Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings. Neural operator (NO) surrogates offer fast inference across parametric and functional inputs; however, most autoregressive NO frameworks remain vulnerable to compounding errors, and ensemble-averaged metrics provide limited guarantees for individual inference trajectories. In practice, error accumulation can become unacceptable beyond the training horizon, and existing methods lack mechanisms for online monitoring or correction. To address this gap, we propose ANCHOR (Adaptive Numerical Correction for High-fidelity Operator Rollouts), an online, instance-aware hybrid inference framework for stable long-horizon prediction of nonlinear, time-dependent PDEs. ANCHOR treats a pretrained NO as the primary inference engine and adaptively couples it with a classical numerical solver using a physics-informed, residual-based error estimator. Inspired by adaptive time-stepping in numerical analysis, ANCHOR monitors an exponential moving average (EMA) of the normalized PDE residual to detect accumulating error and trigger corrective solver interventions without requiring access to ground-truth solutions. We show that the EMA-based estimator correlates strongly with the true relative L2 error, enabling data-free, instance-aware error control during inference. Evaluations on six canonical PDEs: 1D and 2D Burgers', 2D Allen-Cahn, 2D Cahn-Hilliard, 2D Navier-Stokes, and 3D heat conduction, demonstrate that ANCHOR reliably bounds long-horizon error growth, stabilizes extrapolative rollouts, and significantly improves robustness over standalone neural operators, while remaining substantially more efficient than high-fidelity numerical solvers.

翻译：含时偏微分方程的数值模拟是科学与工程应用的核心，但高保真求解器在长时间或时间关键场景下的计算成本往往过高。神经算子代理模型可对参数化和函数化输入实现快速推理，然而大多数自回归神经算子框架仍易受累积误差影响，且集合平均指标对单次推理轨迹的保障有限。实践中，当预测时间超出训练时域时误差累积可能变得不可接受，现有方法又缺乏在线监测或校正机制。针对这一缺口，我们提出ANCHOR（面向高保真算子滚动的自适应数值校正）——一种面向非线性含时偏微分方程稳定长时域预测的在线、实例感知混合推理框架。ANCHOR将预训练神经算子作为主推理引擎，通过基于物理信息的残差误差估计器自适应地耦合经典数值求解器。受数值分析中自适应时间步长启发，ANCHOR监测归一化偏微分方程残差的指数移动平均（EMA）以检测累积误差，并在无需真解的条件下触发校正性求解器干预。研究表明，基于EMA的估计器与真实相对L2误差高度相关，可在推理过程中实现无数据、实例感知的误差控制。在六个典型偏微分方程（一维/二维Burgers方程、二维Allen-Cahn方程、二维Cahn-Hilliard方程、二维Navier-Stokes方程及三维热传导方程）上的评估表明，ANCHOR能可靠约束长时域误差增长、稳定外推滚动轨迹，并显著提升对独立神经算子的鲁棒性，同时保持远高于高保真数值求解器的计算效率。