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

TL;DR

TI-DeepONet introduces a physics-informed neural operator framework that combines adaptive numerical time-stepping with derivative approximation, enabling stable, accurate long-term predictions of complex dynamical systems beyond training horizons.

Contribution

The paper proposes TI-DeepONet, integrating neural operators with adaptive numerical solvers and derivative-based learning, to improve long-term extrapolation stability and accuracy in dynamical system modeling.

Findings

Achieves approximately 96.3% reduction in extrapolation error compared to autoregressive methods.

Maintains stable predictions over nearly twice the training time span.

Outperforms baseline methods across six diverse PDE benchmarks.

Abstract

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 stages in multi-stage integration, adapting to solution-specific dynamics and enhancing fidelity. Across six canonical PDEs featuring chaotic, dissipative, dispersive, and high-dimensional behavior, TI(L)-DeepONet maeginally outperforms TI-DeepONet, and both achieve major reductions in relative L2 extrapolation error: about 96.3% compared to autoregressive methods and 83.6% 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.