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

arXiv: https://arxiv.org/abs/2505.17341 · HTML: https://mlsanity.com/paper/2505.17341/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 31 references that are in the map.
- First posted on arXiv 22 May 2025.

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Learning Dissipative Dynamics in Chaotic Systems](https://mlsanity.com/paper/2106.06898/) (2021)
- [Applications of physics informed neural operators](https://mlsanity.com/paper/2203.12634/) (2022)
- [Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2303.02243/) (2023)
- [On the Benefits of Memory for Modeling Time-Dependent PDEs](https://mlsanity.com/paper/2409.02313/) (2024)
- [Transfer Learning Enhanced DeepONet for Long-Time Prediction of Evolution Equations](https://mlsanity.com/paper/2212.04663/) (2022)
- [Improved generalization with deep neural operators for engineering systems: Path towards digital twin](https://mlsanity.com/paper/2301.06701/) (2023)
- [A multi-fidelity deep operator network (DeepONet) for fusing simulation and monitoring data: Application to real-time settlement prediction during tunnel construction](https://mlsanity.com/paper/2310.00057/) (2023)
- [Advanced Deep Operator Networks to Predict Multiphysics Solution Fields in Materials Processing and Additive Manufacturing](https://mlsanity.com/paper/2403.14795/) (2024)
- [Solving High-Dimensional Inverse Problems with Auxiliary Uncertainty via Operator Learning with Limited Data](https://mlsanity.com/paper/2303.11379/) (2023)

## Built on by (most cited first, 14 of 14)

- [Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference](https://mlsanity.com/paper/2508.05190/) (2025)
- [Data-Efficient Time-Dependent PDE Surrogates: Graph Neural Simulators vs. Neural Operators](https://mlsanity.com/paper/2509.06154/) (2025)
- [Multimodal Neural Operators for Real-Time Biomechanical Modelling of Traumatic Brain Injury](https://mlsanity.com/paper/2510.03248/) (2025)
- [Learning Transient Convective Heat Transfer with Geometry Aware World Models](https://mlsanity.com/paper/2601.22086/) (2026)
- [Enhanced accuracy through ensembling of randomly initialized auto-regressive models for time-dependent PDEs](https://mlsanity.com/paper/2507.03863/) (2025)
- [Sensing Without Colocation: Operator-Based Virtual Instrumentation for Domains Beyond Physical Reach](https://mlsanity.com/paper/2510.18041/) (2025)
- [ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching](https://mlsanity.com/paper/2512.19643/) (2025)
- [Learning Heat-based Equations in Self-similar variables](https://mlsanity.com/paper/2602.00872/) (2026)
- [The Error of Deep Operator Networks Is the Sum of Its Parts: Branch-Trunk and Mode Error Decompositions](https://mlsanity.com/paper/2602.21910/) (2026)
- [Hybrid Neural World Models](https://mlsanity.com/paper/2605.28317/) (2026)
- [A Trainable-by-Parts Operator Learning Framework: Bridging DeepONet and Karhunen-Loeve Expansions for Large-Scale Applications](https://mlsanity.com/paper/2606.28519/) (2026)
- [No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study](https://mlsanity.com/paper/2607.23667/) (2026)
- [LSR-Net: Learning the Forward Evolution Operator for Nonlinear Fluid Dynamics](https://mlsanity.com/paper/2609.19039/) (2026)
- [Neural Constitutive Learning for Generalized Reaction-Diffusion Systems](https://mlsanity.com/paper/2609.37113/) (2026)

## Abstract (arXiv)

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.