long-horizon autoregressive prediction.
Jiaquan Zhang, Shuxu Chen, Haifan Meng et al.· 0 citations