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 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 demonstrates enhanced accuracy and stability over extended inference time horizons when compared to traditional methods. Mean relative L2 errors reduced by 84% (versus FR) and 79% (versus AR) for 1D heat equation; by 87% (versus FR) and 98% (versus AR) for the 1D Burgers equation; by 42% (versus FR) and 89% (versus AR) for the 2D Allen-Cahn equation; and by 58% (vs. FR) and 61% (vs. AR) for the 1D Kuramoto-Sivashinsky 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.