When Data Alone Isn't Enough

The basin's newest well, GW-22, sits close to an old agricultural pumping well that draws groundwater for irrigation. Your supervisor wants a model of groundwater head, the water table elevation, at any distance from the pump, but the team has only eight scattered measurements along the transect, taken over a rushed field day before a storm.

You fit a flexible curve to the eight points. It matches them almost perfectly, and predicts the water table rising the further you get from an actively pumping well. Your supervisor doesn't even look at the R²: 'That's impossible. Water doesn't do that near a pump. Throw more flexible curves at this and you'll always find one more way to be wrong.'

With only eight data points, an enormous number of curves fit almost equally well, some sensible, some physically absurd. Purely data-driven models have no way to know that water can't rise as you move away from a well that's actively drawing it down; they only know the numbers you gave them.

Physics-informed modelling (teaching a model rules from real physical laws so it can never break them, even in gaps with no data) means baking known physical laws, like Darcy's Law from earlier hydrogeology work, directly into a model, so it literally cannot produce answers that violate them, no matter how sparse or noisy the data is.

Imagine asking a kid to connect eight dots on a page into a smooth curve, without telling them anything else. They might draw wild loops that technically touch every dot. Now tell them one extra rule: the line can only go downhill, never up. Suddenly almost all of those wild loops are impossible, and you are left with something much more sensible.

This exact problem, too little data, too much flexibility, appears throughout environmental science: climate models, disease spread models, and groundwater models all combine sparse observations with known physical or biological constraints. Physics-informed machine learning is a fast-growing research area precisely because pure data-driven models struggle when observations are limited.

Explain to a teammate why 'the curve fits the data well' is not the same as 'the curve is physically plausible'.