Unsupervised operator learning approach for dissipative equations via Onsager principle

2025-08-10

AI Summary

To address the reliance of operator learning on costly high-fidelity simulation data for dissipative partial differential equations (PDEs), this paper proposes DOOL—an unsupervised operator learning framework grounded in the Onsager variational principle, with Rayleighian functional minimization as its optimization objective and no requirement for labeled data. DOOL introduces a spatiotemporal decoupled architecture: a trunk network encodes only spatial coordinates, while an explicit time integrator enforcing physical conservation laws enables efficient training and long-term extrapolation. Notably, it is the first work to extend the Onsager framework to non-standard second-order dissipative wave systems. Experiments demonstrate that DOOL significantly outperforms supervised DeepONet and MIONet on canonical dissipative PDEs, achieving superior accuracy, strong generalization across domains and parameters, and high computational efficiency.

Technology Category

• Search and Optimization: Learning to Search
• Computer Vision: Learning & Optimization for CV
• Machine Learning: Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Existing operator learning methods rely on supervised training with high-fidelity simulation data, introducing significant computational cost. In this work, we propose the deep Onsager operator learning (DOOL) method, a novel unsupervised framework for solving dissipative equations. Rooted in the Onsager variational principle (OVP), DOOL trains a deep operator network by directly minimizing the OVP-defined Rayleighian functional, requiring no labeled data, and then proceeds in time explicitly through conservation/change laws for the solution. Another key innovation here lies in the spatiotemporal decoupling strategy: the operator's trunk network processes spatial coordinates exclusively, thereby enhancing training efficiency, while integrated external time stepping enables temporal extrapolation. Numerical experiments on typical dissipative equations validate the effectiveness of the DOOL method, and systematic comparisons with supervised DeepONet and MIONet demonstrate its enhanced performance. Extensions are made to cover the second-order wave models with dissipation that do not directly follow OVP.

Problem

• Unsupervised operator learning for dissipative equations
• Eliminating need for labeled simulation data
• Spatiotemporal decoupling for efficient training

Innovation

• Unsupervised operator learning via Onsager principle
• Minimizing Rayleighian functional without labeled data
• Spatiotemporal decoupling with trunk network efficiency