We introduce Hidden Physics Models, which are essentially data-efficient learning machines capable of leveraging the underlying laws of physics, expressed by time dependent and nonlinear partial differential equations, to extract patterns from high-dimensional data generated from experiments. The proposed methodology may be applied to the problem of learning, system identification, or data-driven discovery of partial differential equations. Our framework relies on Gaussian Processes, a powerful tool for probabilistic inference over functions, that enables us to strike a balance between model complexity and data fitting. The effectiveness of the proposed approach is demonstrated through a variety of canonical problems, spanning a number of scientific domains, including the Navier-Stokes, Schrödinger, Kuramoto-Sivashinsky, and time dependent linear fractional equations. The methodology provides a promising new direction for harnessing the long-standing developments of classical methods in applied mathematics and mathematical physics to design learning machines with the ability to operate in complex domains without requiring large quantities of data.

For more details, please refer to the following: (https://maziarraissi.github.io/HPM/)

• Raissi, Maziar, and George Em Karniadakis. "Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations." arXiv preprint arXiv:1708.00588 (2017).

• Raissi, Maziar, and George Em Karniadakis. "Hidden physics models: Machine learning of nonlinear partial differential equations." Journal of Computational Physics 357 (2018): 125-141.

@article{raissi2017hidden,
  title={Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations},
  author={Raissi, Maziar and Karniadakis, George Em},
  journal={arXiv preprint arXiv:1708.00588},
  year={2017}
}

@article{raissi2017hidden,
  title={Hidden physics models: Machine learning of nonlinear partial differential equations},
  author={Raissi, Maziar and Karniadakis, George Em},
  journal={Journal of Computational Physics},
  year={2017},
  publisher={Elsevier}
}