发现在刚体运动下等变的偏微分方程（PDEs）
Discovering PDEs equivariant under rigid motions
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中文总结 AI 辅助
该研究针对含噪声观测下的PDE发现问题，利用刚体运动对称性构建等变候选项库，结合稀疏回归测试五种方程，在特定噪声和高维场景下优势明显且资源消耗低。
中文摘要 AI 辅助
我们研究在潜在动力学对所有刚体运动对称的假设下，从可能含噪声的观测中发现偏微分方程（PDE）的问题。我们未使用通用的导数单项式库，而是利用该假设构建候选项自身为刚体运动等变的库，并结合稀疏回归，在五种不同方程上对这类库进行基准测试。当噪声本身破坏刚体运动对称性（如径向或轴向变化的噪声），且环境空间维度增加时，该方法的优势最为显著，此时它的资源消耗也更低。
英文摘要
We consider the problem of PDE discovery from possibly noisy observations under the hypothesis that the underlying dynamic is symmetric in all rigid motions. Rather than using a generic library of derivative monomials, we leverage this assumption to construct libraries whose candidate terms are themselves rigid-motion-equivariant, and combine them with sparse regression to benchmark such libraries over five different equations. The advantage is most pronounced when the noise itself breaks rigid-motion symmetry (e.g., radially or axially varying noise), and when the ambient spatial dimension increases, in which case they are also less resource intensive.
