Fig. 3: Summary of validation cases.
Schematic representations of the physical systems (a–c), the corresponding systems of equations, and the optimal dimensionless inputs and outputs discovered by IT-π, are shown alongside the inferred physical properties, including self-similarity, distinct physical regimes, characteristic scales, relevant dimensionless parameters (where applicable), and the irreducible model error. d Rayleigh problem: Optimal dimensionless velocity profiles \({\Pi }_{o}^{ * }\) as a function of Π*, with colors indicating different times t. e Colebrook equation: Contour plot of the prediction score R1 (i.e., the relative importance of \({\Pi }_{1}^{ * }\)) as a function of the dimensionless inputs. The result is obtained by dividing the dimensionless inputs Π* into 10 clusters using the KNN method. f, g, i Normalized irreducible error (\({\tilde{\epsilon }}_{LB}\)) for individual components of Π* and for all components combined, for (f) the Rayleigh problem, (g) the Colebrook equation, and (i) the Malkus waterwheel. Error bars indicate the uncertainty in the normalized irreducible error, computed as the difference between the error bound estimated using the full dataset and that estimated using half the data. Additional details regarding the symbols and definitions of all variables are provided in the main text.
