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Modeling Components

Observations

Measurements we can actually observe.

y=y(s,t),y:RDs×R+→RDys∈Ωy⊆RDst∈Ty⊆R+\begin{aligned} \boldsymbol{y} &= \boldsymbol{y}(\mathbf{s},t), && && \boldsymbol{y}: \mathbb{R}^{D_s}\times\mathbb{R}^+\rightarrow\mathbb{R}^{D_y} && && \mathbf{s}\in\Omega_y\subseteq\mathbb{R}^{D_s} && && t \in \mathcal{T}_y \subseteq \mathbb{R}^+ \end{aligned}
(1)

Covariate

Data which we believe is conditionally important for our model.

x=x(s,t),x:RDs×R+→RDxs∈Ωx⊆RDst∈Tx⊆R+\begin{aligned} \boldsymbol{x} &= \boldsymbol{x}(\mathbf{s},t), && && \boldsymbol{x}: \mathbb{R}^{D_s}\times\mathbb{R}^+\rightarrow\mathbb{R}^{D_x} && && \mathbf{s}\in\Omega_x\subseteq\mathbb{R}^{D_s} && && t \in \mathcal{T}_x \subseteq \mathbb{R}^+ \end{aligned}
(2)

Quantity of Interest (QoI)

The true quantity we are interested in estimating.

u=u(s,t),u:RDs×R+→RDxs∈Ωu⊆RDst∈Tu⊆R+\begin{aligned} \boldsymbol{u} &= \boldsymbol{u}(\mathbf{s},t), && && \boldsymbol{u}: \mathbb{R}^{D_s}\times\mathbb{R}^+\rightarrow\mathbb{R}^{D_x} && && \mathbf{s}\in\Omega_u\subseteq\mathbb{R}^{D_s} && && t \in \mathcal{T}_u \subseteq \mathbb{R}^+ \end{aligned}
(3)

Latent Variables

Unknown, unobserved variables

z=z(s,t),z:RDs×R+→RDxs∈Ωz⊆RDst∈Tz⊆R+\begin{aligned} \boldsymbol{z} &= \boldsymbol{z}(\mathbf{s},t), && && \boldsymbol{z}: \mathbb{R}^{D_s}\times\mathbb{R}^+\rightarrow\mathbb{R}^{D_x} && && \mathbf{s}\in\Omega_z\subseteq\mathbb{R}^{D_s} && && t \in \mathcal{T}_z \subseteq \mathbb{R}^+ \end{aligned}
(4)

Parameters

Unknown, unobserved quantities to be estimated.

θ∈Θ⊆RDθ\begin{aligned} \boldsymbol{\theta} &\in\boldsymbol{\Theta}\subseteq\mathbb{R}^{D_\theta} \end{aligned}
(5)

Operators

f:x(s,t)→u(s,t)\boldsymbol{f}: \boldsymbol{x}(\mathbf{s},t) \rightarrow \boldsymbol{u}(\mathbf{s},t)
(6)

Parameterized

f∗:x(s,t)×Θ→u(s,t)\boldsymbol{f}^*: \boldsymbol{x}(\mathbf{s},t)\times\boldsymbol{\Theta} \rightarrow \boldsymbol{u}(\mathbf{s},t)
(7)

Criteria

Loss Function

L:RDθ×RDy→R\boldsymbol{L}: \mathbb{R}^{D_\theta}\times\mathbb{R}^{D_y} \rightarrow \mathbb{R}
(8)

Objective Function

J:RDu×RDθ→R\boldsymbol{J}: \mathbb{R}^{D_u}\times\mathbb{R}^{D_\theta} \rightarrow \mathbb{R}
(9)

Tasks

Parameter Learning

θ∗=argminθL(θ)\boldsymbol{\theta}^* = \underset{\boldsymbol{\theta}}{\text{argmin}}\hspace{2mm} \boldsymbol{L}(\boldsymbol{\theta})
(10)

Estimation

u∗(θ)=argminuJ(u;θ)\boldsymbol{u}^*(\boldsymbol{\theta}) = \underset{\boldsymbol{u}}{\text{argmin}}\hspace{2mm} \boldsymbol{J}(\boldsymbol{u};\boldsymbol{\theta})
(11)

Bi-Level Optimization

θ∗=argminθL(θ)u∗(θ)=argminuJ(u;θ)\begin{aligned} \boldsymbol{\theta}^* &= \underset{\boldsymbol{\theta}}{\text{argmin}}\hspace{2mm} \boldsymbol{L}(\boldsymbol{\theta}) \\ \boldsymbol{u}^*(\boldsymbol{\theta}) &= \underset{\boldsymbol{u}}{\text{argmin}}\hspace{2mm} \boldsymbol{J}(\boldsymbol{u};\boldsymbol{\theta}) \end{aligned}
(12)
Notation
Operators
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