参数化线性偏微分方程的非线性流形降阶模型的残差驱动提升识别
Residual-Driven Lifting Identification for Nonlinear-Manifold Reduced-Order Models of Parametrized Linear PDEs
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中文总结 AI 辅助
该研究针对参数化线性偏微分方程，提出残差驱动的非线性流形降阶模型提升识别方法，无需高保真快照即可实现，在平流-扩散等问题上精度优于线性子空间，与快照驱动方法精度相当。
中文摘要 AI 辅助
我们提出一种残差驱动的方法，用于训练参数化线性偏微分方程的非线性流形降阶模型，该方法在给定潜在空间和提升空间的情况下，无需高保真解快照即可识别非线性提升。近似由低维潜在坐标和向更丰富的降阶空间的非线性提升表示。我们不将提升拟合到快照数据，而是通过最小化可计算的基于残差的状态误差上界来确定它。对于仿射参数化算子，所得训练目标允许高效的离线-在线分解，且提升更新简化为一系列低维加权最小二乘问题。对平流-扩散问题和平面应变弹性基准的数值评估表明，所提方法相比线性子空间显著提高了精度。所得非线性模型达到与快照驱动的非线性流形训练相当的精度，同时在提升识别阶段避免了高保真快照。
英文摘要
We introduce a residual-driven procedure for training nonlinear-manifold reduced-order models for parametrized linear partial differential equations that, given prescribed latent and lifting spaces, identifies the nonlinear lifting without high-fidelity solution snapshots. The approximation is represented by a low-dimensional latent coordinate together with a nonlinear lifting into a richer reduced space. Rather than fitting the lifting to snapshot data, we determine it by minimizing a computable residual-based upper bound for the state error. For affinely parametrized operators, the resulting training objective admits an efficient offline--online decomposition, and the lifting update reduces to a sequence of low-dimensional weighted least-squares problems. Numerical evaluations on an advection--diffusion problem and a plane-strain elasticity benchmark show that the proposed approach substantially improves accuracy over linear subspaces. The resulting nonlinear models achieve accuracy comparable to snapshot-driven nonlinear-manifold training while avoiding high-fidelity snapshots in the lifting-identification stage.
