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

TL;DR

The paper introduces PITI-DeepONet, a physics-informed time-integrated neural operator that learns a continuous temporal tangent space and advances PDE solutions via standard time-stepping, enabling reliable long-horizon predictions. It unifies six losses (PDE, IC, BC, reconstruction, consistency, and data) within a physics-informed training framework and supports residual-based quality monitoring to detect out-of-distribution states. Through experiments on 1D Heat, 1D Burgers, and 2D Allen-Cahn, the method achieves superior long-term extrapolation compared to Full Rollout and Autoregressive baselines, with residuals serving as a robust error proxy. The approach demonstrates robustness to sampling and random seeds, and its residual-driven monitoring opens avenues for adaptive sampling and multi-fidelity extensions in challenging or stiff PDE regimes.

Abstract

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.

Figures (16)

• Figure 1: A schematic of the proposed physics-informed time-integrated deep operator network (PITI-DeepONet) architecture, which learns a continuous temporal tangent space operator that enables efficient and accurate time-stepping during inference.
• Figure 2: Relative $\mathcal{L}_2$ error over time for the three PDEs in comparison of FR, AR, and time integration using explicit Euler with $\Delta t = 0.01$. Depicted are the mean as well as minimum and maximum value for all test examples. Only inference with explicit Euler is depicted, but all methods show similar values in inference also as shown in Tab. \ref{['tab:result_overview']}.
• Figure 3: 1D Heat equation (Mean rel. $\mathcal{L}_2$ error $7.60\mathrm{e}\text{-} 2$): Comparison of reference data and PITI-DeepONet with explicit Euler alongside the squared difference of the computed field and the residuals during inference. Mean absolute error between squared difference and predicted residual for this example is $3.72\mathrm{e}\text{-} 5$ and Pearson's correlation coefficient is $\rho=0.9985$.
• Figure 4: 1D Heat equation: Total loss during training alongside mean relative $\mathcal{L}_2$ error on the train and validation set for full rollout (FR), autoregressive (AR), and time integration (PITI) model. The relative $\mathcal{L}_2$ error on both network outputs is provided for the PITI model. Hyperparameters were adapted per model as per table \ref{['tab:hyperparameters']}.
• Figure 5: 1D Heat equation: Comparison of time integration models in the vanilla form and special form for time-evolution partial differential equations including total loss and mean relative $\mathcal{L}_2$ error on train and validation set during training. Only the relative $\mathcal{L}_2$ error on the time derivative is provided here. The right plot shows the performance during inference with the fully trained model using an explicit Euler scheme while depicting minimum, mean, and maximum value of the relative $\mathcal{L}_2$ error per time step.