讨论:Koopman算子理论

Koopman算子定义

• 生成元 [math]\displaystyle{ \mathcal{L} g }[/math]表示观测函数 [math]\displaystyle{ g }[/math]随状态演化的瞬时变化率，即[math]\displaystyle{ \mathcal{L} g(x) = \frac{d}{dt} g(x(t)) }[/math] 。

ymz：最后的这个公式其实是最直观的定义，可不可以往前放，单独列一行

pll：和上面的变为一行了

• 结合伊藤公式（Ito's lemma）

ymz：这具体是什么公式，另外中英文是这样对应的吗，lemma直译过来是引理。

pll：“伊藤公式是随机微积分里的 “链式法则”—— 专门用来算 “随布朗运动（一种随机过程）变化的函数” 的微分。”

[math]\displaystyle{ \begin{align*}\mathcal{L}_{\text{stoch}} g(x) &= \lim_{t \to 0} \frac{\mathbb{E}[g(x(t)) \mid x(0)=x] - g(x)}{t} \quad (\text{生成元定义}) \\ &= \lim_{t \to 0} \frac{\mathbb{E}\left[ g(x) + \int_0^t dg(x(s)) \mid x(0)=x \right] - g(x)}{t} \quad (\text{积分基本定理：}g(x(t)) - g(x) = \int_0^t dg(x(s))) \\ &= \lim_{t \to 0} \frac{\mathbb{E}\left[ \int_0^t \left( \sum_{i=1}^n \frac{\partial g}{\partial x_i}(x(s))dx_i(s) + \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2 g}{\partial x_i \partial x_j}(x(s))dx_i(s)dx_j(s) \right) \mid x(0)=x \right]}{t} \quad (\text{多元伊藤公式}) \\ &= \lim_{t \to 0} \frac{\mathbb{E}\left[ \int_0^t \sum_{i=1}^n \frac{\partial g}{\partial x_i}(x(s))\left( f_i(x(s))dt + \sum_{k=1}^m \sigma_{ik}(x(s))dW_k(s) \right) + \frac{1}{2}\sum_{i,j=1}^n \frac{\partial^2 g}{\partial x_i \partial x_j}(x(s))\left( \sum_{k=1}^m \sigma_{ik}(x(s))\sigma_{jk}(x(s))dt \right) ds \mid x(0)=x \right]}{t} \quad (\text{SDE分量代入+Wiener性质：}dW_k dW_l=\delta_{kl}dt, dt\cdot dt=dt\cdot dW_k=0) \\ &= \lim_{t \to 0} \frac{\mathbb{E}\left[ \int_0^t \sum_{i=1}^n \frac{\partial g}{\partial x_i}f_i(x(s))dt + \int_0^t \sum_{i=1}^n \sum_{k=1}^m \frac{\partial g}{\partial x_i}\sigma_{ik}(x(s))dW_k(s) + \int_0^t \frac{1}{2}\sum_{i,j=1}^n \sum_{k=1}^m \frac{\partial^2 g}{\partial x_i \partial x_j}\sigma_{ik}\sigma_{jk}(x(s))dt \mid x(0)=x \right]}{t} \quad (\text{积分项拆分+求和与积分交换}) \\ &= \lim_{t \to 0} \frac{\int_0^t \mathbb{E}\left[ \sum_{i=1}^n \frac{\partial g}{\partial x_i}f_i(x(s)) + \frac{1}{2}\sum_{i,j=1}^n \sum_{k=1}^m \frac{\partial^2 g}{\partial x_i \partial x_j}\sigma_{ik}\sigma_{jk}(x(s)) \mid x(0)=x \right] dt + \mathbb{E}\left[ \int_0^t \sum_{i,k} \frac{\partial g}{\partial x_i}\sigma_{ik}(x(s))dW_k(s) \mid x(0)=x \right]}{t} \quad (\text{Fubini定理：期望与积分交换}) \\ &= \lim_{t \to 0} \frac{\int_0^t \mathbb{E}\left[ \nabla g(x(s)) \cdot f(x(s)) + \frac{1}{2}\text{Tr}\left( \sigma(x(s))\sigma(x(s))^T \nabla^2 g(x(s)) \right) \mid x(0)=x \right] dt}{t} \quad (\text{向量点积定义+迹的定义；伊藤积分期望为0}) \\ &= \lim_{t \to 0} \frac{\int_0^t \left( \nabla g(x) \cdot f(x) + \frac{1}{2}\text{Tr}\left( \sigma(x)\sigma(x)^T \nabla^2 g(x) \right) \right) dt + o(t)}{t} \quad (\text{$t\to0$时$x(s)\to x$，被积函数连续+积分中值定理}) \\ &= \lim_{t \to 0} \frac{\left( \nabla g(x) \cdot f(x) + \frac{1}{2}\text{Tr}\left( \sigma(x)\sigma(x)^T \nabla^2 g(x) \right) \right) t + o(t)}{t} \quad (\text{积分结果展开}) \\ &= \nabla g(x) \cdot f(x) + \frac{1}{2}\text{Tr}\left( \sigma(x)\sigma(x)^T \nabla^2 g(x) \right) \quad (\text{约分$t$，$\lim_{t\to0}\frac{o(t)}{t}=0$}) \end{align*} }[/math]

残差DMD

• 1. Colbrook, M. J., Ayton, L. J. & Szőke, M. Residual dynamic mode decomposition: Robust and verified koopmanism. J. Fluid Mech. 955, A21 (2023).

2. Colbrook, M. J. & Townsend, A. Rigorous data‐driven computation of spectral properties of koopman operators for dynamical systems. Commun. Pure Appl. Math. 77, 221–283 (2024).

ymz：这里的文献是干什么的

pll：我引用文献习惯用编辑模式，这个模式现在用不了，就先把文献放着，防止忘记，现在已经删了，文献也加上了