Single-condition neural solvers encode transferable response spaces for parametric differential equations

2026-09-14

AI Summary

该研究通过利用单条件神经求解器的输出雅可比矩阵定义可重用响应空间，并采用线性子空间转移和主动转移建模方法，解决了参数化偏微分方程跨条件求解的问题。

Abstract

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.

Problem

• parametric partial differential equations
• cross-condition transfer
• neural solution model

Innovation

• Linearized Subspace Transfer
• Active Transfer Modeling
• neural solvers
• parametric PDEs
• response space