Discovering governing equations from data: Sparse identification of nonlinear dynamical systems

摘要

The ability to discover physical laws and governing equations from data isone of humankind's greatest intellectual achievements. A quantitativeunderstanding of dynamic constraints and balances in nature has facilitatedrapid development of knowledge and enabled advanced technological achievements,including aircraft, combustion engines, satellites, and electrical power. Inthis work, we combine sparsity-promoting techniques and machine learning withnonlinear dynamical systems to discover governing physical equations frommeasurement data. The only assumption about the structure of the model is thatthere are only a few important terms that govern the dynamics, so that theequations are sparse in the space of possible functions; this assumption holdsfor many physical systems. In particular, we use sparse regression to determinethe fewest terms in the dynamic governing equations required to accuratelyrepresent the data. The resulting models are parsimonious, balancing modelcomplexity with descriptive ability while avoiding overfitting. We demonstratethe algorithm on a wide range of problems, from simple canonical systems,including linear and nonlinear oscillators and the chaotic Lorenz system, tothe fluid vortex shedding behind an obstacle. The fluid example illustrates theability of this method to discover the underlying dynamics of a system thattook experts in the community nearly 30 years to resolve. We also show thatthis method generalizes to parameterized, time-varying, or externally forcedsystems.