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

2025-08-07

AI Summary

To address the challenge of long-horizon, high-accuracy inference for time-dependent partial differential equations (PDEs), this paper proposes PITI-DeepONet—a dual-output operator learning architecture. Instead of directly predicting future states, it learns a physics-consistent operator mapping system states to their temporal derivatives and couples this with classical time-integration schemes to ensure stable evolution. Crucially, physical information is embedded into the integration process via a residual monitoring mechanism that quantifies prediction fidelity and detects out-of-distribution generalization failure, significantly enhancing extrapolation robustness. The model supports fully physics-driven or physics-informed data-driven training, enforced by physically constrained loss functions. Evaluated on the heat equation, Burgers equation, and Allen–Cahn equation, PITI-DeepONet achieves 42%–87% lower relative L₂ error than full rollout baselines and 79%–98% lower error than autoregressive methods, markedly improving long-term predictive accuracy and reliability.

Technology Category

• Search and Optimization: Learning to Search
• Machine Learning: Deep Neural Architectures and Foundation Models
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

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 fully 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 shows improved accuracy over extended inference time horizons when compared to traditional methods. Mean relative $mathcal{L}_2$ errors reduced by 84% (vs. FR) and 79% (vs. AR) for the one-dimensional heat equation; by 87% (vs. FR) and 98% (vs. AR) for the one-dimensional Burgers equation; and by 42% (vs. FR) and 89% (vs. AR) for the two-dimensional Allen-Cahn 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.

Problem

• Accurately modeling time-dependent PDEs over long horizons
• Reducing error accumulation in autoregressive PDE solutions
• Improving long-term inference reliability in scientific machine learning

Innovation

• Dual-output architecture for stable long-term evolution
• Learns time-derivative operator with classical integration
• Residual monitoring for prediction quality estimation