Motivating Examples

application overview

Six problem domains that have the same shape, and why that shape is what this project is built for.

These are motivating sketches, not case studies. Nothing here has been implemented; the library is a prototype (Related Julia Projects §10 is the honest state of it). The point is to show what kind of problem the machinery is aimed at.

Sources: original to this vault (design and analysis; no single paper).

domain	the relations are	what is learned
SLAM and Sensor Fusion	geometry: poses, landmarks, motion	visual front-ends, depth priors
Trading and Financial Markets	no-arbitrage: parity, triangular, index–constituent	volatility surfaces, illiquid instruments
Energy Markets and Power Grids	Kirchhoff + market clearing	renewable and demand forecasts
Metabolomics and Proteomics	stoichiometry: $Sv=0$	enzyme kinetics, regulation
Molecular Dynamics	force fields, bond constraints	ML potentials
Climate and Dynamical Systems	conservation laws, discretised PDEs	subgrid parametrisations
Language Models †	agreement between experts; grammar and type constraints	the experts themselves

† A different shape from the other six — see the note at the end of §6.

The shape they share

1. A network of relations, not a pipeline

In every one of these, the model is a set of constraints among quantities with no natural input and output. Kirchhoff’s law does not say “current causes voltage”; put–call parity does not say which of the four prices is the answer; $Sv=0$ does not designate an output flux.

A neural network is a pipeline and needs one. A factor graph does not — which is the whole content of Implicit Learners and the reason a Polarity is chosen per call rather than baked in at construction.

2. Observations are sparse, heterogeneous and asynchronous

Nothing in these domains is fully observed. A grid has meters on some buses; a metabolic network has concentrations for some metabolites; a trading venue quotes some instruments and not others, at times that do not line up.

Partial observation is the normal case, and it is exactly what Observed / Unobserved / Latent encodes. Asynchrony is what Time as a Base is for.

3. Some of the model is known and some must be fitted

This is the one that rules out both alternatives. Pure physics tools (ModelingToolkit, FBA solvers, MD engines) cannot express a fitted component. Pure ML tools cannot express a conservation law that must hold exactly.

Every domain below is grey-box: a stoichiometry you trust beside kinetics you do not; an exact power-flow equation beside a wind forecast; a geometric constraint beside a learned visual odometry front-end. A factor graph does not care which kind a node is, which is the point of the factor interface.

4. The decisions are made under uncertainty

Dispatch a generator, trade a spread, refine a structure, publish a projection. Point estimates are not enough and everyone in these fields knows it — which is why each has grown its own uncertainty machinery (EnKF, bootstrapped SLAM covariances, ensemble forecasts).

5. One model, several questions

Each domain asks its model more than one thing, and they are the same relations with different quantities held fixed:

domain	one question	the other question
SLAM	given the map, where am I?	given my pose, what is the map?
grids	state estimation	contingency analysis
metabolism	fluxes from concentrations	concentrations from fluxes
climate	assimilation	prediction

That is Channels and Polarity doing the work it exists for. SLAM has the two directions in its own name.

6. The residual means something

This is the observation that ties the six together and it is easy to miss.

In ordinary machine learning a residual is an error — something to minimise and then forget. In every domain below, “how badly is this relation violated” is itself a quantity of interest:

domain	a nonzero residual is
SLAM	a loop-closure inconsistency; a bad data association
trading	a mispricing — the trade signal itself
power grids	bad data, a failed sensor, or an undetected topology change
metabolism	an unmodelled reaction or a measurement error
MD / structure	strain; incompatibility between experiment and force field
climate	model misfit; where the parametrisation is failing

So the graded energy of Scalar and Multivariate Energy is not bookkeeping. It is per-relation attribution of disagreement, and in at least two of these domains it is the output rather than a diagnostic.

Language Models shares this property and few of the others

A product of expert language models fits §3 (some parts known, some fitted), §4 (uncertainty matters) and §6 (the residual — here, disagreement between experts — is a usable abstention signal). It does not fit §1: for $k$ experts on one sequence the graph is a star, and message passing degenerates to adding the energies.

It is listed because the framing is clarifying — it is a product of experts, not a mixture — rather than because the machinery earns its keep. That note says where it would.

What would have to be true

Being honest about the gap between the shape and the software:

• Scale. Climate and MD are $10^6$–$10^9$ variables. Metabolic networks and grids are $10^3$–$10^5$. SLAM and trading graphs are the ones the current implementation could plausibly reach. See Parallelism and Compilation.
• Loops. Meshed grids, metabolic cycles and SLAM loop closures are all loopy, and loopy message passing gets exact means with wrong variances (Loopy Message Passing). For domains where the uncertainty is the product, that is the blocking problem.
• Non-Gaussian everything. Financial returns are heavy-tailed; molecular configurations are multimodal; flux distributions are constrained to a polytope. The Gaussian fragment is where the exactness results live, and none of these are in it — which is messages §1’s gap in applied clothing.
• Nonlinearity. AC power flow, enzyme kinetics, atmospheric dynamics. The exact results are linear-Gaussian; everything else is approximate.

None of that makes the shape wrong. It makes the six notes below a description of where the work would pay off, not a claim that it already has.

Related: Implicit Learners, Geometric Deep Learning and Physical Laws, Channels and Polarity, Time as a Base, Scalar and Multivariate Energy, Related Julia Projects, Parallelism and Compilation, The Linear Gaussian Chain