用预训练提升神经算子求解时变偏微分方程的精度与效率
Latent Neural Operator Pretraining for Solving Time-Dependent PDEs
- 在混合时变方程数据上预训练，提取物理系统通用表征
- 微调后四类问题误差降低31.7%，最高达57.1%
- 对分布外数据误差减半，数据效率提升三倍，适合小样本场景
近期预训练方法在神经算子求解偏微分方程（PDE）中受到关注，能缓解单个PDE训练时的数据稀缺问题。本文提出基于潜在神经算子（LNO）架构的潜在神经算子预训练框架（LNOP），通过在混合时变PDE数据集上预训练，实现对不同物理系统的通用表征提取，并在单个PDE数据集上微调，于潜在空间求解各类时变PDE。所提框架在四个问题上将解误差降低31.7%，微调后进一步降至57.1%。在分布外数据集上，平均误差降低约50%，数据效率提升3倍。结果表明，该方法在解精度、迁移能力和数据效率方面均优于非预训练神经算子。
原文摘要 · Abstract (English)
Pretraining methods gain increasing attraction recently for solving PDEs with neural operators. It alleviates the data scarcity problem encountered by neural operator learning when solving single PDE via training on large-scale datasets consisting of various PDEs and utilizing shared patterns among different PDEs to improve the solution precision. In this work, we propose the Latent Neural Operator Pretraining (LNOP) framework based on the Latent Neural Operator (LNO) backbone. We achieve universal transformation through pretraining on hybrid time-dependent PDE dataset to extract representations of different physical systems and solve various time-dependent PDEs in the latent space through finetuning on single PDE dataset. Our proposed LNOP framework reduces the solution error by 31.7% on four problems and can be further improved to 57.1% after finetuning. On out-of-distribution dataset, our LNOP model achieves roughly 50% lower error and 3$\times$ data efficiency on average across different dataset sizes. These results show that our method is more competitive in terms of solution precision, transfer capability and data efficiency compared to non-pretrained neural operators.
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