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285 technical terms and definitions
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Filter that removes 99.97% of particles 0.3 microns and larger.
Create airtight seal.
GNNs for graphs with different node/edge types.
Heterogeneous graphs contain multiple node types and edge types requiring specialized message passing for different relation semantics.
Heterogeneous information networks integrate multiple entity types and relations for unified recommendation frameworks.
Combine different technologies in one package.
Heterogeneous integration combines different technologies materials or functions in single package.
Combine dies from different technologies or materials in one package.
Heterogeneous skip-gram predicts context nodes of different types given target nodes.
Heterogeneous Self-Attention Neural Network adaptively learns importance of different metapaths and neighbors.
Second-order ODE solver.
Simple quality indicators.
Hydrofluoric acid to remove native oxide and etch oxide.
# Heterogeneous Graph Transformer (HGT) ## HGT Graph Neural Networks **HGT (Heterogeneous Graph Transformer)** is a graph neural network architecture designed specifically for **heterogeneous graphs** — graphs where nodes and edges can have different types. It was introduced by Hu et al. in 2020. ## 1. Problem Setting ### 1.1 Heterogeneous Graph Definition A heterogeneous graph is defined as: $$ G = (V, E, \tau, \phi) $$ Where: - $V$ — Set of nodes - $E$ — Set of edges - $\tau: V \rightarrow \mathcal{T}$ — Node type mapping function - $\phi: E \rightarrow \mathcal{R}$ — Edge type mapping function - $\mathcal{T}$ — Set of node types - $\mathcal{R}$ — Set of edge/relation types ### 1.2 Real-World Examples - **Academic Networks**: - Node types: `Paper`, `Author`, `Venue`, `Institution` - Edge types: `writes`, `cites`, `published_in`, `affiliated_with` - **E-commerce Graphs**: - Node types: `User`, `Product`, `Brand`, `Category` - Edge types: `purchases`, `reviews`, `belongs_to`, `manufactures` - **Knowledge Graphs**: - Node types: `Person`, `Organization`, `Location`, `Event` - Edge types: `works_at`, `located_in`, `participated_in` ## 2. HGT Architecture ### 2.1 Core Components The HGT layer consists of three main operations: 1. **Heterogeneous Mutual Attention** 2. **Heterogeneous Message Passing** 3. **Target-Specific Aggregation** ### 2.2 Type-Dependent Linear Projections For each node type $\tau \in \mathcal{T}$, HGT defines separate projection matrices: $$ Q_{\tau}^{(i)} \in \mathbb{R}^{d \times \frac{d}{h}}, \quad K_{\tau}^{(i)} \in \mathbb{R}^{d \times \frac{d}{h}}, \quad V_{\tau}^{(i)} \in \mathbb{R}^{d \times \frac{d}{h}} $$ Where: - $d$ — Hidden dimension - $h$ — Number of attention heads - $i$ — Attention head index $(i = 1, 2, \ldots, h)$ ## 3. Mathematical Formulation ### 3.1 Attention Mechanism For a source node $s$ and target node $t$ connected by edge $e$: #### Step 1: Compute Query and Key $$ \text{Query}^{(i)}(t) = Q_{\tau(t)}^{(i)} \cdot H^{(l-1)}[t] $$ $$ \text{Key}^{(i)}(s) = K_{\tau(s)}^{(i)} \cdot H^{(l-1)}[s] $$ #### Step 2: Compute Attention Score $$ \text{ATT-head}^{(i)}(s, e, t) = \left( \text{Key}^{(i)}(s) \cdot W_{\phi(e)}^{\text{ATT}} \cdot \text{Query}^{(i)}(t)^T \right) \cdot \frac{\mu_{\langle \tau(s), \phi(e), \tau(t) \rangle}}{\sqrt{d}} $$ Where: - $W_{\phi(e)}^{\text{ATT}} \in \mathbb{R}^{\frac{d}{h} \times \frac{d}{h}}$ — Edge-type-specific attention matrix - $\mu_{\langle \tau(s), \phi(e), \tau(t) \rangle}$ — Prior importance of meta-relation (learnable scalar) #### Step 3: Softmax Normalization $$ \text{Attention}(s, e, t) = \text{softmax}_{s \in \mathcal{N}(t)} \left( \text{ATT-head}^{(i)}(s, e, t) \right) $$ ### 3.2 Message Computation $$ \text{Message}^{(i)}(s, e, t) = V_{\tau(s)}^{(i)} \cdot H^{(l-1)}[s] \cdot W_{\phi(e)}^{\text{MSG}} $$ Where: - $W_{\phi(e)}^{\text{MSG}} \in \mathbb{R}^{\frac{d}{h} \times \frac{d}{h}}$ — Edge-type-specific message matrix ### 3.3 Multi-Head Aggregation $$ \tilde{H}^{(l)}[t] = \bigoplus_{i=1}^{h} \left( \sum_{s \in \mathcal{N}(t)} \text{Attention}^{(i)}(s, e, t) \cdot \text{Message}^{(i)}(s, e, t) \right) $$ Where $\bigoplus$ denotes concatenation across heads. ### 3.4 Final Output with Residual Connection $$ H^{(l)}[t] = \sigma \left( W_{\tau(t)}^{\text{OUT}} \cdot \tilde{H}^{(l)}[t] + H^{(l-1)}[t] \right) $$ Where: - $W_{\tau(t)}^{\text{OUT}} \in \mathbb{R}^{d \times d}$ — Target-type-specific output projection - $\sigma$ — Activation function (e.g., ReLU, GELU) ## 4. Relative Temporal Encoding (RTE) For temporal/dynamic graphs, HGT incorporates time information: $$ \text{RTE}(\Delta t) = \text{Linear}\left( \text{T2V}(\Delta t) \right) $$ Where $\Delta t = t_{\text{target}} - t_{\text{source}}$ is the time difference. ### Time2Vec Encoding $$ \text{T2V}(\Delta t)[i] = \begin{cases} \omega_i \cdot \Delta t + \varphi_i & \text{if } i = 0 \\ \sin(\omega_i \cdot \Delta t + \varphi_i) & \text{if } i > 0 \end{cases} $$ The temporal attention becomes: $$ \text{ATT-head}^{(i)}(s, e, t) = \left( \text{Key}^{(i)}(s) + \text{RTE}(\Delta t) \right) \cdot W_{\phi(e)}^{\text{ATT}} \cdot \text{Query}^{(i)}(t)^T $$ ## 5. Comparison | Method | Heterogeneity Handling | Metapaths Required | Parameter Efficiency | |--------|----------------------|-------------------|---------------------| | **GCN** | ❌ Homogeneous only | N/A | ✅ High | | **GAT** | ❌ Homogeneous only | N/A | ✅ High | | **R-GCN** | ✅ Yes | ❌ No | ❌ Low (separate weights per relation) | | **HAN** | ✅ Yes | ✅ Yes (manual design) | ⚠️ Medium | | **HGT** | ✅ Yes | ❌ No (automatic) | ✅ High (decomposition) | ## 6. Implementation ### 6.1 PyTorch Geometric Implementation ```python import torch import torch.nn as nn from torch_geometric.nn import HGTConv, Linear class HGT(nn.Module): def __init__(self, metadata, hidden_channels, out_channels, num_heads, num_layers): super().__init__() self.node_types = metadata[0] self.edge_types = metadata[1] # Linear projections for each node type self.lin_dict = nn.ModuleDict() for node_type in self.node_types: self.lin_dict[node_type] = Linear(-1, hidden_channels) # HGT convolutional layers self.convs = nn.ModuleList() for _ in range(num_layers): conv = HGTConv( in_channels=hidden_channels, out_channels=hidden_channels, metadata=metadata, heads=num_heads, group='sum' ) self.convs.append(conv) # Output projection self.out_lin = Linear(hidden_channels, out_channels) def forward(self, x_dict, edge_index_dict): # Initial projection x_dict = { node_type: self.lin_dict[node_type](x).relu() for node_type, x in x_dict.items() } # HGT layers for conv in self.convs: x_dict = conv(x_dict, edge_index_dict) return x_dict ``` ### 6.2 Usage Example ```python # Define metadata metadata = ( ['paper', 'author', 'venue'], # Node types [ ('author', 'writes', 'paper'), ('paper', 'cites', 'paper'), ('paper', 'published_in', 'venue'), ] # Edge types as (src, relation, dst) ) # Initialize model model = HGT( metadata=metadata, hidden_channels=64, out_channels=16, num_heads=4, num_layers=2 ) # Forward pass out_dict = model(x_dict, edge_index_dict) ``` ## 7. Training Objective ### 7.1 Node Classification $$ \mathcal{L}_{\text{node}} = -\sum_{v \in V_{\text{labeled}}} \sum_{c=1}^{C} y_{v,c} \log(\hat{y}_{v,c}) $$ Where: - $y_{v,c}$ — Ground truth label (one-hot) - $\hat{y}_{v,c} = \text{softmax}(H^{(L)}[v])_c$ — Predicted probability ### 7.2 Link Prediction $$ \mathcal{L}_{\text{link}} = -\sum_{(s,e,t) \in E} \log \sigma(H^{(L)}[s]^T \cdot W_{\phi(e)} \cdot H^{(L)}[t]) - \sum_{(s,e,t') \in E^{-}} \log \sigma(-H^{(L)}[s]^T \cdot W_{\phi(e)} \cdot H^{(L)}[t']) $$ Where: - $E^{-}$ — Negative edge samples - $\sigma$ — Sigmoid function ## 8. Complexity Analysis ### 8.1 Time Complexity $$ O\left( |E| \cdot d^2 / h + |V| \cdot d^2 \right) $$ Where: - $|E|$ — Number of edges - $|V|$ — Number of nodes - $d$ — Hidden dimension - $h$ — Number of heads ### 8.2 Space Complexity (Parameters) $$ O\left( |\mathcal{T}| \cdot d^2 + |\mathcal{R}| \cdot d^2 / h \right) $$ This is more efficient than R-GCN which requires $O(|\mathcal{R}| \cdot d^2)$. ## 9. Key Advantages - **No Manual Metapath Design**: Unlike HAN, HGT automatically learns the importance of different meta-relations - **Parameter Efficient**: Uses decomposition to avoid parameter explosion with many relation types - **Unified Framework**: Handles any heterogeneous graph schema - **Temporal Support**: Can incorporate relative time encoding for dynamic graphs - **Interpretable**: Attention weights reveal learned importance of different relations ## 10. Limitations - **Computational Overhead**: More complex than homogeneous GNNs - **Data Requirements**: Needs sufficient examples per node/edge type - **Memory Usage**: Multi-head attention increases memory consumption - **Hyperparameter Sensitivity**: Performance depends on number of heads, layers, hidden dimensions ## 12. Reference | Symbol | Description | |--------|-------------| | $G = (V, E, \tau, \phi)$ | Heterogeneous graph | | $\tau(v)$ | Type of node $v$ | | $\phi(e)$ | Type of edge $e$ | | $H^{(l)}[v]$ | Node $v$ representation at layer $l$ | | $\mathcal{N}(t)$ | Neighbors of target node $t$ | | $Q, K, V$ | Query, Key, Value projections | | $W^{\text{ATT}}, W^{\text{MSG}}$ | Attention and Message weight matrices | | $\mu$ | Learnable meta-relation prior |
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Rework not visible in metrics.
Hidden factory represents rework and waste consumed fixing defects reducing effective capacity.
Hidden losses are inefficiencies not immediately apparent requiring detailed analysis.
Multi-level aggregation.
Multi-level attention structure.
Hierarchical clustering creates tree-structured groupings at multiple similarity levels.
Multi-level context organization.
Multi-level federation structure.
Multi-level fusion strategy.
Multi-level expert organization.
Optimize at different levels sequentially.
Hierarchical planning operates at multiple abstraction levels from high-level goals to low-level actions.
Hierarchical pooling creates multi-resolution graph representations through successive coarsening operations.
Decompose tasks into subtasks.
Hierarchical reinforcement learning decomposes tasks into subtasks with multiple levels of temporal abstraction.
Sample coarse then fine.
HiFi-GAN is a generative adversarial network vocoder that synthesizes high-fidelity audio efficiently using multi-scale discriminators.
System remains operational despite failures.
Stacked DRAM with wide interface.
# Semiconductor Manufacturing Process Recipe Optimization: Mathematical Modeling
## 1. Problem Context
A semiconductor **recipe** is a vector of controllable parameters:
$$
\mathbf{x} = \begin{bmatrix} T \\ P \\ Q_1 \\ Q_2 \\ \vdots \\ t \\ P_{\text{RF}} \end{bmatrix} \in \mathbb{R}^n
$$
Where:
- $T$ = Temperature (°C or K)
- $P$ = Pressure (mTorr or Pa)
- $Q_i$ = Gas flow rates (sccm)
- $t$ = Process time (seconds)
- $P_{\text{RF}}$ = RF power (Watts)
**Goal**: Find optimal $\mathbf{x}$ such that output properties $\mathbf{y}$ meet specifications while accounting for variability.
## 2. Mathematical Modeling Approaches
### 2.1 Physics-Based (First-Principles) Models
#### Chemical Vapor Deposition (CVD) Example
**Mass transport and reaction equation:**
$$
\frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{u}C) = D\nabla^2 C + R(C, T)
$$
Where:
- $C$ = Species concentration
- $\mathbf{u}$ = Velocity field
- $D$ = Diffusion coefficient
- $R(C, T)$ = Reaction rate
**Surface reaction kinetics (Arrhenius form):**
$$
k_s = A \exp\left(-\frac{E_a}{RT}\right)
$$
Where:
- $A$ = Pre-exponential factor
- $E_a$ = Activation energy
- $R$ = Gas constant
- $T$ = Temperature
**Deposition rate (transport-limited regime):**
$$
r = \frac{k_s C_s}{1 + \frac{k_s}{h_g}}
$$
Where:
- $C_s$ = Surface concentration
- $h_g$ = Gas-phase mass transfer coefficient
**Characteristics:**
- **Advantages**: Extrapolates outside training data, physically interpretable
- **Disadvantages**: Computationally expensive, requires detailed mechanism knowledge
### 2.2 Empirical/Statistical Models (Response Surface Methodology)
**Second-order polynomial model:**
$$
y = \beta_0 + \sum_{i=1}^{n}\beta_i x_i + \sum_{i=1}^{n}\beta_{ii}x_i^2 + \sum_{i
More random generation.
High vacuum pumps achieve extremely low pressures for critical processes.
Z-contrast imaging in STEM.
Large misorientation.
High-aspect-ratio contacts and vias in scaled nodes challenge gap fill and reliability requiring advanced processes.
Dielectric with high dielectric constant.
Deposit high-k before poly gate.
Deposit high-k after removing poly.
High dielectric constant gate oxide and metal gate for better performance.
High-k metal gate technology replaces silicon dioxide and polysilicon with high dielectric constant materials and metal electrodes reducing leakage.
Higher numerical aperture for better resolution.
Overlay beyond simple X-Y shift (rotation scaling).
High-power probing tests devices at elevated current and voltage levels requiring specialized probe tips and thermal management.
Adapt to higher resolution images.