ReAl: Refine neural operators via aligning dynamical flow

Autoregressive neural PDE surrogates predict future states by repeatedly applying a learned one-step operator. This is a simple and widely used method, but small one-step errors can accumulate during long rollouts. The resulting drift often appears as spectral amplitude distortion, phase misalignment, and nonlinear mode-interaction error. These effects are especially important for time-dependent PDEs with clear Fourier structure. We introduce the Spectral Generator Neural Operator (SGNO), a structured autoregressive neural operator for long-horizon PDE forecasting. SGNO organizes each learned one-step map as a structured spectral evolution update. A real-valued nonpositive diagonal generator provides a gain-controlled spectral backbone, while a learned correction pathway with complex-valued spectral mixing completes the residual evolution. This design gives the autoregressive step an evolution-like structure while retaining the flexibility needed for dissipative, dispersive, transport-dominated, and nonlinear PDEs. SGNO is designed for periodic linear and semilinear evolution PDEs with Fourier multiplier linear dynamics. Across ten mechanism-matched APEBench tasks spanning this regime, SGNO consistently outperforms strong single-step autoregressive baselines in long-horizon rollout accuracy, reducing GMean100 by a median of 74.8% relative to the strongest available non-SGNO baseline, with per-task reductions ranging from 13.6% to 92.9%. The gains are strongest on dispersive and transport-dominated tasks, as well as tasks involving nonlinear closure and mode coupling. Spectral diagnostics show lower spectral energy error and improved rollout-level phase fidelity. Ablations show that the constrained generator, the structured update, and the learned correction pathway each contribute to performance. The code is available at https://github.com/cruiseresearchgroup/SGNO.

Fast surrogate modeling for high-dimensional physical dynamics requires more than low short-term error: useful models must roll out efficiently while preserving the statistical structure of long trajectories. Neural operators provide inexpensive autoregressive forecasts but can drift in turbulent regimes, whereas rolling diffusion and latent generative surrogates can represent stochastic transitions at the cost of multi-step denoising, noise-schedule design, or auxiliary compression models. We propose MeanFlow Long-term Invariant Spatiotemporal Consistency Autoregressive Models (MeLISA), a latent-free autoregressive generative surrogate built on pixel-space MeanFlow. MeLISA defines a blockwise stochastic transition kernel that generates each forecast block with a single model evaluation, avoiding latent encoders and iterative diffusion solvers at inference time. To stabilize long-horizon rollouts, MeLISA combines a Window-Consistency MeanFlow objective that learns conditional spatiotemporal generation from partially observed temporal windows with a Time Increment Consistency loss that constrains multi-lag finite increments and targets temporal-correlation structure. We evaluate MeLISA with compact UNet and scalable DiT backbones on two high-resolution benchmarks, extended 2D Kolmogorov flow at $256 \times 256$ and turbulent channel-flow slice at $192 \times 192$. MeLISA outperforms neural-operator baselines on short-term forecasting accuracy and long-horizon statistical metrics, including energy spectra, turbulent kinetic energy, and mixing-rate-related dynamics, while achieving inference speeds comparable to, and in some cases faster than, neural operators. Compact 3.7-5.7M-parameter variants already deliver strong parameter efficiency, and DiT variants provide a scalable path up to 150M parameters. Overall, MeLISA benefits both rollout efficiency and long-horizon statistical accuracy.

Recent advances in autoregressive neural surrogate models have enabled orders-of-magnitude speedups in simulating dynamical systems. However, autoregressive models are generally prone to distribution drift: compounding errors in autoregressive rollouts that severely degrade generation quality over long time horizons. Existing work attempts to address this issue by implicitly leveraging the inherent trade-off between short-time accuracy and long-time consistency through hyperparameter tuning. In this work, we introduce a unifying mathematical framework that makes this tradeoff explicit, formalizing and generalizing hyperparameter-based strategies in existing approaches. Within this framework, we propose a robust, hyperparameter-free model implemented as a conditional diffusion model that balances short-time fidelity with long-time consistency by construction. Our model, Self-refining Neural Surrogate model (SNS), can be implemented as a standalone model that refines its own autoregressive outputs or as a complementary model to existing neural surrogates to ensure long-time consistency. We also demonstrate the numerical feasibility of SNS through high-fidelity simulations of complex dynamical systems over arbitrarily long time horizons.