## 0th-order theory

order of magnitude physics:

- dimensional analysis
- back-of-envelope calculation

numerical approximation to one/two number of significance

## 1st-order theory

- (single factor) linear theory
- linear elasticity theory
- Fick’s law
- Darcy’s law
- Newton–Raphson method
- asymptotic homogenization
- linearization

## 2nd-order theory

Forchheimer equation

## Higher-order theory: Asymptotics

所有**渐进展开**方法都是一个思路: "分解" (可分空间上对一组基投影, 不可分空间上的核变换), (排序), 截断近似。

- Taylor expansion
- Spectral methods
- PCA (Principal Component Analysis) of matrices
- K-L expansion, PCE (polynomial chaos expansion)
- Fourier analysis
- Fourier series
- Fourier transform
- generalized Fourier transforms

## General non-linear theory

Navier-Stokes equation

## Not-categorized

- coupling of mechanisms (multi-physics)
- relaxation of assumptions
- generalization:

### Generalization

Countability

- single - multiple - countably infinite - continuous (bounded) - continuous (unbounded)
- scalar- vector - sequence - function

Compactness

- scalar - vector - tensor (matrix)
- real number - complex number
- linear equation - system of linear equations
- linear function - linear transformation - linear functional - linear operator

Complexity of function

- algebraic equation - transcendental equation - ...

Constant to variable

Deterministic to stochastic

- random variable - random vector - random sequence - random waveform
- ODE - SDE; PDE - SPDE

Point to space

- vector - vector space
- matrix - matrix analysis
- function - functional analysis (Hilbert space, Sobolev space)
- random variable - space of random variables (Gaussian Hilbert space)

## Comments

- Nonlinear models seems to be residues, as common phenomena are well approximated by linear models.
- 定量学科是个细活，娇贵得很。粗糙的问题用不上它。

🏷 Category=Modeling