Fig. 5: Discovering the underlying physical laws with experimental data.
From: Discovering physical laws with parallel symbolic enumeration
a, The set-up of the EMPS experiment. b, Collected displacement and input force data. c, The prediction performance along with a typical governing equation discovered by PSE. Herein, RMSE stands for root mean square error. d, The prediction performance along with a typical governing equation discovered by PySR. e, The prediction performance along with a typical governing equation discovered by BMS. f, The prediction performance along with a typical governing equation discovered by wAIC. Additional results for other methods on the EMPS experiment can be found in Supplementary Fig. 3. g, The prediction error versus model complexity for different SR methods on the EMPS dataset, averaged over 20 independent runs. The circle size indicates MSE uncertainty (interquartile range). h, The transformed (or collapsed) Nikuradse’s dataset and discovered equations of turbulent friction by different SR methods. The legend shows the median reward models (equation (6)), among 20 trials, obtained by PSE (red), PySR (green)37, NGGP (blue)32, DGSR (sky blue)22, uDSR (orange)38, TPSR (brown)30, BMS (purple)9 and Operon (pink)39. i, The fitting performance of each SR method on the original Nikuradse’s data. j, The predicted MSE versus model complexity for different SR methods, averaged over 20 independent runs. The circle size indicates MSE uncertainty (interquartile range). Notably, our PSE method excels at fitting turbulent friction data rapidly (for example, within only 1.5 minutes), meanwhile discovering a more parsimonious and accurate equation.
