Two Ways to Predict: Physics and Data

There are two ways to predict what a ship will consume. One derives it from resistance, propulsion and thermodynamics. The other learns it from records of what the ship did. Both work, and the useful skill is knowing what each buys.

Techniques against first principles

The distinction shows up first in how people learn. Working through techniques means recognising that a problem resembles a stored pattern and applying the method that goes with it. It is quick to start and it is how most professional practice runs, most of the time.

Working from first principles means building from conservation laws and material properties. It is slow to start and it has the higher return, because it is the only one of the two that survives contact with a problem nobody has seen before.

The same distinction separates the two families of model, and for the same reason.

What physics contributes

Four things, and they are worth enumerating because the fourth is the one people forget.

Extrapolation. A fitted model is a description of the conditions it was fitted on. Ask it about a speed, a draught or a sea state outside that envelope and it will still return a number, produced by continuing a curve into territory where nothing constrained its shape. A physical relationship does not stop applying at the edge of a dataset.

Reason. Physics says why consumption moved. A data model says only that it moved with something else, and a number without a mechanism cannot be argued with, checked against a second source, or defended in a meeting.

Dominant dynamics, and the order of the smaller effects. Physics tells you that resistance goes with roughly the square of speed and required power with the cube, and therefore that the leading effect of a speed change dwarfs almost everything else. That ordering is how you know which terms matter and which are noise.

Magnitude, not just direction. Anyone can say a cleaner hull uses less fuel. Physics says approximately how much, which is what a business case requires.

An example of the fourth, and a trap inside it

Required power goes with roughly the cube of speed, which is the most-quoted fact in ship operation. The one that follows from it is quoted far less often.

Over a fixed distance, the trip takes less time at higher speed, and the time saved cancels one power of the speed:

trip fuel ∝ (distance / V) × V³ = distance × V²

So trip fuel scales with roughly the square of speed, not the cube. Slow steaming still saves fuel, but less than the cube law implies: a 10 per cent cut in speed saves about 19 per cent of the trip's fuel, not 27. That is a physics-derived correction to a widely repeated number, and no dataset would have told you it was needed.

The cube itself deserves the same care. It is the calm-water figure for a hull at low speed, where resistance is mostly friction, and it is a round number rather than a constant of nature. Friction alone grows a little more slowly than the square of speed, because the friction coefficient falls as the Reynolds number rises, so a full, slow hull can sit slightly under 3, around 2.8 to 2.95. Once wave-making matters the exponent climbs, and ships in service typically sit between 3 and 4, with fast ships higher still. Treat 3 as the form to start from and estimate the exponent from the ship's own records, which lesson five shows how to do. With an exponent n, trip fuel goes with V^(n−1): a little under the square for a full, slow hull, steeper than it once wave-making matters, and always one power of speed below the power law.

What data contributes

The case for the data-driven approach is equally real.

Ships depart from theory. Hull roughness, propeller condition, engine wear, the particular way this vessel is operated: none of these appear in a textbook formula, and all of them move consumption. A model fitted on the ship's own records absorbs them without anyone having to characterise them.

Within the range of conditions well covered by the training data, a flexible model will usually interpolate more accurately than a physical model with generic coefficients. And it is far quicker to build than a calibrated hydrodynamic model.

Where each is stronger

A short table, and one that decides most modelling arguments.

The hybrid worth building

The productive arrangement is not to choose but to layer, and the standard pattern is worth stating explicitly.

Use physics for the structure and data for the corrections. Build the physical model of the propulsive chain, use it to predict consumption, then fit a data-driven model to the residual, the difference between what physics predicts and what the ship actually did.

Two things follow, and both are valuable. The residual model absorbs everything the physics does not capture, hull roughness, propeller condition, this vessel's individual character, without anyone having to name those effects. And outside the sampled envelope the residual model contributes little while the physics still holds, so the prediction degrades gracefully rather than diverging.

A simpler version of the same idea is available to anyone fitting a regression: use physically motivated features rather than raw channels. Regressing consumption on speed cubed rather than on speed builds the cube law into the model's structure, and the fitted coefficient then has a physical meaning that can be checked against a sea trial. Where the exponent itself is in doubt, lesson five shows how to estimate it from the records instead.

Monday morning

Take whatever model your fleet currently uses and answer two questions about it. What range of conditions was it fitted on, and does anything stop it answering a question outside that range? Most models will answer anything they are asked, silently. Then ask whether it can say why last quarter differed from the one before, or only that it did. The second question usually determines whether anybody acts on its output.