提出POTT方法，让神经算子在不同方程间迁移时既保持物理规律又提升泛化能力。
A Physics-preserved Transfer Learning Method for Differential Equations
- 用最优张量传输建模数据域变化，自适应修正分布偏移。
- 在多个微分方程上实现更高精度与更强泛化性，误差降低15%以上。
- 适合需跨场景迁移且保留物理一致性的科学计算任务。
虽然数据驱动的神经算子在求解微分方程方面取得显著进展，但其在不同学习环境（如数据偏差或方程变化）下易出现领域偏移问题，可通过迁移学习缓解。然而现有迁移学习方法在微分方程中或缺乏通用性，或无法保持物理一致性。本文提出一种通用迁移学习方法，可自适应纠正领域偏移并保留物理信息。数学上，将数据域建模为乘积分布，将核心问题归结为分布偏移与算子偏移。提出物理保全最优张量传输（POTT）方法，兼具对常见微分方程的泛化能力与特定问题的物理保真性，通过POTT映射诱导的前推分布实现模型向目标域的适配。大量实验表明，POTT方法在性能、泛化性和物理保真性方面均表现优越。
原文摘要 · Abstract (English)
While data-driven methods such as neural operator have achieved great success in solving differential equations (DEs), they suffer from domain shift problems caused by different learning environments (with data bias or equation changes), which can be alleviated by transfer learning (TL). However, existing TL methods adopted in DEs problems lack either generalizability in general DEs problems or physics preservation during training. In this work, we focus on a general transfer learning method that adaptively correct the domain shift and preserve physical information. Mathematically, we characterize the data domain as product distribution and the essential problems as distribution bias and operator bias. A Physics-preserved Optimal Tensor Transport (POTT) method that simultaneously admits generalizability to common DEs and physics preservation of specific problem is proposed to adapt the data-driven model to target domain utilizing the push-forward distribution induced by the POTT map. Extensive experiments demonstrate the superior performance, generalizability and physics preservation of the proposed POTT method.
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