세미나

MatterGen

Let us define a material $\boldsymbol{M}$ as a triple: $$ \boldsymbol{M} = (\boldsymbol{A}, \boldsymbol{X}, \boldsymbol{L}) $$ where $\boldsymbol{A} = (a^{1}, a^{2}, \dots, a^{n})^{\mathsf{T}} \in \mathbb{A}^{n}$ are the atomic species of the atoms inside the unit cell; $\boldsymbol{L} = (\boldsymbol{l}^{1}, \boldsymbol{l}^{2}, \boldsymbol{l}^{3}) \in \mathbb{R}^{3 \times 3}$ is the lattice, i.e., the shape of the repeating unit cell; and $\boldsymbol{X} = (\boldsymbol{x}^{1}, \boldsymbol{x}^{2}, \dots, \boldsymbol{x}^{n}) \in [0, 1]^{3 \times n}$ are the fractional coordinates of the atoms inside the unit cell.

$$ q(\boldsymbol{A}_{t+1}, \boldsymbol{X}_{t+1}, \boldsymbol{L}_{t+1} | \boldsymbol{A}_{t}, \boldsymbol{X}_{t}, \boldsymbol{L}_{t}) = q(\boldsymbol{A}_{t+1} | \boldsymbol{A}_{t}) q(\boldsymbol{X}_{t+1} | \boldsymbol{X}_{t}) q(\boldsymbol{L}_{t+1} | \boldsymbol{L}_{t}) \qquad t = 0, 1, \dots, T-1. $$

A.5 Atom type diffusion

For the diffusion of the (discrete) atom species A, we use the discrete denoising diffusion probabilistic model (D3PM) approach ¹, which is a generalization of DDPMs to discrete data problems.

$$ q(a_{1:T} | a_{0}) = \prod_{t=1}^{T} q(a_{t} | a_{t-1}). $$

원소의 종류가 $K$개인 경우 $\mathbb{A}^{n} = (\mathbb{Z}_{K})^{n}$이라 생각하자. 실제로는 $i = \mathbf{e}_{i}$와 같이 one-hot encoding을 사용하여 구현한다. 즉

$$ \boldsymbol{a}_{t-1} = \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix} $$

$$ \mathbf{Q}_{t} = \begin{bmatrix} q(1|1) & q(2|1) & \cdots & q(K|1) \\ q(1|2) & q(2|2) & \cdots & q(K|2) \\ \vdots & \vdots & \ddots & \vdots \\ q(1|K) & q(2|K) & \cdots & q(K|K) \end{bmatrix} $$

$$ \begin{align} \boldsymbol{a}_{t-1} \mathbf{Q}_{t} &= \begin{bmatrix} 1 & 0 & \cdots & 0 \end{bmatrix} \begin{bmatrix} q(1|1) & q(2|1) & \cdots & q(K|1) \\ q(1|2) & q(2|2) & \cdots & q(K|2) \\ \vdots & \vdots & \ddots & \vdots \\ q(1|K) & q(2|K) & \cdots & q(K|K) \end{bmatrix} \\ &= \begin{bmatrix} q(1|1) & q(2|1) & \cdots & q(K|1) \end{bmatrix} \end{align} $$

previous limitations

$$ L_{\text{vb}} = \mathbb{E}{q(\boldsymbol{a}{0})} \left[ -\mathbb{E}{q(\boldsymbol{a}{1} | \boldsymbol{a}{0})} \log p{\boldsymbol{\theta}} (\boldsymbol{a}{0} | \boldsymbol{a}{1}, 1) + D_{\text{KL}} \left[q(\boldsymbol{a}{T} | \boldsymbol{a}{0}) | q(\boldsymbol{a}{T}) \right] + \sum\limits{t=2}^{T} \mathbb{E}{q(\boldsymbol{a}{t} | \boldsymbol{a}{0})} D{\text{KL}} \left[ q(\boldsymbol{a}{t-1} | \boldsymbol{a}{t}, \boldsymbol{a}{0}) | p{\boldsymbol{\theta}} (\boldsymbol{a}{t-1} | \boldsymbol{a}{t} )\right] \right] $$

$$ L_{CE} = -\mathbb{E}{q(\boldsymbol{a}{0})} \left[ \sum\limits_{t=2}^{T} \mathbb{E}{q(\boldsymbol{a}{t} | \boldsymbol{a}{0})} \log p{\boldsymbol{\theta}} (\boldsymbol{a}{0} | \boldsymbol{a}{t}, t) \right] $$

$$ L = L_{\text{vb}} + \lambda_{\text{CE}} L_{CE} $$

$$ \left[ \boldsymbol{Q}_{t}^{\text{absorbing}} \right]_{ij} = \begin{cases} 1 & \text{if } i = j = m, \\ 1 - \beta_{t} & \text{if } i = j \ne m, \\ \beta_{t} & \text{if } j = m \ne i, \\ 0 & \text{otherwise}. \end{cases} $$

$$ \boldsymbol{Q}_{t}^{\text{absorbing}} = \begin{bmatrix} 1 - \beta_{t} & 0 & \cdots & \beta_{t} & \cdots & 0 \\ 0 & 1 - \beta_{t} & \cdots & \beta_{t} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \beta_{t} & \cdots & 1 - \beta_{t} \end{bmatrix} $$