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Abstract

In computer science, we often can “bin” all problems into a series of sub-problems that we can iterate over until convergence. The same can be said for data-driven geoscience! In fact, I claim that most problems can be put into a series of 4 categories: 1) data acquisition, 2) learning, 3) estimating, and/or 4) predictions.

Keywords:taskslearningestimationpredictions

Overview

  • Data Acquisition
  • Learn
  • Estimate
  • Predict

Data

D={xn,yn,zn∗,}n=1N\mathcal{D} = \left\{ \mathbf{x}_n, \mathbf{y}_n, \mathbf{z}_n^*, \right\}_{n=1}^N
(1)
Measurements:yn∈Y⊆RDyCovariates:xn∈X⊆RDxSimulationed States:znsim∈Zsim⊆RDzReanalysis States:zn∗∈Z∗⊆RDz\begin{aligned} \text{Measurements}: && && \mathbf{y}_n &\in\mathcal{Y}\subseteq\mathbb{R}^{D_y} \\ \text{Covariates}: && && \mathbf{x}_n &\in\mathcal{X}\subseteq\mathbb{R}^{D_x} \\ \text{Simulationed States}: && && \mathbf{z}_n^{sim} &\in\mathcal{Z}^{sim}\subseteq\mathbb{R}^{D_z} \\ \text{Reanalysis States}: && && \mathbf{z}_n^{*} &\in\mathcal{Z}^{*}\subseteq\mathbb{R}^{D_z} \\ \end{aligned}
(2)

Learning

I have data, D\mathcal{D}, which captures the phenomena that I want to learn.

I want to learn a model, ff, with the associated parameters, θ, give the data, D\mathcal{D}.

θ∗=arg min⁡θL(θ;D)\boldsymbol{\theta}^* = \underset{\boldsymbol{\theta}}{\argmin} \hspace{2mm} \boldsymbol{L}(\boldsymbol{\theta};\mathcal{D})
(3)

where L(⋅)\boldsymbol{L}(\cdot) is our loss function.

L:RDθ×D→R\begin{aligned} \boldsymbol{L} : \mathbb{R}^{D_\theta} \times \mathcal{D} \rightarrow \mathbb{R} \end{aligned}
(4)

Estimation

I have a model, ff, and parameters, θ.

I have some measurements, yy.

I want to estimate a state, zz.

z∗(θ)=arg min⁡zJ(z;θ,D)\mathbf{z}^*(\boldsymbol{\theta}) = \underset{\mathbf{z}}{\argmin} \hspace{2mm} \boldsymbol{J}(\mathbf{z};\boldsymbol{\theta},\mathcal{D})
(5)

where J(⋅)\boldsymbol{J}(\cdot) is our objective function defined as:

J:RDz×RDθ×D→R\begin{aligned} \boldsymbol{J} : \mathbb{R}^{D_z} \times \mathbb{R}^{D_\theta}\times\mathcal{D} \rightarrow \mathbb{R} \end{aligned}
(6)

Parameter & State Estimation

Parameter Estimation:θ∗=arg min⁡θL(θ;D)State Estimation:z∗(θ)=arg min⁡zJ(z;θ,D)\begin{aligned} \text{Parameter Estimation}: && && \boldsymbol{\theta}^* = \underset{\boldsymbol{\theta}}{\argmin} \hspace{2mm} \boldsymbol{L}(\boldsymbol{\theta};\mathcal{D}) \\ \text{State Estimation}: && && \mathbf{z}^*(\boldsymbol{\theta}) = \underset{\mathbf{z}}{\argmin} \hspace{2mm} \boldsymbol{J}(\mathbf{z};\boldsymbol{\theta},\mathcal{D}) \end{aligned}
(7)

This is akin to the:

  • Approximate Inference methods - expectaction maximization, variational inference
  • Bi-Level Optimization
  • Data Assimilation

Prediction

I have my model, parameters, and state estimation.

I want to make a prediction for my QoI, uu.

u∗=f(z∗,θ)u^* = \boldsymbol{f}(\mathbf{z}^*, \boldsymbol{\theta})
(8)

In this case, we never have access to any sort of validation, uu. We are simply making a prediction.

Concepts
Problem Category
Concepts
Modeling