One-Step Evolution for Long-Time Extrapolation: An Error-Bound-Informed and Prior-Guided Neural Residual Framework for Autonomous PDEs

Maqun Zhang; Feng Gao; Wankun Chen; Hui Yu; Yanhai Gan; Junyu Dong

cs.AIAug 22, 2026

One-Step Evolution for Long-Time Extrapolation: An Error-Bound-Informed and Prior-Guided Neural Residual Framework for Autonomous PDEs

Authors: Maqun Zhang, Feng Gao, Wankun Chen, Hui Yu, Yanhai Gan, Junyu Dong

Organizations: State Key Laboratory of Physical Oceanography and Faculty of Information Science and Engineering, Ocean University of China, Qingdao 266100, China · School of Psychology and Neuroscience, University of Glasgow, Glasgow G12 8QQ, U.K.

Abstract

Accurate simulation of the long-time evolution of systems governed by partial differential equations (PDEs) is central to scientific computing. Among existing deep learning?based approaches for solving PDEs, neural operators typically rely on extensive trajectory data, whereas physics-informed meth?ods often exhibit limited stability during long-time extrapolation. For a well-posed autonomous PDE, long-time trajectories can be generated by repeated composition of a fixed-step evolution operator; hence, long-time extrapolation depends on controlling the approximation error of this operator and the propagation of that error under recursive composition. Accordingly, we propose a numerical-prior-guided, physics-constrained method trained without ground-truth trajectory supervision: a low-cost numerical prior reduces the difficulty of approximating the one?step evolution operator, while a weak-form PDE residual provides a computable proxy for the one-step error term in the error?propagation bound. We validate the method on five benchmark cases spanning four PDE classes and compare it with ten physics?informed learning methods under a unified protocol that excludes ground-truth trajectories from training and model selection. The results indicate that, in all five cases, the proposed method reduces long-time extrapolation error relative to the numerical prior and outperforms the best competing baseline in each case, thereby improving long-time simulation accuracy across different PDEs without ground-truth trajectory supervision. The source code developed for this paper will be made publicly available upon acceptance of the manuscript.

Figures & tables

Fig. 1: Overall architecture and training workflow of the proposed method. The theoretical relations at the top connect recursive fixed-step evolution, one-step error propagation, and weak-form error constraints, with colored terms corresponding to their modules. Low-Cost Numerical Prior obtains the prior prediction through restriction, prior stepping, and prolongation. Residual Compensation Network concatenates the current state and prior prediction at $C$ and outputs a correction. Residual Fusion & Hard Physics scales the correction by $s$ , adds it to the prior prediction, and applies $\Pi_{\mathcal{M}}$ to produce the next state for recursive rollout. Weak-Form Physics-Unsupervised Optimization constructs $\mathcal{L}_{\mathrm{weak}}$ from the PDE residuals of short predicted trajectories through test-function projection and weighted aggregation, enabling training without ground-truth trajectory supervision.

Fig. 2: Full-state RMSE for TGV over steps 0–200. Training ends at step 20.

Fig. 3: TGV vorticity field at the representative extrapolation step 100. Because the analytical solution decays rapidly over time, the field at step 200 is provided in Fig. 19.

Fig. 4: RMSE for the smooth heat equation over steps 0–200. Training ends at step 20.

Fig. 5: Temperature field for the smooth heat equation at the representative extrapolation step 120. The field at step 200 is provided in Fig. 20.

Fig. 6: Full-state RMSE for DSL over steps 0–500. Both the raw curve for Ours and its 15-step centered moving average are shown, with the latter included only for readability. All statistics are computed from the unsmoothed data.

Fig. 7: DSL vorticity field at the final extrapolation step 500.

Fig. 8: Full-state RMSE for Gaussian Mound over steps 0–200.

Fig. 9: Free-surface perturbation and velocity field for Gaussian Mound at step 200.

Case	Governing equation	Training/test steps	Evaluation grid	Validation target
Taylor–Green vortex (TGV)	Incompressible Navier–Stokes equations	0–20 / 0–200	$128^{2}$	Smooth analytical flow
Smooth Heat Equation	Heat equation	0–20 / 0–200	$128^{2}$	Dissipation and near-zero stability
Double shear layer (DSL)	Incompressible Navier–Stokes equations	0–150 / 0–500	$128^{2}$	Nonlinear instability and vortex structures
Gaussian Mound	Shallow-water equations	0–20 / 0–200	$256^{2}$	Wave propagation and variable coupling
Cahn–Hilliard (CH) Two-Droplet	Cahn–Hilliard equation	0–20 / 0–260	$256^{2}$	Fourth-order conservation and interface evolution

TABLE I: Benchmark Settings and Validation Targets

Fig. 10: RMSE for CH Two-Droplet over steps 0–260.

Fig. 11: Phase-field distribution for CH Two-Droplet at the representative extrapolation step 200.

Fig. 12: Per-step RMSE reduction of the proposed method relative to the numerical prior. The circles mark the end of training for each case, with the training interval on the left and the extrapolation interval on the right; positive values indicate that Ours outperforms the numerical prior. The corresponding MAE results are provided in Appendix D.

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Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs). For time-dependent PDEs, existing methods typically perform long-horizon prediction through autoregressive rollout directly in high-dimensional physical field spaces, where each predicted state is recursively fed back as the input for the next step. Although effective for short-term prediction, this autoregressive rollout and the lack of continuous-time modeling lead to progressive error accumulation over long-horizon rollouts. In this work, we propose Autoregression-Free Neural Operators (AFNO), which map the time evolution of PDEs into a latent space and model continuous-time vector fields within it. AFNO uses flow matching to learn the latent vector field, thereby enabling continuous evolution over extended horizons, avoiding autoregressive rollout and capturing dynamics under varying parameter configurations through explicit conditioning on physical parameters. Theoretical analysis and extensive experiments on six PDEs demonstrate that AFNO improves long-horizon prediction stability and consistently reduces rollout errors compared with the baselines.

Abstract

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