单条件神经求解器为参数化微分方程编码可迁移响应空间

中文摘要 AI 辅助

针对参数化偏微分方程（PDE）的算子学习通常在预定域上构建全局模型，这需要跨条件数据或昂贵的物理约束训练。本文表明，在单一条件下训练的神经求解模型的输出雅可比矩阵定义了一个可复用的响应空间，用于跨条件的解变化。我们提出了线性化子空间迁移（LST）以利用该空间，并通过在响应空间坐标上最小化目标PDE系统残差来恢复目标解。由于任何单一响应空间的覆盖范围有限，主动迁移建模（ATM）将迁移后的残差作为覆盖指标，从额外的单条件模型中选择性获取响应空间。在六个系统中，单条件响应空间支持跨条件迁移，当添加的空间扩展了表示能力时，富集可提高准确性。与评估的物理信息算子基线相比，ATM降低了误差和离线构建成本，在代表性案例中实现了数量级的精度提升，并将目标适应时间缩短至毫秒到秒级。这些结果确立了神经求解器作为可复用的局部参数模型。

英文摘要

Operator learning for parametric partial differential equations (PDEs) typically builds global models over prescribed domains, requiring cross-condition data or costly physics-constrained training. Here we show that the output Jacobian of a neural solution model trained at one condition defines a reusable response space for cross-condition solution variations. We introduce Linearized Subspace Transfer (LST) to exploit this space and recover target solutions by minimizing the target PDE-system residual over response-space coordinates. Because any single response space has finite coverage, Active Transfer Modeling (ATM) uses post-transfer residuals as coverage indicators to selectively acquire response spaces from additional single-condition models. Across six systems, single-condition response spaces supported cross-condition transfer, with enrichment improving accuracy when added spaces expanded representation capacity. Relative to evaluated physics-informed operator baselines, ATM reduced error and offline construction cost, with orders-of-magnitude accuracy gains in representative cases and millisecond-to-second target adaptation. These results establish neural solvers as reusable local parametric models.

发表机构

• Institute of AI for Industries, Chinese Academy of Sciences（中国科学院工业人工智能研究所）
• Institute of Computing Technology, Chinese Academy of Sciences（中国科学院计算技术研究所）
• Northwestern Polytechnical University（西北工业大学）

机构由 AI 辅助整理，请以论文原文为准。