AI 中文总结
该研究针对一维守恒偏微分方程提出基于累积分布变换的降阶建模框架，通过分析不同方程在CDT坐标下的解流形得到估计，据此开发CDT - POD数值方案，实验表明其能用更少模态捕捉解流形。
AI 中文摘要
我们提出了一种基于累积分布变换（CDT）的一维守恒偏微分方程降阶建模（ROM）框架。CDT将非负、等质量状态映射到一个希尔伯特空间，其中一维瓦瑟斯坦距离变为加权\(L^2\)距离，平移变为仿射变换。这使得该变换特别适用于以输运为主导的动力学，在这种情况下，欧拉线性子空间ROM通常存在柯尔莫哥洛夫宽度衰减缓慢的问题。我们通过在CDT坐标中分析解流形来研究标量守恒动力学中的这一现象。对于线性输运，变换后的解流形包含在由变换后的初始数据和常数函数所张成的二维空间中，且柯尔莫哥洛夫2宽度为零。对于非线性双曲守恒律，我们证明了两种互补类型的估计：仅依赖于守恒输运结构且在激波形成后仍有意义的稳健\(O(n^{-1})\)界，以及在光滑的激波前区域更精确的\(O(n^{-2})\)界。对于守恒平流扩散，我们表明CDT轨迹与纯输运平面的距离保持在\(O(\sqrt{DT})\)以内，并且在额外的正则性条件下或远离初始层时，我们还得到了更精确的\(O(D^2T^2)\)估计。在这两种情况下，当扩散系数趋于零时，线性输运的零2宽度行为得以恢复。基于这些估计，我们开发了一种CDT - POD数值方案：将快照映射到CDT空间，在变换后的坐标中进行本征正交分解（POD），并使用逆CDT来重建物理状态。对几种以输运为主导的动力学进行的数值实验表明，CDT - POD能够用比欧拉POD少得多的模态来捕捉解流形。
英文摘要
We propose a reduced order modeling (ROM) framework for 1D conservative PDEs based on the cumulative distribution transform (CDT). The CDT maps nonnegative, equal-mass states into a Hilbert space in which 1D Wasserstein distances become weighted $L^2$ distances and translations become affine shifts. This makes the transform especially suited for transport-dominated dynamics, where Eulerian linear-subspace ROMs often suffer from slow decay of Kolmogorov widths. We study this phenomenon for scalar conservative dynamics by analyzing the solution manifold in CDT coordinates. For linear transport, the transformed solution manifold is contained in the 2-dimensional space spanned by the transformed initial datum and the constant function, and has zero Kolmogorov $2$-width. For nonlinear hyperbolic conservation laws, we prove two complementary types of estimates: robust $O(n^{-1})$ bounds that rely only on the conservative transport structure and remain meaningful after shock formation, and sharper $O(n^{-2})$ bounds in smooth pre-shock regimes. For conservative advection-diffusion, we show that the CDT trajectory remains within distance $O(\sqrt{DT})$ of the pure-transport plane, and we also obtain sharper $O(D^2T^2)$ estimates under additional regularity or away from initial layers. In both cases, the zero 2-width behavior of linear transport is recovered as the diffusion coefficient tends to zero. Motivated by these estimates, we develop a CDT-POD numerical scheme: snapshots are mapped to CDT space, Proper Orthogonal Decomposition (POD) is performed in transformed coordinates, and the inverse CDT is used to reconstruct physical states. Numerical experiments for several transport-dominated dynamics show that CDT-POD can capture solution manifolds with substantially fewer modes than Eulerian POD.
