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

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.

Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference

Accurately modeling and inferring solutions to time-dependent partial differential equations (PDEs) over extended horizons remains a core challenge in scientific machine learning. Traditional full rollout (FR) methods, which predict entire trajectories in one pass, often fail to capture the causal dependencies and generalize poorly outside the training time horizon. Autoregressive (AR) approaches, evolving the system step by step, suffer from error accumulation, limiting long-term accuracy. These shortcomings limit the long-term accuracy and reliability of both strategies. To address these issues, we introduce the Physics-Informed Time-Integrated Deep Operator Network (PITI-DeepONet), a dual-output architecture trained via fully physics-informed or hybrid physics- and data-driven objectives to ensure stable, accurate long-term evolution well beyond the training horizon. Instead of forecasting future states, the network learns the time-derivative operator from the current state, integrating it using classical time-stepping schemes to advance the solution in time. Additionally, the framework can leverage residual monitoring during inference to estimate prediction quality and detect when the system transitions outside the training domain. Applied to benchmark problems, PITI-DeepONet shows improved accuracy over extended inference time horizons when compared to traditional methods. Mean relative $\mathcal{L}_2$ errors reduced by 84% (vs. FR) and 79% (vs. AR) for the one-dimensional heat equation; by 87% (vs. FR) and 98% (vs. AR) for the one-dimensional Burgers equation; and by 42% (vs. FR) and 89% (vs. AR) for the two-dimensional Allen-Cahn equation. By moving beyond classic FR and AR schemes, PITI-DeepONet paves the way for more reliable, long-term integration of complex, time-dependent PDEs.

TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation

Accurate temporal extrapolation presents a fundamental challenge for neural operators in modeling dynamical systems, where reliable predictions must extend significantly beyond the training time horizon. Conventional Deep Operator Network (DeepONet) approaches employ two inherently limited training paradigms - fixed-horizon rollouts that predict complete spatiotemporal solutions while disregarding temporal causality, and autoregressive formulations that accumulate errors through sequential predictions. We introduce TI-DeepONet, a framework that integrates neural operators with adaptive numerical time-stepping techniques to preserve the Markovian structure of dynamical systems while mitigating error propagation in extended temporal forecasting. Our approach reformulates the learning objective from direct state prediction to the approximation of instantaneous time-derivative fields, which are then integrated using established numerical schemes. This architecture supports continuous-time prediction and enables deployment of higher-precision integrators during inference than those used during training, balancing computational efficiency with predictive accuracy. We further develop TI(L)-DeepONet, which incorporates learnable coefficients for intermediate slopes in the integration process, adapting to solution-specific variations and enhancing fidelity. Evaluation across three canonical PDEs shows that TI(L)-DeepONet marginally outperforms TI-DeepONet, with both reducing relative L2 extrapolation errors: approximately 81% over autoregressive and 70% over fixed-horizon methods. Notably, both maintain prediction stability for temporal domains extending to about twice the training interval. This research establishes a physics-aware operator learning paradigm that bridges neural approximation with numerical analysis while preserving the causal structure of dynamical systems.