Advanced Optimal Transport Strategies for Efficient Computation and Reduced Order Modeling in Complex Systems

The simulation of complex systems has become a pillar of modern scientific discoveries, allowing researchers to access and investigate scenarios that are impossible or impractical to obtain from classical experiments. Commonly, these models are described by partial differential equations (PDEs), that encode the physical knowledge about the problem at hand. However, such simulations often involve unbearable computational costs, dictated by the necessity to use a high number of degrees of freedom and repeated evaluations for many different configurations of the problem. Reduced Order Models (ROMs) have emerged to face the need to drastically reduce the computational resources, trying to capture the essential behaviour of a complex system with a much faster (surrogate) model. However, traditional linear ROMs often struggle with systems characterized by strong nonlinearities, advection-dominated phenomena, or solutions lying on intrinsically nonlinear manifolds, leading to slow decay of the Kolmogorov n-width and consequently poor approximation quality. This thesis explores the integration of the theory of Optimal Transport (OT) within the broader landscape of ROMs to address such limitations. OT provides a powerful geometric tool to analyze, compare, and transform probability distributions. This can be used to better understand and manipulate complex data features coming from numerical simulations. In particular, we exploit the core OT concepts to develop new algorithms and methodologies that aim to strengthen and complement the capabilities of traditional ROMs from different points of view. Firstly, we propose an OT-inspired deep learning framework that allows the construction of a nonlinear ROM, particularly suited for solving problems with slow-decaying Kolmogorov n-width. This framework utilizes the Kernel Proper Orthogonal Decomposition (kPOD) with a new Wasserstein-based kernel (the “Sinkhorn kernel”) to capture the nonlinear relationships within the data. Autoencoder neural networks are then trained to learn the direct and inverse maps of the kPOD, using the Sinkhorn divergence as a robust loss function. This approach proves to be more accurate and effective in terms of reduction when compared to traditional methods on a wide range of PDE problems. Secondly, we develop a new ROM model that leverages interpolation with respect to the Wasserstein distance, exploiting the so-called displacement interpolation. This model can be exploited in several context, e.g.\ to perform data augmentation and continuous time prediction starting from sparse measurements of nonlinear dynamical systems. By generating physically consistent interpolated snapshots along geodesic paths in the Wasserstein space, this method enriches sparse training datasets, improves the expressiveness of the reduced basis, and enables accurate solution reconstruction at arbitrary temporal resolutions. The efficacy of this approach is demonstrated on challenging high-dimensional and multiscale atmospheric flow simulations. Thirdly, we address the efficient computation of semi-discrete OT, a common variant in which a continuous measure is transported to a discrete one, which underpins the practical application of the aforementioned OT-based ROMs. We present a comprehensive set of numerical strategies for solving the dual formulation of entropically regularized semi-discrete OT. These strategies include adaptive interaction truncation using spatial data structures (R-trees), multilevel acceleration schemes for both source and target measures, and regularization parameter scheduling. The synergistic combination of these techniques significantly reduces the computational cost, making large-scale semi-discrete OT problems tractable. Collectively, the contributions presented in this thesis demonstrate the transformative potential of Optimal Transport theory in advancing the field of reduced order modeling. By providing efficient computational tools for OT, innovative data augmentation techniques, and novel OT-infused deep learning architectures, this work paves the way for more accurate, robust, and efficient ROMs for a wide range of complex scientific and engineering applications.