Deep learning for real-time inverse problems and data assimilation with uncertainty quantification for digital twins

We develop novel approaches to enhance the functionality and efficiency of digital twins through deep learning techniques. Digital twins, sophisticated virtual models of physical systems, serve as dynamic counterparts, mirroring real-world entities' behavior and performance. Their primary role includes real-time monitoring, flaw detection, automated control, and proactive maintenance. To perform such tasks it is necessary to use various methods from scientific computing to combine data with physics models. Furthermore, to assess the reliability of a digital twin, it is important to not only make predictions, but also quantify the uncertainty of estimations and predictions. In this regard, Bayesian methods serve as a natural framework to pose the problems. However, the complexity and real-time operation requirements pose significant computational challenges. The thesis explores the integration of deep learning in scientific computing for enabling digital twins. The focus is on overcoming the computational hurdles in real-time data assimilation and inverse problem-solving through surrogate modeling. Deep learning, with its capability to handle high-dimensional functions in both deterministic and stochastic settings is shown to be a viable solution to these challenges. This work investigates several key areas, beginning with a brief introduction to digital twins, data assimilation, and inverse problems. It proceeds to elaborate on the role of Bayesian inversion problems in static and dynamic contexts, highlighting the importance of uncertainty quantification. The focus is on the challenges of real-time computation and the potential of Bayesian methods, despite their computational intensity. The thesis investigates deep learning architectures like dense and residual neural networks, convolutional neural networks, recurrent neural networks and transformers. The core contributions are presented in the four main chapters of the thesis:

1) Reduced Order Modeling for Parameterized Time-Dependent Partial Differential Equations: Introducing a deep learning framework for solving parameterized PDEs using spatially and memory-aware neural networks, demonstrated on linear advection and incompressible Navier-Stokes equations.

2) Markov Chain Generative Adversarial Neural Networks: Enhancing Bayesian inversion for static problems with sparse observations using a unique algorithm combining MCMC and GANs, demonstrated on a Darcy flow problem and pipe flow leakage detection.

3) Probabilistic Digital Twin for Leak Localization: Developing a probabilistic framework using supervised Wasserstein Autoencoders for leak localization in water distribution networks, tested on several network models.

4) The Deep Latent Space Particle Filter: Proposing a real-time nonlinear data assimilation method for dynamic PDEs with uncertainty quantification, using transformer-based dimensionality reduction and time-stepping in particle filters, demonstrated on various dynamic systems. This thesis contributes significantly to digital twins, addressing computational limitations and opening new avenues for practical applications in various industries