21,689 research outputs found
Graph Element Networks: adaptive, structured computation and memory
We explore the use of graph neural networks (GNNs) to model spatial processes
in which there is no a priori graphical structure. Similar to finite element
analysis, we assign nodes of a GNN to spatial locations and use a computational
process defined on the graph to model the relationship between an initial
function defined over a space and a resulting function in the same space. We
use GNNs as a computational substrate, and show that the locations of the nodes
in space as well as their connectivity can be optimized to focus on the most
complex parts of the space. Moreover, this representational strategy allows the
learned input-output relationship to generalize over the size of the underlying
space and run the same model at different levels of precision, trading
computation for accuracy. We demonstrate this method on a traditional PDE
problem, a physical prediction problem from robotics, and learning to predict
scene images from novel viewpoints.Comment: Accepted to ICML 201
Data-driven discovery of coordinates and governing equations
The discovery of governing equations from scientific data has the potential
to transform data-rich fields that lack well-characterized quantitative
descriptions. Advances in sparse regression are currently enabling the
tractable identification of both the structure and parameters of a nonlinear
dynamical system from data. The resulting models have the fewest terms
necessary to describe the dynamics, balancing model complexity with descriptive
ability, and thus promoting interpretability and generalizability. This
provides an algorithmic approach to Occam's razor for model discovery. However,
this approach fundamentally relies on an effective coordinate system in which
the dynamics have a simple representation. In this work, we design a custom
autoencoder to discover a coordinate transformation into a reduced space where
the dynamics may be sparsely represented. Thus, we simultaneously learn the
governing equations and the associated coordinate system. We demonstrate this
approach on several example high-dimensional dynamical systems with
low-dimensional behavior. The resulting modeling framework combines the
strengths of deep neural networks for flexible representation and sparse
identification of nonlinear dynamics (SINDy) for parsimonious models. It is the
first method of its kind to place the discovery of coordinates and models on an
equal footing.Comment: 25 pages, 6 figures; added acknowledgment
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