Stochastic processes as surrogate models for dynamical systems in magnetic confinement fusion

Röck, Katharina (2024): Stochastic processes as surrogate models for dynamical systems in magnetic confinement fusion. Dissertation, LMU München: Fakultät für Mathematik, Informatik und Statistik

Abstract

When designing machine learning (ML) models for scientific applications, a key point is to incorporate a priori domain-specific information in the model. Especially, when constructing reduced complexity models as surrogates, we need to ensure that the mathematical and physical properties of the underlying system are reflected correctly by the ML model. The first part of this thesis focuses on physics-consistent Gaussian processes (GPs) that respect laws of physics by design. This stands in contrast to so-called physics-informed regressors that incorporate physical constraints weakly through the loss function. In scenarios where data originate from underlying linear partial differential equations (PDEs) with localized sources, the proposed model is a superposition of a Gaussian process with a specialized kernel that is constructed to exactly fulfill the homogeneous part of the PDE while a linear model is used for sources. The specialized kernel ensures an exact correspondence and physical interpretability of hyperparameters allowing insights into the underlying physical characteristics. Physics-consistent GPs are then extended to model mappings in the phase space of Hamiltonian systems. Here, we propose a surrogate model based on multi-output GPs deploying derivative information with a matrix-valued covariance function to fully preserve the symplecticity of the Hamiltonian flow and thus conserve integrals of motion. The proposed method is related to geometric integration methods, but models the flow map with larger time steps, accelerating long-term computations. In chaotic systems, the symplectic surrogate model can not only be used for faster computations but also for early classification of chaotic versus regular trajectories, based on the calculation of Lyapunov exponents directly available from the surrogate model. One particular challenge in applying ML models to problems in plasma physics is the lack of labeled data for training larger models. Usually, physical experiments are extremely expensive and with regard to future fusion reactors, sufficient data will not be available until operations start. The second part of this thesis treats data augmentation via robust surrogate models of multivariate time series data to mitigate this problem. We apply Student-$t$ process regression in a state space formulation to ensure reliable uncertainty estimates despite outliers. This reduces computational complexity and allows us to use the model for high-resolution time series. We are using different approaches in this regard. One approach assumes uncorrelated input signals and induces correlations and cross-correlations via coloring transformations in a post-processing step. Another technique immediately incorporates correlations by using a multivariate Matérn kernel. Both approaches are found to be well-suited for data imputation and augmentation for multichannel time series sensor data with outliers.

• Dokumententyp: Dissertationen (Dissertation, LMU München)
• Themengebiete: 500 Naturwissenschaften und Mathematik ; 500 Naturwissenschaften und Mathematik > 510 Mathematik
• Fakultäten: Fakultät für Mathematik, Informatik und Statistik
• Sprache der Hochschulschrift: Englisch
• Datum der mündlichen Prüfung: 8. Mai 2024
• 1. Bericht­erstatter:in: Bischl, Bernd