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Data-driven discovery of differential equations has been an emerging researchtopic. We propose a novel algorithm subsampling-based threshold sparse Bayesianregression (SubTSBR) to tackle high noise and outliers. The subsamplingtechnique is used for improving the accuracy of the Bayesian learningalgorithm. It has two parameters: subsampling size and the number ofsubsamples. When the subsampling size increases with fixed total sample size,the accuracy of our algorithm goes up and then down. When the number ofsubsamples increases, the accuracy of our algorithm keeps going up. Wedemonstrate how to use our algorithm step by step and compare our algorithmwith threshold sparse Bayesian regression (TSBR) for the discovery ofdifferential equations. We show that our algorithm produces better results. Wealso discuss the merits of discovering differential equations from data anddemonstrate how to discover models with random initial and boundary conditionas well as models with bifurcations. The numerical examples are: (1)predator-prey model with noise, (2) shallow water equations with outliers, (3)heat diffusion with random initial and boundary condition, and (4)fish-harvesting problem with bifurcations.

SubTSBR to tackle high noise and outliers for data-driven discovery of differential equations