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Harnessing data to discover the underlying governing laws or equations thatdescribe the behavior of complex physical systems can significantly advance ourmodeling, simulation and understanding of such systems in various science andengineering disciplines. This work introduces a novel physics-informed deeplearning framework to discover governing partial differential equations (PDEs)from scarce and noisy data for nonlinear spatiotemporal systems. In particular,this approach seamlessly integrates the strengths of deep neural networks forrich representation learning, physics embedding, automatic differentiation andsparse regression to (1) approximate the solution of system variables, (2)compute essential derivatives, as well as (3) identify the key derivative termsand parameters that form the structure and explicit expression of the PDEs. Theefficacy and robustness of this method are demonstrated, both numerically andexperimentally, on discovering a variety of PDE systems with different levelsof data scarcity and noise accounting for different initial/boundaryconditions. The resulting computational framework shows the potential forclosed-form model discovery in practical applications where large and accuratedatasets are intractable to capture.

Physics-informed learning of governing equations from scarce data