Learning signals defined on graphs with optimal transport and Gaussian process regression
This addresses uncertainty quantification for computational physics applications where outputs are mesh-based signals, though it appears incremental as it combines existing techniques like optimal transport and Gaussian processes.
The paper tackles the problem of extending Gaussian process regression to predict signals defined on graph nodes when inputs are large sparse graphs with varying sizes and adjacency structures, achieving efficient performance in fluid dynamics and solid mechanics applications.
In computational physics, machine learning has now emerged as a powerful complementary tool to explore efficiently candidate designs in engineering studies. Outputs in such supervised problems are signals defined on meshes, and a natural question is the extension of general scalar output regression models to such complex outputs. Changes between input geometries in terms of both size and adjacency structure in particular make this transition non-trivial. In this work, we propose an innovative strategy for Gaussian process regression where inputs are large and sparse graphs with continuous node attributes and outputs are signals defined on the nodes of the associated inputs. The methodology relies on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. In addition to enabling signal prediction, the main point of our proposal is to come with confidence intervals on node values, which is crucial for uncertainty quantification and active learning. Numerical experiments highlight the efficiency of the method to solve real problems in fluid dynamics and solid mechanics.