The Best of Both Worlds: Hybridizing Neural Operators and Solvers for Stable Long-Horizon Inference

TL;DR

<3-5 sentence high-level summary> The paper addresses the challenge of reliable long-horizon inference for time-dependent PDEs when using fast neural operator surrogates, which can accumulate error and lose fidelity. It proposes ANCHOR, a hybrid framework that couples TI-DeepONet with a high-fidelity solver, guided by a physics-informed EMA-based residual error estimator and an adaptive, time-decaying error threshold to trigger corrections. The authors demonstrate strong correlations between the EMA estimator and true relative $L_2$ error across four canonical PDEs (1D and 2D Burgers', 2D Allen–Cahn, 3D heat conduction) and show that ANCHOR bounds error growth while delivering substantial speedups (up to 2x–3x in higher dimensions) compared to purely numerical solvers. This approach provides a practical path toward trustworthy, efficient surrogate-based simulations suitable for design, optimization, and decision-making in engineering contexts.

Abstract

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 four canonical PDEs: 1D and 2D Burgers', 2D Allen-Cahn, 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.

Figures (12)

• Figure 1: Schematic of the proposed ANCHOR framework, which couples a neural operator with a high-fidelity numerical solver. ANCHOR leverages the complementary strengths of both approaches: the efficiency of the neural surrogate and the fidelity of the numerical solver.
• Figure 2: For all PDEs considered in this work. Column 1: Temporal evolution of error accumulation across different frameworks for a representative sample. Column 2: Stabilization of relative $L_2$ error growth achieved by intermittently invoking the numerical solver, for the same samples; green shaded regions indicate timesteps handled by the high-fidelity numerical solver. Column 3: EMA-based error estimator over time, with the corresponding Pearson correlation coefficient ($\rho_{corr}$) between the estimator and the underlying relative $L_2$ error mentioned in the boxes. Column 4: Distribution of $\rho_{corr}$ across all test samples.
• Figure 3: 1D Burgers’ equation: Spatiotemporal error distributions over $t \in [0,1]$ for all frameworks, illustrated using two representative samples. The error color bar for AR-DON is shown separately from those of TI-DON and ANCHOR, since the errors of TI-DON and ANCHOR are approximately two orders of magnitude lower than those of AR-DON. The color bar on the far right indicates the timesteps solved by TI-DON (blue) and by the high-fidelity numerical solver (pink). Here, $t \in [0,0.5]$ corresponds to the interpolation regime, while $t \in [0.5,1.0]$ denotes the extrapolation regime.
• Figure 4: 2D Burgers' equation: Spatial error distributions over $t\in[0,1]$, where $t\in[0,0.33]$ corresponds to the interpolation regime and $t\in[0.33,1.0]$ to extrapolation for all frameworks, illustrated using two representative samples. The error color bars for each framework are shown separately due to significant differences in the ranges of error magnitudes. The color bar below each set of contours indicates timesteps solved by TI-DON (green) and by the high-fidelity numerical solver (magenta).
• Figure 5: 2D Allen-Cahn equation: Spatial error distributions over $t\in[0,1]$, where $t\in[0,0.33]$ corresponds to the interpolation regime and $t\in[0.33,1.0]$ to extrapolation for all frameworks, illustrated using two representative samples. The error color bars for each framework are shown separately due to significant differences in the ranges of error magnitudes. The color bar below each set of contours indicates timesteps solved by TI-DON (green) and by the high-fidelity numerical solver (magenta).