Koopman operator-based discussion on partial observation in stochastic systems

Consider a state space $\mathcal{M}\subseteq\mathbb{R}^{D}$ and an adequate functional space $\mathcal{F}$; an element in $\mathcal{F}$ is called an observable function, and $\mathcal{F}\ni\phi:\mathcal{M}\to\mathbb{R}$. To emphasize the distinctions between the state space and the functional space, we add underlines to denote state vectors on $\mathcal{M}$, e.g., $\underline{\bm{x}}$. The $d$-th element of $\underline{\bm{x}}$ is denoted as $\underline{x_{d}}\in\mathbb{R}$. By contrast, an observable function, which yields the value of the $d$-th element $\underline{x_{d}}$, is denoted as $x_{d}(\underline{\bm{x}})=\underline{x_{d}}$. Note that it is common in many papers on the Koopman operator approach to use abbreviations, e.g., $x_{d}\in\mathcal{F}$, for notational simplicity. While the abbreviations are convenient, we again emphasize the importance of distinguishing between elements in the state space and those in the function space. Hence, we employ the notation with underlines for the state space.

Let $\underline{\bm{x}(t)}$ be the state vector of the system at time $t$. The deterministic time-evolution equation for $\underline{\bm{x}(t)}$ is given by

$\displaystyle\frac{\rmd}{\rmd t}\underline{\bm{x}(t)}=\bm{a}(\underline{\bm{x}(t)}),$

(1)

where $\bm{a}(\underline{\bm{x}})$ is a vector of functions giving the time-evolution for each coordinate $\underline{x_{d}(t)}$. While it could be possible to include the time variable $t$ in the functions, we assume that $\bm{a}(\underline{\bm{x}})$ is time-independent for simplicity.

It is beneficial to see the coupled ordinary differential equations in (1) from the viewpoint of a dynamical system with discrete-time steps. Setting a time interval for observation as $\Delta t_{\mathrm{obs}}$, the state vector evolves in time as follows:

$\displaystyle\underline{\bm{x}(t+\Delta t_{\mathrm{obs}})}=\bm{F}_{\Delta t_{\mathrm{obs}}}(\underline{\bm{x}(t)}),$

(2)

where $\bm{F}_{\Delta t_{\mathrm{obs}}}:\mathcal{M}\to\mathcal{M}$ yields the state vector after the time-evolution with $\Delta t_{\mathrm{obs}}$.

In the Koopman operator approach, we consider a time-evolution of observable functions instead of that of the state vector. Consider the observable function $x_{d}:\mathcal{M}\to\mathbb{R}$, which yields the $d$-th element of the state vector. Then, we consider a function that gives the $d$-th element of the state vector after the time-evolution with $\Delta t_{\mathrm{obs}}$. As we will see later, it is possible to obtain such a function by employing a map $\mathcal{K}_{\Delta t_{\mathrm{obs}}}:\mathcal{F}\to\mathcal{F}$:

$\displaystyle\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{d}=x_{d}\circ\bm{F}_{\Delta t_{\mathrm{obs}}},$

(3)

which leads to

$\displaystyle x_{d}(\underline{\bm{x}(t+\Delta t_{\mathrm{obs}})})=\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{d}(\underline{\bm{x}(t)}).$

(4)

Note that the Koopman operator is linear even if the original system is nonlinear.

Figure 1 shows the correspondence between the state space $\mathcal{M}$ and the function space $\mathcal{F}$. Although the observable function $x_{1}$ is the simple flat plane, the action of the Koopman operator yields the curved surface $\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{1}$. The function $\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{1}:\mathcal{M}\to\mathbb{R}$ gives the coordinate value after the time-evolution, i.e., $\underline{x_{1}(t+\Delta t_{\mathrm{obs}})}$.

Note that while it is possible to consider various types of observable functions, we focus only on the simple observables $\{x_{d}\}$ in the present work. A collection of the observable functions for the coordinates is called a full-state observable $\bm{g}:\mathcal{M}\to\mathbb{R}^{D}$, which is a vector-valued observable function and

$\displaystyle\bm{g}=[x_{1}\quad x_{2}\quad\dots\quad x_{D}]^{\top}.$

(5)

Again, we note that it is crucial to distinguish $\bm{g}$ and $\underline{\bm{x}}=[\underline{x_{1}}\,\,\underline{x_{2}}\,\,\dots\,\,\underline{x_{D}}]^{\top}\in\mathbb{R}^{D}$.

As discussed in [23], we can obtain an approximate representation of the Koopman operator $\mathcal{K}_{\Delta t_{\mathrm{obs}}}$ by a data-driven approach. That is, introducing a set of basis functions, called a dictionary, leads to a Koopman matrix $K_{\Delta t_{\mathrm{obs}}}$ for the time interval $\Delta t_{\mathrm{obs}}$. The data-driven method is called the EDMD algorithm.

The EDMD algorithm requires a data set of snapshot pairs. The snapshot pairs are denoted as $\{(\underline{\bm{x}_{n}},\underline{\bm{y}_{n}})\}_{n=1}^{N_{\mathrm{data}}}$, where

$\displaystyle\underline{\bm{x}_{n}}=[\underline{x_{n1}(t)}\quad\underline{x_{n2}(t)}\quad\cdots\quad\underline{x_{nD}(t)}]^{\top}$

(6)

and

	$\displaystyle\underline{\bm{y}_{n}}$	$\displaystyle=\bm{F}_{\Delta t_{\mathrm{obs}}}(\underline{\bm{x}_{n}})$			(7)
		$\displaystyle=[\underline{x_{n1}(t+\Delta t_{\mathrm{obs}})}\quad\underline{x_{n2}(t+\Delta t_{\mathrm{obs}})}\quad\cdots\quad\underline{x_{nD}(t+\Delta t_{\mathrm{obs}})}]^{\top}.$

$N_{\mathrm{data}}$ represents the number of snapshot pairs. The dictionary, $\bm{\psi}$, consists of $N_{\mathrm{dic}}$ basis functions, i.e.,

$\displaystyle\bm{\psi}(\underline{\bm{x}})=[\psi_{1}(\underline{\bm{x}})\quad\psi_{2}(\underline{\bm{x}})\quad\cdots\quad\psi_{N_{\mathrm{dic}}}(\underline{\bm{x}})]^{\top},$

(8)

where $\psi_{i}:\mathcal{M}\to\mathbb{R}$ for $i=1,\dots,N_{\mathrm{dic}}$. For example, a dictionary with monomial functions for $D=2$ is given as

$\displaystyle\bm{\psi}(\underline{\bm{x}})=[\underline{1}\quad x_{1}(\underline{\bm{x}})\quad x_{2}(\underline{\bm{x}})\quad x_{1}^{2}(\underline{\bm{x}})\quad x_{1}x_{2}(\underline{\bm{x}})\quad x_{2}^{2}(\underline{\bm{x}})\quad x_{1}^{3}(\underline{\bm{x}})\quad\dots\quad x_{2}^{5}(\underline{\bm{x}})]^{\top},$

(9)

where the maximum degree of the monomial functions is $5$, and $\underline{1}$ means a scalar value $1$. The dictionary spans a subspace $\mathcal{F}_{\bm{\psi}}\subseteq\mathcal{F}$.

Note that the dictionary with the monomial functions contains the full-state observable $\bm{g}$; see the second and third elements on the right-hand side of (9). However, the action of the Koopman operator yields the function $\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{d}$, which may not be in the spanned subspace $\mathcal{F}_{\bm{\psi}}$. In this case, we have the following expansion:

$\displaystyle\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{d}(\underline{\bm{x}})\simeq\sum_{i=1}^{N_{\mathrm{dic}}}c^{\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{d}}_{i}\psi_{i}(\underline{\bm{x}}),$

(10)

where $\{c^{\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{d}}_{i}\}_{i=1}^{N_{\mathrm{dic}}}$ are the expansion coefficients. Similarly, we can consider the action for an arbitrary function $\phi\in\mathcal{F}$, $\mathcal{K}_{\Delta t_{\mathrm{obs}}}\phi$, and its coefficients $\{c^{\mathcal{K}_{\Delta t_{\mathrm{obs}}}\phi}_{i}\}_{i=1}^{N_{\mathrm{dic}}}$. Note that the function $\phi$ before the time-evolution may not be in the spanned subspace $\mathcal{F}_{\bm{\psi}}$, and the function $\phi$ is also approximately written as follows:

$\displaystyle\phi(\underline{\bm{x}})\simeq\sum_{i=1}^{N_{\mathrm{dic}}}c^{\phi}_{i}\psi_{i}(\underline{\bm{x}}).$

(11)

Since a linear combination of the dictionary functions yields an arbitrary function and its time-evolved function approximately, it is enough for us to consider the action of the Koopman operator to the dictionary. Then, the Koopman operator $\mathcal{K}_{\Delta t_{\mathrm{obs}}}$ is approximated by a finite-size matrix $K_{\Delta t_{\mathrm{obs}}}$ as follows:

$\displaystyle\left[\begin{array}[]{c}\mathcal{K}_{\Delta t_{\mathrm{obs}}}\psi_{1}(\underline{\bm{x}})\\ \mathcal{K}_{\Delta t_{\mathrm{obs}}}\psi_{2}(\underline{\bm{x}})\\ \vdots\\ \mathcal{K}_{\Delta t_{\mathrm{obs}}}\psi_{N_{\mathrm{dic}}}(\underline{\bm{x}})\end{array}\right]=\mathcal{K}_{\Delta t_{\mathrm{obs}}}\bm{\psi}(\underline{\bm{x}})\simeq K_{\Delta t_{\mathrm{obs}}}\bm{\psi}(\underline{\bm{x}}).$

(16)

It is easy to determine the Koopman matrix $K_{\Delta t_{\mathrm{obs}}}$ from the dataset; we solve the following least squares problem numerically:

$\displaystyle K_{\Delta t_{\mathrm{obs}}}=\mathop{\rm arg~{}min}\limits_{\widetilde{K}_{\Delta t_{\mathrm{obs}}}}\sum_{n=1}^{N_{\mathrm{data}}}\left\|\bm{\psi}(\underline{\bm{y}_{n}})-\widetilde{K}_{\Delta t_{\mathrm{obs}}}\bm{\psi}(\underline{\bm{x}_{n}})\right\|^{2}.$

(17)

Note that a comparison with (10) and (17) immediately leads to the element at the $d$-th row and the $i$-th column of the Koopman matrix $K_{\Delta t_{\mathrm{obs}}}$ as follows:

	$\displaystyle\left[K_{\Delta t_{\mathrm{obs}}}\right]_{di}=c^{\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{d-1}}_{i}\quad(\textrm{for $2\leq d\leq(D+1)$}),$
	$\displaystyle\left[K_{\Delta t_{\mathrm{obs}}}\right]_{di}=c^{\mathcal{K}_{\Delta t_{\mathrm{obs}}}\psi_{d}}_{i}\quad(\textrm{for $d>(D+1)$}).$		(18)

Since the first dictionary function $\psi_{1}(\bm{\underline{x}})=1(\bm{\underline{x}})$ is time-invariant, $[K_{\Delta t_{\mathrm{obs}}}]_{1i}=1$ for all $i$.

In short, one can consider the linear action of $\mathcal{K}_{\Delta t_{\mathrm{obs}}}$ in the function space instead of the nonlinear time-evolution on the state space, $\bm{F}_{\Delta t_{\mathrm{obs}}}$, in (2). This linearity naturally leads to the use of a linear combination of basis functions, which facilitates estimation from the data via the least squares method. The usage of linearity is one of the benefits of the Koopman operator approach.

As denoted in Sec. 1, the Koopman operator approach is also available for stochastic systems; for example, see [23, 32, 33]. In the stochastic cases, the action of the Koopman operator yields an expected value of a statistic for the state variable rather than the state variable itself. For example,

$\displaystyle\mathcal{K}_{\Delta t_{\mathrm{obs}}}x_{1}(\underline{\bm{x}_{\mathrm{ini}}})=\mathbb{E}[x_{1}(t+\Delta t_{\mathrm{obs}})|\underline{\bm{x}(t)}=\underline{\bm{x}_{\mathrm{ini}}}].$

(19)

In the following, we briefly explain why the Koopman operator approach yields the expectation values from the viewpoint of the Fokker-Planck and the backward Kolmogorov equations.

Consider a $D$-dimensional vector of stochastic variables, $\underline{\bm{X}}\in\mathcal{M}$, and assume that the vector $\underline{\bm{X}(t)}$ at time $t$ obeys the following stochastic differential equation:

$\displaystyle\rmd\underline{\bm{X}(t)}=\bm{a}(\underline{\bm{X}(t)})\,\rmd t+B(\underline{\bm{X}(t)})\,\rmd\underline{\bm{W}(t)},$

(20)

where $\bm{a}(\underline{\bm{X}})$ is a vector of drift coefficient functions, $B(\underline{\bm{X}})$ is a matrix of diffusion coefficient functions, and $\underline{\bm{W}(t)}$ is a vector of Wiener processes $\{\underline{W_{i}(t)}\}$. The Wiener processes satisfy

$\displaystyle\mathbb{E}[\underline{W_{i}(t)}]=0,\qquad\mathbb{E}[(\underline{W_{i}(t)}-\underline{W_{i}(s)})(\underline{W_{j}(t)}-\underline{W_{j}(s)})]=(t-s)\delta_{ij},$

(21)

for $0\leq s\leq t$. Let $\underline{\bm{X}(0)}=\underline{\bm{x}_{\mathrm{ini}}}$ be the initial condition for the stochastic differential equation. The stochastic differential equation in (20) has a corresponding Fokker-Planck equation [34], which describes the time-evolution of the probability density function $p(\underline{\bm{x}},t)$:

$\displaystyle\frac{\partial}{\partial t}p(\underline{\bm{x}},t)=\mathcal{L}^{\dagger}\,p(\underline{\bm{x}},t),$

(22)

where

$\displaystyle\mathcal{L}^{\dagger}=-\sum_{d}\frac{\partial}{\partial\underline{x_{d}}}a_{d}(\underline{\bm{x}})+\frac{1}{2}\sum_{i,j}\frac{\partial^{2}}{\partial\underline{x_{i}}\partial\underline{x_{j}}}\left[B(\underline{\bm{x}})B(\underline{\bm{x}})^{\top}\right]_{ij}$

(23)

is the time-evolution operator for the Fokker-Planck equation. The initial condition at time $t$ is

$\displaystyle p(\underline{\bm{x}},t)=\delta(\underline{\bm{x}}-\underline{\bm{x}_{\mathrm{ini}}}),$

(24)

where $\delta(\cdot)$ is the Dirac delta function.

Note that the time-evolved probability density function, $p(\underline{\bm{x}},t+\Delta t_{\mathrm{obs}})$, is formally written as

$\displaystyle p(\underline{\bm{x}},t+\Delta t_{\mathrm{obs}})=\rme^{\mathcal{L}^{\dagger}\Delta t_{\mathrm{obs}}}p(\underline{\bm{x}},t).$

(25)

Then, the expectation for the observable function $x_{d}$ is given as follows:

	$\displaystyle\mathbb{E}[x_{d}(\underline{\bm{x}(t+\Delta t_{\mathrm{obs}})})|\underline{\bm{x}(t)}=\underline{\bm{x}_{\mathrm{ini}}}]$
	$\displaystyle=\int_{\mathcal{M}}x_{d}(\underline{\bm{x}})\,p(\underline{\bm{x}},t+\Delta t_{\mathrm{obs}})\rmd\underline{\bm{x}}$
	$\displaystyle=\int_{\mathcal{M}}x_{d}(\underline{\bm{x}})\left(e^{\mathcal{L}^{\dagger}\Delta t_{\mathrm{obs}}}\delta(\underline{\bm{x}}-\underline{\bm{x}_{\mathrm{ini}}})\right)\rmd\underline{\bm{x}}$
	$\displaystyle=\int_{\mathcal{M}}\left(e^{\mathcal{L}\Delta t_{\mathrm{obs}}}x_{d}(\underline{\bm{x}})\right)\delta(\underline{\bm{x}}-\underline{\bm{x}_{\mathrm{ini}}})\,\rmd\underline{\bm{x}}$
	$\displaystyle=\int_{\mathcal{M}}\varphi_{d}(\underline{\bm{x}},t+\Delta t_{\mathrm{obs}})\delta(\underline{\bm{x}}-\underline{\bm{x}_{\mathrm{ini}}})\,\rmd\underline{\bm{x}}$
	$\displaystyle=\varphi_{d}(\underline{\bm{x}_{\mathrm{ini}}},t+\Delta t_{\mathrm{obs}}),$		(26)

where

$\displaystyle\mathcal{L}=\sum_{d}a_{d}(\bm{x})\frac{\partial}{\partial x_{d}}+\frac{1}{2}\sum_{i,j}\left[\bm{B}(\bm{x})\bm{B}(\bm{x})^{\top}\right]_{ij}\frac{\partial^{2}}{\partial x_{i}\partial x_{j}},$

(27)

and $\varphi_{d}(\underline{\bm{x}},t)$ is the solution of the following partial differential equation, i.e., the so-called the backward Kolmogorov equation:

$\displaystyle\frac{\partial}{\partial t}\varphi_{d}(\underline{\bm{x}},t)=\mathcal{L}\varphi_{d}(\underline{\bm{x}},t),$

(28)

which leads to

$\displaystyle\varphi_{d}(\underline{\bm{x}},t+\Delta t_{\mathrm{obs}})=\rme^{\mathcal{L}\Delta t_{\mathrm{obs}}}\varphi_{d}(\underline{\bm{x}},t).$

(29)

Note that $\varphi_{d}(\underline{\bm{x}},t+\Delta t_{\mathrm{obs}})\in\mathcal{F}$, and the initial condition is

$\displaystyle\varphi_{d}(\underline{\bm{x}},t)=x_{d}(\underline{\bm{x}}).$

(30)

Here, $\mathcal{L}^{\dagger}$ in (23) yields the time-evolution of the probability density function on the state space $\mathcal{M}$. By contrast, $\mathcal{L}$ in (27) corresponds to the time-evolution of functions on the function space $\mathcal{F}$. In addition, (29) indicates that

$\displaystyle\mathcal{K}=\rme^{\mathcal{L}\Delta t_{\mathrm{obs}}},$

(31)

and it is straightforwardly understandable that the Koopman operator $\mathcal{K}$ yields the expectation after the time-evolution.

Note that the above discussion is also available to the deterministic cases. When there is no diffusion, i.e., $B(\underline{\bm{x}})=0$, the time-evolution operator $\mathcal{L}^{\dagger}$ includes only the first-order derivatives. While the second-order derivatives lead to the effect of broadening the probability density function, the first-order derivatives correspond to the shift of the coordinates. Then, the zero diffusion cases retain the probability density function as the Dirac delta function; $\delta(\underline{\bm{x}}-\underline{\bm{x}_{\mathrm{ini}}})\to\delta(\underline{\bm{x}}-\underline{\bm{x}(t+\Delta t_{\mathrm{obs}})})$. Hence, the expectation yields the value of $\underline{\bm{x}(t+\Delta t_{\mathrm{obs}})}$, and the previous discussions on the deterministic cases are recovered adequately.

Here, we follow [18] and revisit the effects of partial observations in deterministic systems. Note that the following discussions are slightly different from [18]. As discussed below, the absence of diffusion coefficients $B(\bm{x})$ is crucial to connect the Koopman operator approach and the conventional discussions for partial observations, i.e., the Mori-Zwanzig formalism.

Assume that only some of the coordinates are observable, and we denote the index set as $\mathcal{O}$ and the partial observables as $\bm{g}_{\mathcal{O}}$ whose elements are $\{x_{d}|d\in\mathcal{O}\}$; the number of partial observables is $N_{\mathrm{obs}}$. For example, when we observe $\underline{x_{1}}$ and $\underline{x_{3}}$ in a $D=3$ case, $\mathcal{O}=\{1,3\}$, $\bm{g}_{\mathcal{O}}=[x_{1}\,\,x_{3}]^{\top}$, and $N_{\mathrm{obs}}=2$. We also introduce a dictionary $\bm{\psi}_{\overline{\mathcal{O}}}$ that includes many basis functions $\psi_{i}\in\mathcal{F}$ related to the unobserved variables. We denote the number of dictionary functions as $N_{\mathrm{dic}}^{\overline{\mathcal{O}}}$. The discussions on the function space and the introduction of the basis functions, i.e., the dictionary, lead to a linear algebraic approach for the nonlinear systems; this linearity is characteristic of the Koopman operator approach.

We discuss a case where the basis functions are time-dependent; this case corresponds to the discussion in [18]. In this case, an arbitrary function $\phi\in\mathcal{F}$ is expressed approximately as

$\displaystyle\phi(\underline{\bm{x}},t)=\sum_{d\in\mathcal{O}}c^{\phi,\mathcal{O}}_{d}x_{d}(\underline{\bm{x}},t)+\sum_{i=1}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}c^{\phi,\overline{\mathcal{O}}}_{i}\psi_{i}(\underline{\bm{x}},t).$

(32)

Then, we derive an explicit representation of the time-evolution operator $\mathcal{L}$ in (27). Using the same basis functions above, we approximate the operator $\mathcal{L}$ as matrices; it is enough to consider the actions on $\{x_{d}\}$ and $\{\psi_{i}\}$, and we have

$\displaystyle\mathcal{L}x_{d}(\underline{\bm{x}},t)\simeq\sum_{d^{\prime}\in\mathcal{O}}L^{\mathcal{O}\mathcal{O}}_{dd^{\prime}}x_{d^{\prime}}(\underline{\bm{x}},t)+\sum_{i=1}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}L^{\mathcal{O}\overline{\mathcal{O}}}_{di}\psi_{i}(\underline{\bm{x}},t),$

(33)

and

$\displaystyle\mathcal{L}\psi_{i}(\underline{\bm{x}},t)\simeq\sum_{d\in\mathcal{O}}L^{\overline{\mathcal{O}}\mathcal{O}}_{id}x_{d}(\underline{\bm{x}},t)+\sum_{i^{\prime}=1}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}L^{\overline{\mathcal{O}}\overline{\mathcal{O}}}_{ii^{\prime}}\psi_{i^{\prime}}(\underline{\bm{x}},t).$

(34)

Then, using the matrices, $L^{\mathcal{O}\mathcal{O}}\in\mathbb{R}^{N_{\mathrm{obs}}\times N_{\mathrm{obs}}}$, $L^{\mathcal{O}\overline{\mathcal{O}}}\in\mathbb{R}^{N_{\mathrm{obs}}\times N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}$, $L^{\overline{\mathcal{O}}\mathcal{O}}\in\mathbb{R}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}\times N_{\mathrm{obs}}}$, and $L^{\overline{\mathcal{O}}\overline{\mathcal{O}}}\in\mathbb{R}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}\times N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}$, the time-evolution equation is given as

$\displaystyle\frac{\rmd}{\rmd t}\left[\begin{array}[]{c}\bm{g}_{\mathcal{O}}(t)\\ \bm{\psi}_{\overline{\mathcal{O}}}(t)\end{array}\right]=\left[\begin{array}[]{cc}L^{\mathcal{O}\mathcal{O}}&L^{\mathcal{O}\overline{\mathcal{O}}}\\ L^{\overline{\mathcal{O}}\mathcal{O}}&L^{\overline{\mathcal{O}}\overline{\mathcal{O}}}\end{array}\right]\left[\begin{array}[]{c}\bm{g}_{\mathcal{O}}(t)\\ \bm{\psi}_{\overline{\mathcal{O}}}(t)\end{array}\right].$

(41)

As shown in [18], it is possible to derive the time-evolution equation for $\bm{g}_{\mathcal{O}}(t)$. First, we solve (41) for $\bm{\psi}_{\overline{\mathcal{O}}}(t)$ implicitly;

$\displaystyle\bm{\psi}_{\overline{\mathcal{O}}}(t)=\int_{0}^{t}\rme^{(t-s)L^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}L^{\overline{\mathcal{O}}\mathcal{O}}\bm{g}_{\mathcal{O}}(s)\rmd s+\rme^{tL^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}\bm{\psi}_{\overline{\mathcal{O}}}(0).$

(42)

Then, inserting (42) to (41), we have

$\displaystyle\frac{\rmd}{\rmd t}\bm{g}_{\mathcal{O}}(t)=L^{\mathcal{O}\mathcal{O}}\bm{g}_{\mathcal{O}}(t)+L^{\mathcal{O}\overline{\mathcal{O}}}\int_{0}^{t}\rme^{(t-s)L^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}L^{\overline{\mathcal{O}}\mathcal{O}}\bm{g}_{\mathcal{O}}(s)\rmd s+L^{\mathcal{O}\overline{\mathcal{O}}}\rme^{tL^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}\bm{\psi}_{\overline{\mathcal{O}}}(0).$

(43)

Replacing $L^{\mathcal{O}\mathcal{O}}$ with $M$ and $-L^{\mathcal{O}\overline{\mathcal{O}}}\rme^{sL^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}L^{\overline{\mathcal{O}}\mathcal{O}}$ with $K_{\mathrm{mem}}(s)$, we have

$\displaystyle\frac{\rmd}{\rmd t}\bm{g}_{\mathcal{O}}(t)=M\bm{g}_{\mathcal{O}}(t)-\int_{0}^{t}K_{\mathrm{mem}}(t-s)\bm{g}_{\mathcal{O}}(s)\rmd s+\bm{f}(t),$

(44)

where $\bm{f}(t)=L^{\mathcal{O}\overline{\mathcal{O}}}\rme^{tL^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}\bm{\psi}_{\overline{\mathcal{O}}}(0)$. $M$ is called the Markov transition matrix, which quantifies the interactions between the observables $\bm{g}_{\mathcal{O}}(t)$. $K_{\mathrm{mem}}(s)$ is called the memory kernel and depends on the history of the observed quantity; (43) indicates that the memory effect stems from the path via the unobserved quantities. $\bm{f}(t)$ is referred to as noise since it depends on the initial values of the unobserved quantities $\bm{\psi}_{\overline{\mathcal{O}}}(0)$. That is, there is no information about their explicit values, and then we cannot predict them.

Note that (44) is an equation in the function space $\mathcal{F}$. It is crucial to distinguish the function space $\mathcal{F}$ and the state space $\mathcal{M}$; an element $x_{d}$ in $\bm{g}_{\mathcal{O}}(t)$ is different from an element $\underline{x_{d}}$ in the state vector $\underline{\bm{x}}$ in general. However, deterministic systems have a unique feature that one can interpret (44) as an equation for the state vector.

First, we focus on the time-evolution equation in (26); we see that the final expression $\varphi_{d}(\underline{\bm{x}_{\mathrm{ini}}},t+\Delta t_{\mathrm{obs}})$ yields the expectation values. Although the expectation value $\mathbb{E}[x_{d}]$ is not the state $\underline{x_{d}}$, the absence of the noise term does not change an initial Dirac-delta-type density function even in the time-evolution. That is,

$\displaystyle e^{\mathcal{L}^{\dagger}\Delta t_{\mathrm{obs}}}\delta(\underline{\bm{x}}-\underline{\bm{x}(t)})=\delta(\underline{\bm{x}}-\underline{\bm{x}(t+\Delta t_{\mathrm{obs}})}).$

(45)

Then, the expectation value is equal to the corresponding state;

$\displaystyle\mathbb{E}[x_{d}(\underline{\bm{x}(t+\Delta t_{\mathrm{obs}})})|\underline{\bm{x}(t)}=\underline{\bm{x}_{\mathrm{ini}}}]=\underline{x_{d}(t+\Delta t_{\mathrm{obs}})}.$

(46)

This fact enables us to equate the observable function $x_{d}$ with the state $\underline{x_{d}}$.

Second, we introduce a different perspective on the time-evolution equation. In deterministic systems, $B(\bm{x})=0$, i.e., there is no noise. Hence, the second term on the right-hand side of (27) is absent. Then, the following operator yields the time-evolution in the function space:

$\displaystyle\mathcal{L}=\sum_{d}a_{d}(\bm{x})\frac{\partial}{\partial x_{d}}.$

(47)

When we consider the time-evolution for the observable function $x_{d}$, the action of $\mathcal{L}$ leads to

$\displaystyle\mathcal{L}x_{d}=a_{d}(\bm{x}),$

(48)

and then, we have

$\displaystyle\frac{\rmd}{\rmd t}x_{d}=a_{d}(\bm{x}).$

(49)

Note that (49) is the same form as the original time-evolution equation in the state space:

$\displaystyle\frac{\rmd}{\rmd t}\underline{x_{d}}=a_{d}(\underline{\bm{x}}).$

(50)

This fact also allows us to equate the time-evolution of $x_{d}$ in the function space $\mathcal{F}$ with that of $\underline{x_{d}}$ in the state space $\mathcal{M}$.

In short, although we should distinguish the function space and the state space, the feature of the deterministic system enables us to consider the derived equation (44) in the function space as the one in the state space. This feature justifies the coarse-grained approach based on the Mori-Zwanzig formalism; we can construct time-evolution equations of a smaller set of macroscopic variables that are measurable, and other unmeasured degrees of freedom play the role of noise.

As discussed in Sec. 3, the derived equation in (44) is for the function space. Note that the function at time $t+\Delta t_{\mathrm{obs}}$ yields the expectation values in (26). Due to the presence of the noise term, even if the initial density function is the Dirac delta function, the values of $\mathbb{E}[x_{d}(\underline{\bm{x}(t+\Delta t_{\mathrm{obs}})})]$ and $\underline{x_{d}}(t+\Delta t_{\mathrm{obs}})$ are different because the density function is spread out. Hence, the generalized Langevin equation in (44) is not directly related to the state vector $\underline{\bm{x}}$.

Here, we discuss the generalized Langevin equation in the function space. Although the time-dependent basis in Sec. 3 is beneficial due to the connection with the state vector, it would be preferable to fix the basis functions in the discussion below. Then, we employ the following basis expansion:

$\displaystyle\phi(\underline{\bm{x}},t)=\sum_{d\in\mathcal{O}}c^{\phi,\mathcal{O}}_{d}(t)x_{d}(\underline{\bm{x}})+\sum_{i=1}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}c^{\phi,\overline{\mathcal{O}}}_{i}(t)\psi_{i}(\underline{\bm{x}}),$

(51)

where the expansion coefficients $\{c^{\phi,\mathcal{O}}_{d}(t)\}$ and $c^{\phi,\overline{\mathcal{O}}}_{i}(t)$ are time-dependent, which differs from (32).

Then, we derive the time-evolution equation for the expansion coefficients $\{c^{\phi,\mathcal{O}}_{d}(t)\}$ and $c^{\phi,\overline{\mathcal{O}}}_{i}(t)$. In the derivation, we employ the bra and ket notations for the notational simplicity. Note that the notations are essentially the same as the Doi-Peliti method [35, 36, 37]; see the discussion in [38, 39, 40].

In the function space, we introduce the following ket states $\{|x_{d}\rangle\}$ and $\{|\psi_{i}\rangle\}$:

	$\displaystyle|x_{d}\rangle=x_{d}\qquad\textrm{for }d=1,\dots,D,$		(52)
	$\displaystyle|\psi_{i}\rangle=\psi_{i}\qquad\textrm{for }i=1,\dots,N_{\mathrm{dic}}^{\overline{\mathcal{O}}},$		(53)

where we omit the argument of functions, $\underline{\bm{x}}$. By contrast, the bra states $\{\langle x_{d}|\}$ and $\{\langle\psi_{i}|\}$ are defined as follows:

	$\displaystyle\langle x_{d}|=\int\prod_{d^{\prime}=1}^{D}\rmd x_{d^{\prime}}\,\delta(x_{d^{\prime}})\left(\frac{\partial}{\partial x_{d}}\right)(\cdot),$		(54)
	$\displaystyle\langle\psi_{i}|=\int\rmd\bm{x}\,\rho(\bm{x})\psi_{i}(\bm{x})(\cdot),$		(55)

where $\rho(\bm{x})$ is a suitable measure to yield $L^{2}$ functions. From (52) and (54), we have the following orthonormal relation:

$\displaystyle\langle x_{d}|x_{d^{\prime}}\rangle=\int\rmd x_{d}\,\delta(x_{d})\left(\frac{\partial}{\partial x_{d}}\right)x_{d^{\prime}}=\delta_{d,d^{\prime}},$

(56)

where $\delta_{d,d^{\prime}}$ is the Kronecker delta function.

It would be easy to explain with a concrete example. Hence, we here assume that there are only two observable variables, $\mathcal{O}=\{1,2\}$. Using the above bra and ket notations, (51) is rewritten as

$\displaystyle\phi(\underline{\bm{x}},t)=\sum_{d\in\mathcal{O}}c^{\phi,\mathcal{O}}_{d}(t)|x_{d}\rangle+\sum_{i=1}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}c^{\phi,\overline{\mathcal{O}}}_{i}(t)|\psi_{i}\rangle.$

(57)

The time-derivative of (57) leads to

$\displaystyle\frac{\rmd}{\rmd t}\phi(\underline{\bm{x}},t)=\sum_{d\in\mathcal{O}}\frac{\rmd}{\rmd t}\left(c^{\phi,\mathcal{O}}_{d}(t)\right)|x_{d}\rangle+\sum_{i=1}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}\frac{\rmd}{\rmd t}\left(c^{\phi,\overline{\mathcal{O}}}_{i}(t)\right)|\psi_{i}\rangle.$

(58)

Next, we take the action of $\langle x_{d}|$ on all terms from the left. Similar to the discussion in Sec. 3.1, we have the following time-evolution equation:

$\displaystyle Z\frac{\rmd}{\rmd t}\left[\begin{array}[]{c}\bm{c}^{\phi,\mathcal{O}}(t)\\ \bm{c}^{\phi,\overline{\mathcal{O}}}(t)\\ \end{array}\right]=\left[\begin{array}[]{cc}\widetilde{L}^{\mathcal{O}\mathcal{O}}&\widetilde{L}^{\mathcal{O}\overline{\mathcal{O}}}\\ \widetilde{L}^{\overline{\mathcal{O}}\mathcal{O}}&\widetilde{L}^{\overline{\mathcal{O}}\overline{\mathcal{O}}}\end{array}\right]\left[\begin{array}[]{c}\bm{c}^{\phi,\mathcal{O}}(t)\\ \bm{c}^{\phi,\overline{\mathcal{O}}}(t)\\ \end{array}\right],$

(65)

where we define

$\displaystyle Z=\left[\begin{array}[]{cccccc}1&0&\langle x_{1}|\psi_{1}\rangle&\langle x_{1}|\psi_{2}\rangle&\cdots&\langle x_{1}|\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}\rangle\\ 0&1&\langle x_{2}|\psi_{1}\rangle&\langle x_{2}|\psi_{2}\rangle&\cdots&\langle x_{2}|\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}\rangle\\ \langle\psi_{1}|x_{1}\rangle&\langle\psi_{1}|x_{2}\rangle&\langle\psi_{1}|\psi_{1}\rangle&\langle\psi_{1}|\psi_{2}\rangle&\cdots&\langle\psi_{1}|\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}\rangle\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ \langle\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}|x_{1}\rangle&\langle\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}|x_{2}\rangle&\langle\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}|\psi_{1}\rangle&\langle\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}|\psi_{2}\rangle&\cdots&\langle\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}|\psi_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}\rangle\end{array}\right]$

(71)

and

$\displaystyle\langle x_{d}|\mathcal{L}|x_{d^{\prime}}\rangle=\widetilde{L}^{\mathcal{O}\mathcal{O}}_{dd^{\prime}},\quad\langle x_{d}|\mathcal{L}|\psi_{i}\rangle=\widetilde{L}^{\mathcal{O}\overline{\mathcal{O}}}_{di},\quad\langle\psi_{i}|\mathcal{L}|x_{d^{\prime}}\rangle=\widetilde{L}^{\overline{\mathcal{O}}\mathcal{O}}_{id^{\prime}},\quad\langle\psi_{i}|\mathcal{L}|\psi_{i^{\prime}}\rangle=\widetilde{L}^{\overline{\mathcal{O}}\overline{\mathcal{O}}}_{ii^{\prime}}.$

(72)

Although the matrix $Z$ is not diagonal, the following assumption for the dictionary functions $\{|\psi_{i}\rangle\}$ makes the discussion simple: the functions $\{|\psi_{i}\rangle\}$ consist of monomial basis functions. Then, an element in $\{|\psi_{i}\rangle\}$ has the following form:

$\displaystyle|\bm{n}\rangle\equiv\prod_{d=1}^{D}x_{d}^{n_{d}},$

(73)

where $\bm{n}=\{n_{1},n_{2},\dots,n_{D}\}$. The corresponding bra state is defined as

$\displaystyle\langle\bm{n}|\equiv\int\prod_{d=1}^{D}\rmd x_{d}\,\delta(x_{d})\left(\frac{\partial}{\partial x_{d}}\right)^{n_{d}}(\cdot).$

(74)

Then, we have the following orthogonal relation:

$\displaystyle\langle\bm{n}|\bm{m}\rangle=\int\prod_{d=1}^{D}\rmd x_{d}\,\delta(x_{d})\left(\frac{\partial}{\partial x_{d}}\right)^{n_{d}}x_{d}^{m_{d}}=\prod_{d=1}^{D}n_{d}!\delta_{n_{d},m_{d}}.$

(75)

Note that the monomial functions for the observable states,

$\displaystyle|\{1,0,\dots,0\}\rangle=|x_{1}\rangle\quad\textrm{and}\quad|\{0,1,\dots,0\}\rangle=|x_{2}\rangle,$

(76)

should not be included in $\{|\psi_{i}\rangle\}$. Using the above assumption for the dictionary functions, we have the following matrix $Z$ in (71):

$\displaystyle Z=\left[\begin{array}[]{cccccc}1&0&0&0&\cdots&0\\ 0&1&0&0&\cdots&0\\ 0&0&z_{1}&0&\cdots&0\\ \vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&0&\cdots&z_{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}\end{array}\right],$

(82)

where we denote the diagonal part with $\{z_{i}|i=1,\dots,N_{\mathrm{dic}}^{\overline{\mathcal{O}}}\}$. Since the matrix $Z$ is diagonal, $Z^{-1}$ is also diagonal, which leads to

	$\displaystyle\frac{\rmd}{\rmd t}\left[\begin{array}[]{c}\bm{c}^{\phi,\mathcal{O}}(t)\\ \bm{c}^{\phi,\overline{\mathcal{O}}}(t)\\ \end{array}\right]$	$\displaystyle=Z^{-1}\left[\begin{array}[]{cc}\widetilde{L}^{\mathcal{O}\mathcal{O}}&\widetilde{L}^{\mathcal{O}\overline{\mathcal{O}}}\\ \widetilde{L}^{\overline{\mathcal{O}}\mathcal{O}}&\widetilde{L}^{\overline{\mathcal{O}}\overline{\mathcal{O}}}\end{array}\right]\left[\begin{array}[]{c}\bm{c}^{\phi,\mathcal{O}}(t)\\ \bm{c}^{\phi,\overline{\mathcal{O}}}(t)\\ \end{array}\right]$			(89)
		$\displaystyle=\left[\begin{array}[]{cc}\widetilde{L}^{\mathcal{O}\mathcal{O}}&\widetilde{L^{\prime}}^{\mathcal{O}\overline{\mathcal{O}}}\\ \widetilde{L^{\prime}}^{\overline{\mathcal{O}}\mathcal{O}}&\widetilde{L^{\prime}}^{\overline{\mathcal{O}}\overline{\mathcal{O}}}\end{array}\right]\left[\begin{array}[]{c}\bm{c}^{\phi,\mathcal{O}}(t)\\ \bm{c}^{\phi,\overline{\mathcal{O}}}(t)\\ \end{array}\right].$			(94)

Note that the multiplication of the diagonal matrix $Z^{-1}$ does not change the relationship between observable and unobserved functions. Since (94) has the same form as (41), we finally obtain the following equation similar to (44):

$\displaystyle\frac{\rmd}{\rmd t}\bm{c}^{\phi,\mathcal{O}}(t)=\widetilde{L}^{\mathcal{O}\mathcal{O}}\bm{c}^{\phi,\mathcal{O}}(t)+\widetilde{L^{\prime}}^{\mathcal{O}\overline{\mathcal{O}}}\int_{0}^{t}\rme^{(t-s)\widetilde{L^{\prime}}^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}\widetilde{L^{\prime}}^{\overline{\mathcal{O}}\mathcal{O}}\bm{c}^{\phi,\mathcal{O}}(s)\rmd s+\widetilde{L^{\prime}}^{\mathcal{O}\overline{\mathcal{O}}}\rme^{t\widetilde{L^{\prime}}^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}\bm{c}^{\phi,\overline{\mathcal{O}}}(0).$

(95)

Although the form of (95) is similar to (41), there are differences. First, (95) is a vector-state equation for only one function $\phi$. Second, the initial values of the coefficients $\{\bm{c}^{\phi,\mathcal{O}}(0)\}$ are one-hot because we focus only on the observable functions $\{x_{d}|d\in\mathcal{O}\}$. For example, assume that we focus on $x_{1}$. Hence, we have the following equation for $\phi=x_{1}$ at the initial time $t=0$:

$\displaystyle x_{1}=\phi(\underline{\bm{x}},0)=\sum_{d\in\mathcal{O}}c^{\phi,\mathcal{O}}_{d}(0)|x_{d}\rangle+\sum_{i=1}^{N_{\mathrm{dic}}^{\overline{\mathcal{O}}}}c^{\phi,\overline{\mathcal{O}}}_{i}(0)|\psi_{i}\rangle,$

(96)

which indicates that only the first element, $c^{\phi,\mathcal{O}}_{1}(0)$, is 1, and the other terms become zero because the left-hand side is the function $x_{1}$. Then, different from the time-dependent basis functions, the third term in the right-hand side in (95), i.e., the noise term, vanishes;

$\displaystyle\frac{\rmd}{\rmd t}\bm{c}^{\phi,\mathcal{O}}(t)=\widetilde{M}\bm{c}^{\phi,\mathcal{O}}(t)-\int_{0}^{t}\widetilde{K}_{\mathrm{mem}}(t-s)\bm{c}^{\phi,\mathcal{O}}(s)\rmd s,$

(97)

where $\widetilde{M}\equiv\widetilde{L}^{\mathcal{O}\mathcal{O}}$ and $\widetilde{K}_{\mathrm{mem}}(s)\equiv-\widetilde{L^{\prime}}^{\mathcal{O}\overline{\mathcal{O}}}\rme^{s\widetilde{L^{\prime}}^{\overline{\mathcal{O}}\overline{\mathcal{O}}}}\widetilde{L^{\prime}}^{\overline{\mathcal{O}}\mathcal{O}}$. Note that different from (44), it is not straightforwardly clear that the use of time-delayed states is beneficial. The second term on the right-hand side in (97) is related to the expansion coefficient, which does not directly correspond to the delayed state.