Name
Abstract | ||
|---|---|---|
While there is currently a lot of enthusiasm about “big data”, useful data is usually “small” and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from small data. In particular, 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. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1016/j.jcp.2017.11.039 | Journal of Computational Physics |
Keywords | DocType | Volume |
Probabilistic machine learning,System identification,Bayesian modeling,Uncertainty quantification,Fractional equations,Small data | Journal | 357 |
ISSN | Citations | PageRank |
0021-9991 | 36 | 1.85 |
References | Authors | |
13 | 2 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Maziar Raissi | 1 | 171 | 11.29 |
| George Em Karniadakis | 2 | 1396 | 177.42 |