Christian Donner

Yoeri Poels, Alessandro Pau, Christian Donner et al.

When a plasma disrupts in a tokamak, significant heat and electromagnetic loads are deposited onto the surrounding device components. These forces scale with plasma current and magnetic field strength, making disruptions one of the key challenges for future devices. Unfortunately, disruptions are not fully understood, with many different underlying causes that are difficult to anticipate. Data-driven models have shown success in predicting them, but they only provide limited interpretability. On the other hand, large-scale statistical analyses have been a great asset to understanding disruptive patterns. In this paper, we leverage data-driven methods to find an interpretable representation of the plasma state for disruption characterization. Specifically, we use a latent variable model to represent diagnostic measurements as a low-dimensional, latent representation. We build upon the Variational Autoencoder (VAE) framework, and extend it for (1) continuous projections of plasma trajectories; (2) a multimodal structure to separate operating regimes; and (3) separation with respect to disruptive regimes. Subsequently, we can identify continuous indicators for the disruption rate and the disruptivity based on statistical properties of measurement data. The proposed method is demonstrated using a dataset of approximately 1600 TCV discharges, selecting for flat-top disruptions or regular terminations. We evaluate the method with respect to (1) the identified disruption risk and its correlation with other plasma properties; (2) the ability to distinguish different types of disruptions; and (3) downstream analyses. For the latter, we conduct a demonstrative study on identifying parameters connected to disruptions using counterfactual-like analysis. Overall, the method can adequately identify distinct operating regimes characterized by varying proximity to disruptions in an interpretable manner.

Théo Galy-Fajou, Florian Wenzel, Christian Donner et al.

We propose a new scalable multi-class Gaussian process classification approach building on a novel modified softmax likelihood function. The new likelihood has two benefits: it leads to well-calibrated uncertainty estimates and allows for an efficient latent variable augmentation. The augmented model has the advantage that it is conditionally conjugate leading to a fast variational inference method via block coordinate ascent updates. Previous approaches suffered from a trade-off between uncertainty calibration and speed. Our experiments show that our method leads to well-calibrated uncertainty estimates and competitive predictive performance while being up to two orders faster than the state of the art.

Christian Donner, Manfred Opper

We present an approximate Bayesian inference approach for estimating the intensity of an inhomogeneous Poisson process, where the intensity function is modelled using a Gaussian process (GP) prior via a sigmoid link function. Augmenting the model using a latent marked Poisson process and Pólya--Gamma random variables we obtain a representation of the likelihood which is conjugate to the GP prior. We estimate the posterior using a variational free--form mean field optimisation together with the framework of sparse GPs. Furthermore, as alternative approximation we suggest a sparse Laplace's method for the posterior, for which an efficient expectation--maximisation algorithm is derived to find the posterior's mode. Both algorithms compare well against exact inference obtained by a Markov Chain Monte Carlo sampler and standard variational Gauss approach solving the same model, while being one order of magnitude faster. Furthermore, the performance and speed of our method is competitive with that of another recently proposed Poisson process model based on a quadratic link function, while not being limited to GPs with squared exponential kernels and rectangular domains.

Christian Donner, Manfred Opper

We reconsider a nonparametric density model based on Gaussian processes. By augmenting the model with latent Pólya--Gamma random variables and a latent marked Poisson process we obtain a new likelihood which is conjugate to the model's Gaussian process prior. The augmented posterior allows for efficient inference by Gibbs sampling and an approximate variational mean field approach. For the latter we utilise sparse GP approximations to tackle the infinite dimensionality of the problem. The performance of both algorithms and comparisons with other density estimators are demonstrated on artificial and real datasets with up to several thousand data points.

Christian Donner, Manfred Opper

We consider the inverse Ising problem, i.e. the inference of network couplings from observed spin trajectories for a model with continuous time Glauber dynamics. By introducing two sets of auxiliary latent random variables we render the likelihood into a form, which allows for simple iterative inference algorithms with analytical updates. The variables are: (1) Poisson variables to linearise an exponential term which is typical for point process likelihoods and (2) Pólya-Gamma variables, which make the likelihood quadratic in the coupling parameters. Using the augmented likelihood, we derive an expectation-maximization (EM) algorithm to obtain the maximum likelihood estimate of network parameters. Using a third set of latent variables we extend the EM algorithm to sparse couplings via L1 regularization. Finally, we develop an efficient approximate Bayesian inference algorithm using a variational approach. We demonstrate the performance of our algorithms on data simulated from an Ising model. For data which are simulated from a more biologically plausible network with spiking neurons, we show that the Ising model captures well the low order statistics of the data and how the Ising couplings are related to the underlying synaptic structure of the simulated network.