Math AI - Stochastic Differential Equation Forward

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Reference

PII: 0304-4149(82)90051-5 (core.ac.uk) @andersonReversetimeDiffusion1982 : 專門針對 reverse time SDE equations.

http://pordlabs.ucsd.edu/pcessi/theory2019/gardiner_ito_calculus.pdf

Ito equations

Introduction

隨機過程簡單系統

一個問題是隨機過程簡單系統有什麽實際用途？

非常多，例如 noisy control system, robust control system, finance system

更進一步，隨機過程的時光倒流 (減熵) 有什麽用途？

信號處理的 smoothing 問題

最近大熱門的 diffusion process 就是利用時光倒流來 train 一個 neural network.

之後可以用 noise 產生 image. 更廣汎說，可以從一個機率分佈 transform 到另一個機率分佈。作爲生成式 AI 或是 style transfer AI.

有幾類隨機過程問題：

  • Signal + noise estimate the original signal (Kalman filter, Kalman-Bucy filter)
  • 布朗運動 : 描述現象 only?
  • Distribution A + noise : diffusion denoise

此處我們討論隨機微分方程 (Ito equation) 的 forward-time 和 backward-time 的形式和對應 Fokker-Planck equations。

(Forward) Ito Differential Equation

我們從 general 的 stochastic differential equation (SDE) 開始： \(\begin{align} d x_t=f\left(x_t, t\right) d t+g\left(x_t, t\right) d w_t \end{align}\) where $w_t$ has the usual properties. (就是 Wiener process, white noise)

以上是 random samples or process 的微分方程。可以用來作爲 Monte Carlo 模擬，但無法直接計算機率分佈如何隨時間變化。所以可以導入機率分佈的偏微分方程 Fokker-Planck equation 如下。

Forward unconditioned equation (Fokker-Planck equation)

也就是 Fokker-Planck Equations, yields \(\begin{align} -\frac{\partial p\left(x_t, t\right)}{\partial t}= & \sum_i \frac{\partial}{\partial x_t^i}\left[p\left(x_t, t\right) f^i\left(x_t, t\right)\right] -\frac{1}{2} \sum_{i, j, k} \frac{\partial^2\left[g^{i k}\left(x_t, t\right) g^{j k}\left(x_t, t\right) p\left(x_t, t\right)\right]}{\partial x_t^i \partial x_t^j} . \end{align}\) 以上可以説是 general diffusion equation.

  • 左邊是機率分佈對時間微分。

  • 右邊第二項是機率分佈對空間的二次微分，就是機率分佈 Laplacian (空間 tension)。
  • 右邊第一項是機率分佈的 transient 項？如果 eigenvalue 實部小於 0, 會隨時間 exponential 消失。

Ito Differential Equation Examples

Coefficients Without $x$ Dependence (Forward)

$\mathrm{d}x = a(t) \,\mathrm{d}t + b(t)\, \mathrm{d}W(t)$

此處 $a(t)$ 和 $b(t)$ 是 non-random function of time. 上式兩側積分：

$x(t) = x_0 + \int_{t=t_0}^{t} a(t) \,\mathrm{d}t + \int_{t=t_0}^{t} b(t)\, \mathrm{d}W(t)$

因爲 $W(t)$ 是 Wiener (zero-mean Gaussian) process. $x(t)$ 可以視爲 Gaussian random variable 的 linear combination, 還是 zero-mean Gaussian.

這個 Gaussian process 的 mean:

\(\langle x(t)\rangle=\left\langle x_0\right\rangle+\int_{t_0}^t a(t) d t\) 因爲右邊第二項 Ito 積分的 mean 為 0. 再來計算 auto-correlation (including variance): \(\begin{align} \langle[x(t) & -\langle x(t)\rangle][x(s)-\langle x(s)\rangle]\rangle \equiv\langle x(t), x(s)\rangle \\ & =\left\langle\int_{t_0}^t b\left(t^{\prime}\right) d W\left(t^{\prime}\right) \int_{t_0}^s b\left(s^{\prime}\right) d W\left(s^{\prime}\right)\right\rangle=\int_{t_0}^{\min (t, s)}\left[b\left(t^{\prime}\right)\right]^2 d t^{\prime} \end{align}\) 有了 Gaussin 的 mean and variance 就可以完全決定 $x(t)$.

  • Random process 的 mean depends on $a(t)$ 對時間的積分。
  • Random process 的 variance 隨著時間增加 ( $b(t)^2 > 0$ )。不過是否收斂 depending on $b(t)^2$. 如果是負的 exponential function 則收斂。
  • 非 stationary process!
  • $x(t)$ 的 frequency domain spectrum 就是 FFT of auto-correlation function.

Multiplicative Linear White Noise Process (Forward)

$\mathrm{d}x = c x\, \mathrm{d}W(t)$

其 sample 的解形式如下 (make y = log x) : \(x(t)=x\left(t_0\right) \exp \left\{c\left[W(t)-W\left(t_0\right)\right]-\frac{1}{2} c^2\left(t-t_0\right)\right\}\) We can calculate the mean by using the formula for any Gaussian variable $z$ with zero mean \(\langle\exp z\rangle=\exp \left(\left\langle z^2\right\rangle / 2\right)\) so that \(\begin{aligned} \langle x(t)\rangle=\left\langle x\left(t_0\right)\right\rangle & \exp \left[\frac{1}{2} c^2\left(t-t_0\right)-\frac{1}{2} c^2\left(t-t_0\right)\right] \\ = & \left\langle x\left(t_0\right)\right\rangle . \end{aligned}\) This result is also obvious from definition, since \(\begin{align} & \langle \mathrm{d} x\rangle=\langle c x \, \mathrm{d} W(t)\rangle=0 \quad \text { so that } \\ & \frac{\mathrm{d}\langle x\rangle}{\mathrm{d} t}=0 . \end{align}\) We can also calculate the autocorrelation function \(\begin{aligned} \langle x(t) x(s)\rangle & =\left\langle x\left(t_0\right)^2\right\rangle\left\langle\exp \left\{c\left[W(t)+W(s)-2 W\left(t_0\right)\right]-\frac{1}{2} c^2\left(t+s-2 t_0\right)\right\}\right\rangle \\ & =\left\langle x\left(t_0\right)^2\right\rangle \exp \left\{\frac{1}{2} c^2\left[\left\langle\left[W(t)+W(s)-2 W\left(t_0\right)\right]^2\right\rangle-\left(t+s-2 t_0\right)\right]\right\} \\ & =\left\langle x\left(t_0\right)^2\right\rangle \exp \left\{\frac{1}{2} c^2\left[t+s-2 t_0+2 \min (t, s)-\left(t+s-2 t_0\right)\right]\right\} \\ & =\left\langle x\left(t_0\right)^2\right\rangle \exp \left[c^2 \min \left(t-t_0, s-t_0\right)\right] \end{aligned}\)

  • Random process 的 mean 不隨時間變化。Multiplicative white noise.
  • Random process 的 variance depends on $c > 1$ or $c <1$. 如果大於 1, variance exponentially 增加。
  • 非 stationary process!
  • $x(t)$ 的 frequency domain spectrum 就是 FFT of auto-correlation function.

Ornstein-Uhlenbeck Process (Forward)

假設 $x$ 是 non-deterministic $n$-dimentional process, 並由下述綫性微分方程表示：

\[\begin{align} \mathrm{d} x=A x \mathrm{~d} t+B \mathrm{~d} w . \\ \end{align}\]

其中 $A, B$ 都是常數矩陣。$w$ 是 vector Wiener process.
\(\begin{align} x(t)= x(0) \mathrm{e}^{At} + \int_{0}^t \mathrm{e}^{A(t-s)} B \mathrm{~d} w(s) \end{align}\)

\[\langle x(t)\rangle=\left\langle x_0\right\rangle \mathrm{e}^{At}\]

The correlation function follows similarly \(\begin{aligned} \left\langle\boldsymbol{x}(t), \boldsymbol{x}^{\mathrm{T}}(s)\right\rangle \equiv & \left\langle[\boldsymbol{x}(t)-\langle\boldsymbol{x}(t)\rangle][\boldsymbol{x}(s)-\langle\boldsymbol{x}(s)\rangle]^{\mathrm{T}}\right\rangle \\ = & \mathrm{e}^{A t}\left\langle\boldsymbol{x}(0), \boldsymbol{x}^{\mathrm{T}}(0)\right\rangle \mathrm{e}^{A^T s} +\int_0^{\min (t, s)} \mathrm{e}^{A\left(t-t^{\prime}\right)} B B^{\mathrm{T}} \mathrm{e}^{A^T\left(s-t^{\prime}\right)} d t' \end{aligned}\)

(Variance) Stationary Version ($A$ 的 eigenvalues 實部小於 0)

如果 $A$ 的 eigvalues 實部小於 0，存在 wide-sense stationary solution (mean and variance): \(\begin{align}x(t)=\int_{-\infty}^t \mathrm{e}^{A(t-s)} B \mathrm{~d} w(s) .\end{align}\)

\[\langle x(t)\rangle= 0\] \[\begin{aligned}\left\langle\boldsymbol{x}(t), \boldsymbol{x}^{\mathrm{T}}(s)\right\rangle = \int_{-\infty}^{\min (t, s)} \mathrm{e}^{A\left(t-t^{\prime}\right)} B B^{\mathrm{T}} \mathrm{e}^{A^T\left(s-t^{\prime}\right)} d t'\end{aligned}\]

上式乍看之下很難是 stationary，但是如果我們定義 (time average, 不是 space average!) \(\begin{aligned}\sigma = \left\langle\boldsymbol{x}(t), \boldsymbol{x}^{\mathrm{T}}(t)\right\rangle \end{aligned}\) 可以證明 $A \sigma + \sigma A^T = B B^T$ [reference Gardiner Ito Calculus]

因爲 $A, B$ 都是 constant matrix, 所以 $\sigma$ 本身也是 constant, 也就是 stationary! \(\begin{align}P=E\left[x(t) x^{T}(t)\right] = \left<x(t)x^T(t)\right> = \sigma .\end{align}\)

Appendix