Data assimilation is factor-graph inference with different vocabulary, an enormous state, and a sixty-year head start. The interesting question is not whether the shape fits — it plainly does — but what a relational, grey-box treatment would add.
Sources: original to this vault (design and analysis; no single paper).
Theory (CT-ML wiki): Variational Free Energy
The relations
| relation | says |
|---|---|
| the discretised dynamics | consecutive states are related by a PDE solve |
| conservation laws | mass, energy, momentum are preserved exactly |
| observation operators | a satellite radiance is a functional of the state |
| boundary and forcing | the system is driven by things it does not determine |
The conservation laws are the acausal ones and they are not negotiable: a scheme that violates
mass balance is wrong regardless of how well it fits. That is precisely a hard constraint
alongside soft ones — a LinearConstraintFactor with small noise beside factors with large
noise.
The field already does this, under other names
4D-Var minimises a sum of a background term and observation terms over a time window. That is minimising over a chain — Energy-Based Factor Graphs §1 exactly, and at zero temperature (a MAP estimate, no posterior).
The Ensemble Kalman Filter propagates an ensemble and updates it against observations: a
sample-based belief with a Gaussian update rule, which is a particular answer to the combine
problem messages §1 records as open.
So both dominant methods are instances of the framework, and the framework’s contribution would have to be at the margins the existing methods leave: a genuine posterior rather than a MAP (4D-Var), non-Gaussian updates (EnKF), and learned components inside the model rather than bolted beside it.
What is observed
Satellites, radiosondes, buoys, ships, stations — different instruments, different error characteristics, wildly non-uniform coverage in space and time, and asynchronous by nature. The observation operator is often the hardest part of the model.
Asynchrony over a continuum is the Time as a Base case: state as a trajectory queried at observation times rather than a value at grid times.
What would be learned
This is where the domain has moved and where the fit is sharpest. Subgrid parametrisations — clouds, convection, turbulence, radiation — are the known unknowns of climate modelling, they are the dominant source of inter-model spread, and they are increasingly being replaced by learned emulators.
A learned convection scheme inside a model that must still conserve mass and energy is exactly the grey-box case, and the failure mode of naive ML parametrisations is exactly the one a hard constraint prevents: they drift, because nothing forces conservation.
Learned components that must respect exact conservation laws is not a niche requirement here. It is the central open problem of ML-augmented climate modelling.
What a residual means
Model misfit, localised. Per-factor energies say where and when the model disagrees with observation, which is how parametrisation deficiencies are diagnosed. And the free energy is a model-comparison score: which of two parametrisation schemes explains the observations better, with complexity accounted for rather than penalised by hand.
For a field whose central output is an ensemble of structurally different models, a principled comparison score is not a small thing.
One model, several questions
- Assimilation: observations clamped, state inferred.
- Prediction: state clamped, future propagated.
- Attribution: clamp a counterfactual forcing and ask what the state would have been.
- Observing-system design: which new instrument most reduces posterior uncertainty?
The third is the socially consequential one and it is a polarity choice.
What would be hard
This is the least tractable of the six and it should be said plainly.
- Scale. – state variables. Nothing in this project is within several orders of magnitude (Parallelism and Compilation), and operational assimilation is already at the limit of the largest machines available.
- Chaos. Sensitivity to initial conditions bounds predictability regardless of method, and linearised error propagation is valid only over short windows — which is why 4D-Var uses a window at all.
- Strong nonlinearity, so the exact linear-Gaussian results do not apply.
- The dynamics factor is a PDE solve. Wrapping one as a factor means each message costs a model run, which changes the economics of message passing entirely: you cannot iterate.
The realistic reading is that this domain is a source of structure to learn from rather than a near-term target — 4D-Var and EnKF are worth studying as two well-tested answers to questions this project has open, especially the EnKF’s treatment of sample-based beliefs.
Related: Motivating Examples, Energy-Based Factor Graphs, Time as a Base, messages, Parallelism and Compilation, The Linear Gaussian Chain