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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

arXiv:2608.22026v1cs.AIcs.LG

TL;DR

Long-time PDE extrapolation must control both one-step approximation error and its recursive propagation, while existing neural operators often require trajectories and physics-informed methods can be unstable. The paper combines a low-cost numerical prior with learned residual correction and weak-form physics constraints, training without ground-truth trajectories. Across five benchmark cases, it reduces error relative to the numerical prior and outperforms the best competing baseline in every case.

  • Problem

    Long-time PDE simulation requires stable recursive evolution, but neural operators typically need extensive trajectory data and physics-informed methods can have limited long-time stability.

  • Method

    A low-cost numerical prior provides baseline evolution, a correction network learns unresolved prior error, and weak-form PDE residuals constrain one-step error without ground-truth trajectories.

  • Results

    Across five benchmark cases, the proposed method reduces long-time extrapolation error relative to the numerical prior and outperforms the best competing baseline in each case.

  • Takeaways & Limitations

    The framework improves long-time simulation accuracy across different PDEs without ground-truth trajectory supervision.

  • Takeaways & Limitations

    Experiments focus mainly on two-dimensional regular grids with fixed initial and boundary conditions, requiring further validation for complex geometries, multiphysics, high-dimensional systems, and parametric PDE families.

Abstract

from arXiv · show

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.

I. INTRODUCTION

Long-time PDE extrapolation is difficult because trajectory-based neural operators need extensive supervision, while recursive approximate operators accumulate one-step errors. The proposed framework combines a low-cost numerical prior, learned residual correction, weak-form physics constraints, and hard physical enforcement without ground-truth trajectories.

  • Motivation: Neural operators typically require paired high-fidelity trajectories, whereas physics-informed methods can struggle with stable long-time extrapolation.The paper frames these limitations as the motivation for a trajectory-free, stability-oriented approach.
  • Proposed framework: The method uses a low-cost numerical prior as baseline evolution and trains a correction network to reduce the difficulty of approximating the full evolution operator.The prior is intended to capture dominant dynamics while the network learns the unresolved correction.
  • Motivation: Long-time evolution of a well-posed autonomous PDE can be generated by recursively composing one fixed-step state-transition operator.Approximation errors introduced at each recursive application can propagate over the rollout.
  • Proposed framework: A weak-form PDE residual provides a physics-based proxy for one-step approximation error, while hard constraints preserve initial and boundary conditions.This enables training without ground-truth trajectory supervision.
  • Evaluation: The evaluation compares the method with ten physics-informed learning methods on five benchmark cases under a protocol excluding ground-truth trajectories from training and model selection.The benchmarks span multiple PDE classes and solution structures.

B. Physics-Informed PDE Solvers

The paper situates its method among physics-informed, autoregressive, and hybrid numerical–learning PDE solvers, then formulates autonomous PDE evolution through a reusable fixed-step operator. This formulation clarifies why exact recursive structure does not guarantee stable extrapolation for an approximate learned operator.

  • B. Physics-Informed PDE Solvers: PINN variants impose PDE, initial, and boundary constraints but may suffer gradient imbalance and ill-conditioned optimization from multi-term differential losses.Weak-form variants and adaptive weighting methods address parts of this optimization challenge.
  • B. Physics-Informed PDE Solvers: Autoregressive PDE methods repeatedly apply a fixed-step state-transition model to generate long-time trajectories without ground-truth trajectory supervision.Examples include PDE-Net, AR-DenseED, and PhyCRNet.
  • D. Hybrid Numerical–Learning PDE Solvers: Hybrid numerical–learning methods retain discrete or mechanistic structure while learning unknown dynamics, closure terms, or discretization errors.Representative approaches include Universal Differential Equations, coarse-grid correction methods, and Solver-in-the-Loop.
  • D. Hybrid Numerical–Learning PDE Solvers: The proposed workflow combines a low-cost numerical prior, residual compensation network, hard physical projection, and weak-form optimization for recursive rollout.The architecture links these modules to one-step error propagation and weak-form error constraints.
  • 1) Fixed-Step Evolution Operator for Autonomous PDEs: For autonomous PDEs, time-invariant dynamics permit repeated composition of the same fixed-step evolution operator.The learning target is therefore reformulated from the full spatiotemporal solution to a recursively applied one-step operator.
  • 1) Fixed-Step Evolution Operator for Autonomous PDEs: Recursive structure alone does not ensure stable extrapolation because approximation errors are introduced and propagated at every step.The paper therefore connects one-step approximation error with its long-time propagation before introducing the prior, correction, and physics-based loss.

2) Approximation of the Exact Evolution Operator and the Proposed Method:

The proposed operator approximates exact evolution by correcting a numerical prior rather than learning the full operator. Weak-form residual constraints provide a computable surrogate for unavailable one-step error during recursive short-window training.

  • Error propagation: Long-time error depends on one-step operator error and its propagation under recursive application.The stated error relation assumes local stability, reachable predicted states, and a shared initial state.
  • Prior-guided correction: A low-cost numerical prior supplies baseline evolution, while a convolutional network learns a correction to the prior error.This narrows the approximation range when the prior captures dominant dynamics.
  • Weak-form constraint: Because the exact operator and ideal correction are unavailable, the method uses weak-form PDE residuals instead of directly optimizing the ideal correction.Weak testing is adopted because evolution-problem stability estimates commonly control state error through a dual norm of the spatiotemporal residual.
  • Weak-form constraint: The weak-form loss converts unavailable one-step operator error into a computable residual proxy that constrains the learned operator.The proxy is justified under PDE stability and a sufficiently rich test space.
  • Training: During training, the same approximate operator is recursively applied over a short window while the weak-form residual constrains the predicted trajectory.Parameter sharing makes every step follow the same evolution rule, linking one-step error reduction to its accumulated bound.

B. Prior-Guided Residual Correction Operator

The implementation treats the numerical prior as a fixed baseline and places trainable capacity in a residual correction operator. Cross-resolution computation lets a coarse prior support prediction on the target grid.

  • Forward pass: A fixed numerical prior first produces a baseline prediction, after which a Residual CNN estimates the prior-error correction.Residual fusion and physical constraints then produce the next state.
  • Parameterization: Neither the numerical prior nor the physical constraints are trainable; all trainable parameters reside in the correction operator Cθ.This keeps learning focused on correcting the prior rather than replacing the numerical evolution rule.
  • Cross-resolution computation: The prior is represented as a fixed one-step time-marching operator on a grid with spatial scale hc, while the target grid has scale hf and hc ≥ hf.The method explicitly handles computation across the prior and target resolutions.

1) Cross-Resolution Prior Embedding:

The method evaluates a numerical prior on a coarser grid and uses a lightweight convolutional correction network on the target grid to correct prior error. Residual initialization and shared architecture support progressive, efficient one-step correction during extrapolation.

  • Cross-Resolution Prior Embedding: The prior module is not tied to a specific numerical scheme and only needs to advance the current state by a fixed time step while preserving its physical variables.The cross-resolution operators restrict the target-grid state to the prior grid and prolong the prediction back to the target grid.
  • Cross-Resolution Prior Embedding: The prior operates on a coarser grid, reducing local grid-based degrees of freedom to approximately (Nc/Nf)^d of the target-grid count.Restriction and prolongation, however, remain part of the prior pipeline, while the CNN operates on the target grid.
  • Cross-Resolution Prior Embedding: The correction network receives the current state and upsampled prior prediction, then outputs a correction with matching dimensionality.This input contains the complete state needed to determine the next-step evolution.
  • Cross-Resolution Prior Embedding: A lightweight fully convolutional architecture balances representation capacity, computational cost, and physical identifiability during repeated extrapolation.Its receptive field is designed to cover the dominant spatial scale of one-step prior error without global coupling.
  • Cross-Resolution Prior Embedding: Zero-initialized residual outputs make the initial model equal to a physically valid prior and let training progressively correct it rather than relearn the full evolution.The residual CNN uses convolutional and residual blocks whose capacity and receptive field depend on width, depth, and dilation.

3) Residual Fusion and Physical Constraints:

The predicted correction is scaled and fused with the numerical prior, then projected into the PDE-specific physically admissible set. Weak-form residuals and hard constraints support physics-consistent recursive evolution without requiring exact trajectories.

  • Residual Fusion and Physical Constraints: The correction is scaled channel-wise and additively fused with the prior prediction to account for differences among physical variables.The scale vector is determined by nondimensionalization or prescribed physical magnitudes and does not use ground-truth supervision.
  • Residual Fusion and Physical Constraints: The fused prediction is mapped into the physically admissible set through a PDE-specific projection that can enforce boundaries, incompressibility, or positivity.When explicit projection is unnecessary, the operator is the identity.
  • Residual Fusion and Physical Constraints: The numerical prior, CNN correction, and physical projection together define a complete one-step evolution operator for recursive application.The same composite construction is optimized through weak-form physics and PDE-specific regularization.
  • Weak-Form Physics Constraint for One-Step Evolution: The weak-form PDE residual evaluates short-window predicted sequences against the governing equation because the exact evolution operator is unavailable during training.The residual is used as a conditional a posteriori proxy for one-step approximation error under stated conditions.
  • Residual Fusion and Physical Constraints: Initial and boundary conditions are enforced as hard constraints, so the training objective adds neither initial-condition nor boundary-condition losses.The initial condition initializes recursion, while boundary-consistent operations and projection preserve boundary conditions.

2) Overall Objective and Training without Ground-Truth Trajectory Supervision:

Training uses a composite physics-based objective over short recursively generated windows, excluding ground-truth trajectories from optimization and model selection. After training, frozen parameters are applied repeatedly beyond the training interval, so long-time performance depends on one-step consistency and recursive stability.

  • Overall Objective and Training without Ground-Truth Trajectory Supervision: The overall objective combines the weak-form loss with PDE-specific physical regularization to constrain predictions without ground-truth fitting terms.Regularization can address divergence, conserved quantities, equilibrium states, or correction magnitude after variable scaling.
  • Overall Objective and Training without Ground-Truth Trajectory Supervision: Training recursively generates predicted sequences over prescribed short windows before evaluating the objective and updating parameters.Model selection uses physical residuals on held-out windows and independent test functions, while reference solutions are reserved for error evaluation.
  • Recursive Extrapolation with Frozen Parameters: After training, frozen parameters are repeatedly applied from the prescribed initial state using the same composite operator at every extrapolation step.The state is not reset after the training window, and no further network updates or ground-truth corrections are introduced.
  • Recursive Extrapolation with Frozen Parameters: The evaluation reports RMSE curves and representative fields across long rollouts, with training endpoints marked in the curves.Examples include smooth heat-equation temperature fields and DSL full-state RMSE with raw and moving-average curves whose statistics use unsmoothed data.
  • Recursive Extrapolation with Frozen Parameters: Long-time performance therefore depends on one-step physical consistency and stability under recursive composition beyond the short training interval.The method directly applies the learned one-step operator beyond the training horizon.

IV. NUMERICAL EXPERIMENTS AND RESULTS

The experiments evaluate long-time extrapolation without ground-truth trajectory supervision across five benchmark cases, using a unified comparison protocol and RMSE-based assessment.

  • Results overview: The results section reports long-time extrapolation performance across five benchmark cases and analyzes correction relative to the numerical prior.The protocol excludes ground-truth trajectories from training and model selection.
  • Experimental setup: Five PDE cases with distinct dynamics are evaluated after short-window training, with continuous rollouts from step 0 to each test endpoint.Training steps define the physics-based optimization window, while test steps denote the full rollout interval.
  • Comparison protocol: Ten physics-informed learning methods are compared under matched governing equations, conditions, training windows, and evaluation grids.The baselines include PINN, PINNsFormer, RoPINN, gPINN, Causal PINN, hp-VPINN, SA-PINN, PINO, AR-DenseED, and PhyCRNet.
  • Evaluation metrics: Per-step MAE and RMSE evaluate all state variables, with the main text emphasizing RMSE curves and representative extrapolated fields.Complete MAE results and longest-horizon fields are reported in Appendix D.

1) TGV: Smooth Analytical Flow:

The TGV and heat-equation cases test extrapolation for smooth analytical flow and linear dissipation, respectively, with the proposed method maintaining lower errors and physically consistent fields.

  • Visual evaluation: Representative fields are examined at step 100 for TGV and step 120 for the heat equation, alongside rollout RMSE curves.The supplied figures identify the corresponding rollout and field visualizations.
  • TGV: Smooth Analytical Flow: 3.86 × 10−5 versus 9.47 × 10−2 average RMSE over steps 21–200: the proposed method outperforms PINN on smooth analytical flow.At step 100, it retains a vortex structure consistent with the reference solution while competing methods show clear deviations.
  • Smooth Heat Equation: 9.36 × 10−5 versus 1.88 × 10−2 average RMSE over steps 21–200: the proposed method outperforms PINNsFormer during linear dissipation.At step 120, it preserves smooth temperature distribution and the correct decay trend without evident rebound or nonphysical oscillations.

4) Gaussian Mound: Smooth Hyperbolic Wave Propagation:

The Gaussian Mound and CH Two-Droplet cases examine hyperbolic wave propagation and fourth-order phase-field evolution, while cross-case analysis tracks correction gains against the numerical prior.

  • Gaussian Mound: Smooth Hyperbolic Wave Propagation: 1.71×10−3 versus 3.99×10−3 average RMSE over steps 21–200: the proposed method outperforms gPINN for hyperbolic wave propagation.At step 200, it preserves the annular wave’s propagation position, symmetry, and velocity direction.
  • CH Two-Droplet: 4.29×10−2 versus 6.40×10−2 average RMSE over steps 21–260: the proposed method outperforms gPINN for fourth-order conservative phase-field evolution.At step 200, it preserves phase-separation interfaces and the main droplet structures.
  • Cross-case prior correction: All five cases maintain positive per-step gains during extrapolation relative to the numerical prior.Gains decrease for TGV and Heat, increase generally for DSL and CH, and remain relatively stable for Gaussian Mound.
  • Cross-case prior correction: Correction magnitude depends on numerical-prior accuracy and accumulated error rather than varying monotonically with equation complexity.MAE results follow the RMSE trend, indicating consistent prior-error reduction across extrapolation intervals.

V. CONCLUSION

The study presents a numerical-prior-guided, physics-constrained framework for long-time PDE extrapolation without ground-truth trajectory supervision. Across five benchmarks, it improves on numerical priors and physics-informed baselines, while remaining limited in scope to relatively simple settings.

  • Conclusion: The framework controls recursive extrapolation by approximating a fixed-step evolution operator and constraining its one-step error.It combines a low-cost numerical prior, a correction network, and a weak-form PDE residual for physics-unsupervised training.
  • Conclusion: The method performs long-time rollout recursively from the initial state without ground-truth trajectory supervision.After training, the same unified evolution approximation is repeatedly applied for extrapolation.
  • Conclusion: Across five benchmark cases, the proposed method outperforms the corresponding numerical prior and provides more stable extrapolation than multiple physics-informed learning baselines.The cases span smooth flow, dissipation, shear instability, wave propagation, and conservative phase-field evolution.
  • Conclusion: Learned prior correction consistently reduces prior error throughout the extrapolation interval, so gains do not arise solely from the numerical prior.The framework builds on existing physical evolution rules rather than replacing them, mitigating error accumulation during long-time rollout.
  • Limitations: The experiments mainly cover two-dimensional regular grids with fixed initial and boundary conditions, leaving broader applicability for future validation.Unvalidated settings include complex geometries, multiphysics coupling, high-dimensional systems, and parametric PDE families.
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