Extrapolation
Definition
The ability of a model to remain usable outside its training distribution — more extreme parameters, longer horizons, unseen geometries. It is the biggest bottleneck taking Physics-AI from papers to engineering, and the weakness most easily hidden by evaluation metrics.
What it is
Interpolation predicts inside the hull of training samples; extrapolation predicts outside it. Physics-AI models are almost all interpolators: the Reynolds-number range, geometry family and time horizon seen during training determine how far they can travel at deployment. Beyond that, error does not degrade gently — it usually falls off a cliff.
Key points
- Three kinds: parameter extrapolation (higher Re, larger load), geometry extrapolation (unseen shapes), horizon extrapolation (more rollout steps than trained).
- Evaluation needs a dedicated OOD set: treating in-distribution test error as generalization ability is the most common misjudgment.
- Mitigations: normalize inputs to dimensionless groups (Re, Ma rather than raw units), physics-constrained regularization, domain adaptation, online correction.
- Engineering answer: publish a "model applicability envelope" plus automatic fallback to the classical solver when out of range — more realistic than chasing unlimited generalization.
Common pitfalls / boundaries
- More data does not fix extrapolation: it is a distribution problem, not a sample-count problem.
- Claims that "physics constraints make the model extrapolate" need OOD test results, not reasoning.