TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation
This addresses a key gap in long-term forecasting of complex physical systems for researchers and practitioners in computational physics and machine learning, though it is incremental as it builds on existing DeepONet paradigms.
The paper tackled the challenge of accurate long-term temporal extrapolation in neural operators for dynamical systems by introducing TI-DeepONet and TI(L)-DeepONet, which integrate neural operators with adaptive numerical time-stepping to approximate instantaneous time-derivative fields, resulting in about 81% and 70% reductions in relative L2 extrapolation error compared to autoregressive and fixed-horizon methods, respectively, and stable predictions over temporal domains nearly twice the training interval.
Accurate temporal extrapolation remains a fundamental challenge for neural operators modeling dynamical systems, where predictions must extend far beyond the training horizon. Conventional DeepONet approaches rely on two limited paradigms: fixed-horizon rollouts, which predict full spatiotemporal solutions while ignoring temporal causality, and autoregressive schemes, which accumulate errors through sequential prediction. We introduce TI-DeepONet, a framework that integrates neural operators with adaptive numerical time-stepping to preserve the Markovian structure of dynamical systems while mitigating long-term error growth. Our method shifts the learning objective from direct state prediction to approximating instantaneous time-derivative fields, which are then integrated using standard numerical solvers. This naturally enables continuous-time prediction and allows the use of higher-order integrators at inference than those used in training, improving both efficiency and accuracy. We further propose TI(L)-DeepONet, which incorporates learnable coefficients for intermediate slopes in multi-stage integration, adapting to solution-specific dynamics and enhancing fidelity. Across four canonical PDEs featuring chaotic, dissipative, dispersive, and high-dimensional behavior, TI(L)-DeepONet slightly outperforms TI-DeepONet, and both achieve major reductions in relative L2 extrapolation error: about 81% compared to autoregressive methods and 70% compared to fixed-horizon approaches. Notably, both models maintain stable predictions over temporal domains nearly twice the training interval. This work establishes a physics-aware operator learning framework that bridges neural approximation with numerical analysis principles, addressing a key gap in long-term forecasting of complex physical systems.