用参数化方法提升非线性系统预测精度，无需重新训练即可泛化到新参数。
Parametric Interpolation of Dynamic Mode Decomposition for Predicting Nonlinear Systems
- 将参数相关结构嵌入DMD回归，直接学习参数依赖的低维模型
- 仅需少量训练数据，就能在多维参数空间中实现长期准确预测
- 适用于流体、粒子束等复杂物理系统，尤其适合数据稀疏场景
本文提出参数化插值动态模态分解（piDMD），一种将已知参数仿射结构直接融入DMD回归步骤的参数化降阶建模框架。与现有方法通过插值模态、特征值或降阶算子不同，piDMD在多个训练参数样本上学习单一参数仿射的Koopman代理降阶模型（ROM），并在未见参数值下进行预测而无需重新训练。我们在圆柱绕流、横向磁场中电子束振荡以及虚拟阴极振荡三个基准问题上验证了该方法，后两者采用电磁粒子-网格（EMPIC）方法模拟。在所有测试中，piDMD均实现了高精度长期预测，并在训练样本更少、参数空间维度更高的情况下，优于当前主流基于插值的参数化DMD基线方法。
原文摘要 · Abstract (English)
We present parameter-interpolated dynamic mode decomposition (piDMD), a parametric reduced-order modeling framework that embeds known parameter-affine structure directly into the DMD regression step. Unlike existing parametric DMD methods which interpolate modes, eigenvalues, or reduced operators and can be fragile with sparse training data or multi-dimensional parameter spaces, piDMD learns a single parameter-affine Koopman surrogate reduced order model (ROM) across multiple training parameter samples and predicts at unseen parameter values without retraining. We validate piDMD on fluid flow past a cylinder, electron beam oscillations in transverse magnetic fields, and virtual cathode oscillations -- the latter two being simulated using an electromagnetic particle-in-cell (EMPIC) method. Across all benchmarks, piDMD achieves accurate long-horizon predictions and improved robustness over state-of-the-art interpolation-based parametric DMD baselines, with less training samples and with multi-dimensional parameter spaces.
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