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What PI-GPR Does And Does Not Guarantee

Physics-informed Gaussian Process Regression can make a model more useful and more interpretable, especially in low-data materials problems. It is still a statistical model. A physics term in the mean function is an inductive bias, not proof that the model is physically correct.

What PI-GPR Does

In matgpr, a physics-informed model has the form:

\[ y(\mathbf{x}) = m_{physics}(\mathbf{x}_{phys}) + f_{residual}(\mathbf{x}) + \epsilon \]

where:

  • \(m_{physics}\) is a user-defined or template-based physics mean function,
  • \(\mathbf{x}_{phys}\) are the features used by the physics equation,
  • \(f_{residual}\) is the GP residual learned from data,
  • \(\epsilon\) is observation noise.

This setup does provide a few clear guarantees about the model definition:

  • The physics equation enters the GP mean function, not just a postprocessing plot.
  • Learnable physics parameters are optimized during GP training together with kernel and likelihood parameters.
  • The GP residual remains available to model mismatch between the physics trend and the observations.
  • Predictive uncertainty is still computed from the fitted GP model.
  • The physics equation, features, learned parameters, and validation protocol can be reported explicitly.

What PI-GPR Does Not Guarantee

PI-GPR does not automatically guarantee:

  • lower RMSE than standard GPR,
  • better extrapolation outside the training domain,
  • physically valid predictions for every candidate,
  • calibrated uncertainty intervals,
  • causal interpretation of learned parameters,
  • correct physics if the equation is misspecified,
  • correct units if feature scaling and unit handling are wrong,
  • monotonicity, positivity, boundedness, or conservation laws unless those constraints are encoded separately,
  • automatic discovery of the right physics equation from data.

A physics-informed mean can hurt performance when the equation has the wrong trend, uses weak features, leaks information from test data, or is too complex for the available dataset.

What To Validate

Always compare PI-GPR against a standard GPR baseline using the same data split, features, kernel class, optimizer settings, and evaluation metrics. For a publishable result, report:

  • repeated learning curves,
  • held-out test metrics,
  • parity plots with uncertainty bars,
  • uncertainty diagnostics such as interval coverage and NLPD,
  • learned physics parameters,
  • the exact features used by the physics equation,
  • the applicability domain of the dataset.

The strongest evidence for PI-GPR is not that it contains an equation. The strongest evidence is that the equation improves or stabilizes validation performance in the intended low-data regime while remaining physically interpretable.

When Extra Constraints Are Needed

Use additional matgpr tools when the property has known limits:

  • Use target transforms for positive or bounded targets.
  • Use virtual observations for soft known-limit or monotonicity anchors.
  • Use derivative-constrained GPR when trusted slope information is available.
  • Use physics-aware or learned heteroscedastic noise when uncertainty changes across the materials space.

These tools add more structure than a mean function alone. They still require validation and should be reported separately from the PI-GPR mean equation.