Modern 4DVar

Operators

CNRS
MEOM

In this section, we will break down the

Variable I:a⃗=a⃗(x⃗,t),x⃗∈Ωa⊆RDa,t∈Ta⊆R+Variable II:u⃗=u⃗(x⃗,t),x⃗∈Ωu⊆RDu,t∈Ta⊆R+\begin{aligned} \text{Variable I}: && \vec{\boldsymbol{a}} &= \vec{\boldsymbol{a}}(\vec{\mathbf{x}},t), && \vec{\mathbf{x}}\in\boldsymbol{\Omega}_a\subseteq\mathbb{R}^{D_a}, && t\in\boldsymbol{\mathcal{T}}_a\subseteq\mathbb{R}^+ \\ \text{Variable II}: && \vec{\boldsymbol{u}} &= \vec{\boldsymbol{u}}(\vec{\mathbf{x}},t), && \vec{\mathbf{x}}\in\boldsymbol{\Omega}_u\subseteq\mathbb{R}^{D_u}, && t\in\boldsymbol{\mathcal{T}}_a\subseteq\mathbb{R}^+ \end{aligned}
(1)
T[a](x⃗,t):a⃗(x⃗,t)→u⃗(x⃗,t)\boldsymbol{T}[\boldsymbol{a}](\vec{\mathbf{x}},t): \vec{\boldsymbol{a}}(\vec{\mathbf{x}},t) \rightarrow \vec{\boldsymbol{u}}(\vec{\mathbf{x}},t)
(2)

Simple Example

# obtain values
a: Array = ...
u: Array = ...

# initialize params of transformation
params: Params = Params(w=..., b=...)

# initialize transformation
transform: Callable = lambda a, params: a * params.w + params.b

# apply transformation
u_hat: Array = transform(a, params)

# test to ensure it is equal
np.testing.assert_array_equal(u, u_hat)

Problems:

  • What if the domains are different?
  • What about spatiotemporal data?

Motivating Examples

  • Forecasting Problem
  • Interpolation Problem
  • Multimodal Domain Transformation

Domain

Ωu=Tθ(Ωa)\Omega_u = \boldsymbol{T_\theta}\left(\Omega_a\right)
(3)
# Domain 1
field_a: Field = Field(values_a, domain_a)
field_u: Field = Field(values_u, domain_u)

# initialize interpolator
a_to_u_f: Callable = FieldInterpolator(field_u.values, field_u.coords)

# apply interpolator
field_a_on_u: Field = a_to_u_f(field_a.values, field_a.coords)

Functional

Latent Space

Variable I —> Variable II a = Transformer(u) z_u = Encoder(u) z_a = LatentTransformer(z_u) a’ = DecoderD2(z_a)

Examples:

  • Lift & Learn
  • Koopman Theory
  • Neural Operators
Model Representations
Overview
Model Representations
Space & Time