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

arXiv: https://arxiv.org/abs/2508.05190 · HTML: https://mlsanity.com/paper/2508.05190/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 7 Aug 2025.

## Builds on (most cited first, 21 of 21)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Modeling the Dynamics of PDE Systems with Physics-Constrained Deep Auto-Regressive Networks](https://mlsanity.com/paper/1906.05747/) (2019)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Towards Stability of Autoregressive Neural Operators](https://mlsanity.com/paper/2306.10619/) (2023)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [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)
- [Deep Neural Operator Driven Real Time Inference for Nuclear Systems to Enable Digital Twin Solutions](https://mlsanity.com/paper/2308.07523/) (2023)
- [Improved generalization with deep neural operators for engineering systems: Path towards digital twin](https://mlsanity.com/paper/2301.06701/) (2023)
- [Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning](https://mlsanity.com/paper/2407.15887/) (2024)
- [Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysms](https://mlsanity.com/paper/2205.03780/) (2022)
- [Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturing](https://mlsanity.com/paper/2405.09572/) (2024)
- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble](https://mlsanity.com/paper/2503.06320/) (2025)
- [Advanced Deep Operator Networks to Predict Multiphysics Solution Fields in Materials Processing and Additive Manufacturing](https://mlsanity.com/paper/2403.14795/) (2024)
- [Learning Hidden Physics and System Parameters with Deep Operator Networks](https://mlsanity.com/paper/2412.05133/) (2024)

## Built on by (most cited first, 5 of 5)

- [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)
- [ANCHOR: Error-Controlled Adaptive Numerical Correction for Neural Operator Time Marching](https://mlsanity.com/paper/2512.19643/) (2025)
- [No Free Lunch in Flow Surrogates under Time-Varying Boundary Conditions: A Two-Regime Study](https://mlsanity.com/paper/2607.23667/) (2026)
- [Neural Constitutive Learning for Generalized Reaction-Diffusion Systems](https://mlsanity.com/paper/2609.37113/) (2026)

## Abstract (arXiv)

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.