用高阶时间导数提升降维模型的长期预测能力
Higher-Order LaSDI: Reduced Order Modeling with Multiple Time Derivatives
- 引入高阶有限差分法，高效计算多阶时间导数
- 提出滚动损失函数，使模型在任意时长远期预测更准
- 适合需要长时间模拟的物理系统建模任务
求解复杂的偏微分方程在物理科学中至关重要，但通常需要计算成本高昂的数值方法。降维模型（ROM）通过利用维度约简来构建快速近似解。尽管现代ROM能处理参数化的PDE族，但在长时间预测上性能下降。我们提出：(1) 一种灵活、高阶且低成本的有限差分方案；(2) 一种滚动损失函数，训练ROM在任意时间范围内实现精准预测。我们在二维Burgers方程上验证了该方法的有效性。
原文摘要 · Abstract (English)
Solving complex partial differential equations is vital in the physical sciences, but often requires computationally expensive numerical methods. Reduced-order models (ROMs) address this by exploiting dimensionality reduction to create fast approximations. While modern ROMs can solve parameterized families of PDEs, their predictive power degrades over long time horizons. We address this by (1) introducing a flexible, high-order, yet inexpensive finite-difference scheme and (2) proposing a Rollout loss that trains ROMs to make accurate predictions over arbitrary time horizons. We demonstrate our approach on the 2D Burgers equation.
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