用最优投影修复数据驱动模型的守恒律，精准又不扰动原动力学。
A Frobenius-Optimal Projection for Enforcing Linear Conservation in Learned Dynamical Models
- 通过Frobenius范数最小投影，强制线性守恒律成立
- 修正后矩阵与原模型差异极小，且精确满足守恒条件
- 适用于任意线性学习模型，尤其适合物理建模场景
针对数据驱动线性动力系统中守恒律失效的问题，本文提出一种基于Frobenius范数的最优投影方法。给定学习得到的算子 $\\(widehat{A}$$$ 和编码一个或多个不变量的满秩约束矩阵 $C$，本文证明：在满足 $C^\top A = 0$ 的条件下，最接近 $\\(widehat{A}$$$ 的矩阵 $A^\star$ 为正交投影 $A^\star = \\widehat{A} - C(C^\top C)^{-1}C^\top \\widehat{A}$。该修正具有唯一性、低秩特性，完全由 $C^\top \\widehat{A}$ 决定。单守恒情形下退化为秩一更新。理论证明 $A^\star$ 能精确实现守恒，同时对动力学扰动最小。数值实验在马尔可夫型例子中验证了该性质。该投影为任意学习线性模型嵌入精确不变量提供通用、基础的机制。
原文摘要 · Abstract (English)
We consider the problem of restoring linear conservation laws in data-driven linear dynamical models. Given a learned operator $\widehat{A}$ and a full-rank constraint matrix $C$ encoding one or more invariants, we show that the matrix closest to $\widehat{A}$ in the Frobenius norm and satisfying $C^\top A = 0$ is the orthogonal projection $A^\star = \widehat{A} - C(C^\top C)^{-1}C^\top \widehat{A}$. This correction is uniquely defined, low rank and fully determined by the violation $C^\top \widehat{A}$. In the single-invariant case it reduces to a rank-one update. We prove that $A^\star$ enforces exact conservation while minimally perturbing the dynamics, and we verify these properties numerically on a Markov-type example. The projection provides an elementary and general mechanism for embedding exact invariants into any learned linear model.
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