无需标注数据，用能量原理训练神经网络求解耗散方程。
Unsupervised operator learning approach for dissipative equations via Onsager principle
- 基于昂萨格原理，直接最小化能量泛函训练网络。
- 在典型耗散方程上实现高精度预测，优于有监督方法。
- 适合无高质量仿真数据时的物理建模，尤其适用于动态系统。
现有算子学习方法依赖高保真模拟数据进行有监督训练，带来高昂计算成本。本文提出深度昂萨格算子学习（DOOL）方法，一种新型无监督框架，用于求解耗散方程。该方法基于昂萨格变分原理（OVP），通过直接最小化OVP定义的雷利扬泛函来训练深度算子网络，无需标签数据，并通过解的守恒/变化律显式推进时间。另一关键创新在于时空解耦策略：算子主干网络仅处理空间坐标，提升训练效率；集成外部时间步进实现时间外推。在典型耗散方程上的数值实验验证了方法有效性，与有监督的DeepONet和MIONet系统的对比显示其性能更优。方法还拓展至不直接服从OVP的二阶耗散波动模型。
原文摘要 · Abstract (English)
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.
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