Probabilistic and deterministic kernel methods have proven very useful and versatile for a number of classification,
density estimation, and prediction problems arising in science and society. Yet, these methods are often considered as black boxes, and the fantastic expressiveness allowed by the choice of the underlying positive definite kernel is classically underestimated. This double minisymposium gathers researchers from various horizons who have been investigating the incorporation of physical and other structural information in kernel methods in contexts such as Gaussian Process (GP) modelling, adaptive Bayesian integration, space-filling design with minimum energy measures versus maximum mean discrepancy, and probabilistic prediction of probability density fields. In Part I, the emphasis will be put on the incorporation of physical laws and boundary information in GP-related models, with applications in a number of fields encompassing in particular electromagnetism, mechanics, geophysics and biology. In Part II, the focus will be put more specifically on kernels and distances for space-filling design, image-valued GP modelling, high-dimensional integration, and assessing predictions of probability density fields by spatial logistic Gaussian and related models.
10:30
Linearly constrained Gaussian processes
Thomas Schön | Uppsala University | Sweden
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Thomas Schön | Uppsala University | Sweden
In this talk we want to show that the combined use of data-driven modelling and existing scientific knowledge can be
quite rewarding. We briefly illustrate this using concrete examples from physics, including modelling the ambient magnetic field, neutron diffraction experiments aiming to reconstruct the strain field, and computed tomographic (CT) reconstruction. These are all concrete examples where physics provide us with linear operator constraints that needs to be fulfilled (first example) or alternatively measurements constituted by line integrals (two last). The reason for the usefulness of the Gaussian process is that it offers a probabilistic and non-parametric model of nonlinear functions. When these properties are combined with basic existing scientific knowledge of the phenomenon under study we have a useful mathematical tool capable of fusing existing knowledge with new measured data. We will show how the Gaussian process can be adapted so that it obeys linear operator constraints (including ODEs, PDEs and integrals), motivated for example by the specific examples above. Towards the end we will also (very briefly) sketch how the Gaussian process can incorporate deep neural networks to further enhance its flexibility. These developments opens up for the use of basic scientific knowledge within one of our classic machine learning models.
Joint work with Carl Jidling, Niklas Wahlström, Adrian Wills.
11:00
Physically-inspired Gaussian processes with application to biology
Andrés Felipe López-Lopera | Ecole Nationale Supérieure des Mines de Saint-Etienne | France
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Andrés Felipe López-Lopera | Ecole Nationale Supérieure des Mines de Saint-Etienne | France
Physically-inspired Gaussian processes (GPs) provide a flexible stochastic framework where linear differential
equations are encoded into covariance functions (kernels). As data-driven approaches, they can be established without specifying all the physical interactions from mechanistic processes. By enforcing GPs with physical knowledge, accurate predictions are provided even in regions where data are not available. In this talk, we focus on GPs physically inspired by a reaction-diffusion model where both the decay and diffusion rate constants are encoded as parameters of the kernels. Two types of GP-based models are studied where the main difference lies in where the prior is placed. On a biological application describing the post-transcriptional regulation of Drosophila, we demonstrate the capability and versatility of the models to capture the dynamics of spatio-temporal
interactions between mRNAs and gap proteins.
Joint work with Nicolas Durrande, Mauricio Alexander Álvarez
11:30
BdryGP: a new Gaussian process model for incorporating boundary information
Simon Mak | Duke University | United States
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Simon Mak | Duke University | United States
Gaussian processes (GPs) are widely used as surrogate models for emulating computer code, which simulate
complex physical phenomena. In many problems, additional boundary information (i.e., the behavior of the phenomena along input boundaries) is known beforehand, either from governing physics or scientific knowledge. While there has been recent work on incorporating boundary information within GPs, such models do not provide theoretical insights on improved convergence rates.
To this end, we propose a new GP model, called BdryGP, for incorporating boundary information. We show that BdryGP not only has improved convergence rates over existing GP models (which do not incorporate boundaries), but is also more resistant to the ``curse-of-dimensionality'' in nonparametric regression. Our proofs make use of a novel connection between GP interpolation and finite-element modeling.
Joint work with Liang Ding (HKUST) and C. F. Jeff Wu (Georgia Tech)
12:00
End to end GP-based inversion of a mass density field from gravimetric measurements
Cédric Travelletti | Idiap Research Institute and University of Bern | Switzerland
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Cédric Travelletti | Idiap Research Institute and University of Bern | Switzerland
The use of GP priors in Bayesian inverse problems is well established but resulting implementations are seldom discussed. We provide a detailed description of the whole inversion process on a test case from volcano geophysics, demonstrating in turn how to overcome various practical problems.
The goal in our motivating inverse problem is to reconstruct the mass density field inside a bounded region from measurements of the gravitational field on the outside, i.e. on the surface of the Stromboli volcano in our test case. Those measurements may be viewed as linear forms of the field, a neat framework for GP modelling. Also, considering the GP induced on the surface by the prior mass density field and the forward operator turns out to be fruitful in several respects.
We show how the proposed approach enables enjoying MLE for hyper-parameters such as classically used in GP modelling for computer experiments, yet at the cost of numerical instabilities due to the specific form for the covariance matrix. Using chunking, automatic differentiation and GPU-based libraries, we demonstrate how GP-based inversion scales to grids of several hundreds of thousands of cells. We also appeal to fast k-fold cross-validation and explore how associated outputs can help balance numerical instabilities. We finally highlight how this approach can be used to guide new measurements towards an efficient reconstruction of the mass density field.
Joint work with David Ginsbourger and Niklas Linde