Cascading Approaches - Research Notebook

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Interpolator + DA¶

1. Train NerF on Observations

Assume we have a dataset of independent observations, $y_n$ , on the observation domain $(\Omega_y,\mathcal{T}_y)$ .

\begin{aligned} \mathcal{D}=\{(x_n,t_n),y_n\}_{n=1}^N, && && x\in\Omega_y\subseteq\mathbb{R}^{D_s} && t\in\mathcal{T}_y\subseteq\mathbb{R}^+ \end{aligned}

We train a neural field, $f_\theta$ , to interpolate the observation wrt the spatiotemporal coordinate values.

\begin{aligned} y_n &= f_\theta(x_n, t_n) + \varepsilon_n \end{aligned}

2. Train Strong Constrained DA with NerF

\begin{aligned} \boldsymbol{u}_0 &\sim \mathcal{N}(\boldsymbol{u}_b,\boldsymbol{\Sigma}_b)\\ \boldsymbol{u}_t &= \boldsymbol{f_\theta}\left( \boldsymbol{u}_{t-1},t\right)\\ \boldsymbol{y}_t &= \boldsymbol{h_\theta}(\boldsymbol{u}_t, t) + \boldsymbol{\varepsilon}_y, && && \boldsymbol{\varepsilon}_y \sim \mathcal{N}(0,\boldsymbol{\Sigma_y}) \end{aligned}

We can estimate the state by minimizing the objective function

\mathcal{J}(u) = \sum_{t=1}^T||\boldsymbol{y_\theta}(t) - h_\theta(u_t,t)||_{\boldsymbol{\Sigma_y}^{-1}}^2 + ||\boldsymbol{u}_0 - \boldsymbol{u}_b||_{\boldsymbol{\Sigma}_b^{-1}}^2

3. Train NerF

Assume we have a dataset of independent reanalysis points, $u_n^*$ , on the observation domain $(\Omega_y,\mathcal{T}_u)$ .

\begin{aligned} \mathcal{D}=\{(x_n,t_n),u_n^*\}_{n=1}^N, && && x\in\Omega_z\subseteq\mathbb{R}^{D_s} && t\in\mathcal{T}_z\subseteq\mathbb{R}^+ \end{aligned}

We train a neural field, $f_\theta$ , to interpolate the observation wrt the spatiotemporal coordinate values.

\begin{aligned} \boldsymbol{u}^*(x_n, t_n) &= f_\theta(x_n, t_n) + \varepsilon_n, && && \mathbf{x}\in\Omega_z\subseteq\mathbb{R}^{D_s} \end{aligned}

5. Train Weak-Constrained DA

\begin{aligned} \boldsymbol{u}_0 &\sim \mathcal{N}(\boldsymbol{u}_b,\boldsymbol{\Sigma}_b)\\ \boldsymbol{u}_t &= \boldsymbol{f_\theta}\left( \boldsymbol{u}_{t-1},t\right), && && \boldsymbol{\varepsilon}_u \sim \mathcal{N}(0,\boldsymbol{\Sigma_u}) \\ \boldsymbol{y}_t &= \boldsymbol{h_\theta}(\boldsymbol{u}_t, t) + \boldsymbol{\varepsilon}_y, && && \boldsymbol{\varepsilon}_y \sim \mathcal{N}(0,\boldsymbol{\Sigma_y}) \end{aligned}

We can estimate the state by minimizing the objective function

\mathcal{J}(u) = \sum_{t=1}^T||\boldsymbol{y_\theta}(t) - h_\theta(u_t,t)||_{\boldsymbol{\Sigma_y}^{-1}}^2 + \sum_{t=1}^T||\boldsymbol{u_\theta}(t) - \boldsymbol{f_\theta}\left( \boldsymbol{u}_{t-1},t\right)||_{\boldsymbol{\Sigma_u}^{-1}}^2 + ||\boldsymbol{u}_0 - \boldsymbol{u}_b||_{\boldsymbol{\Sigma}_b^{-1}}^2

Interpolator + Foundational Models¶

2. Train Embedding on NerF

Assume we have a dataset of sequential, independent observations, $y_t$ , which is given by the neural field, $f_\theta$ . However, we query the functa on the latent domain, $(\Omega_z, \mathcal{T}_z)$ .

\begin{aligned} \boldsymbol{y_\theta}(t)&=\boldsymbol{f_\theta}(\mathbf{X}_z,t), && && \mathbf{X}_z\in\mathbb{R}^{D_{\Omega_z}} && t\in\mathcal{T}_z\subseteq\mathbb{R}^+ \end{aligned}

where $\mathbf{X}_z = \{ \mathbf{x}\in\Omega_z\in\mathbb{R}^{D_s}\}$ . We can create a dataset by (quasi-)randomly selecting points

\mathcal{D}=\{ \boldsymbol{y_\theta}(t) \}_{t=1}^T \hspace{10mm}

We train an embedding on the latent domain, $z$ , using the Neural Field. We can also apply a random mask, $\boldsymbol{m}$ , to help augment the data by randomly masking pixels.

\mathcal{L}(\theta) = \frac{1}{|\mathcal{D}|}\sum_{t\in\mathcal{D}} ||\boldsymbol{y_\theta}(t) - T_D\circ T_E\circ \boldsymbol{m}\circ \boldsymbol{y_\theta}(t)||^2_2

Latent Variable¶

Train (Masked) AutoEncoder on Simulations
PnP for Real Observations
Train AutoEncoder on Sparse Observations
Train Variational AutoEncoder (Probabilistic Reconstruction)
Train U-Net (DEQ)

Dynamical Model