High-dimensional data analysis and visualization constitute fundamental challenges in machine learning, where nonlinear dimensionality reduction (NLDR) techniques have proven instrumental in discovering low-dimensional embeddings that preserve essential structural properties of complex datasets. These methods, encompassing techniques such as t-distributed Stochastic Neighbor Embedding (t-SNE) [13], Isometric Mapping (Isomap) [12], Locally Linear Embedding (LLE) [10] and Laplacian Eigenmaps [1] excel at revealing intrinsic data manifolds and facilitating interpretable visualizations of high-dimensional phenomena. However, a critical architectural limitation pervades the entire class of traditional NLDR methods: they inherently lack reconstruction capabilities, operating as one-way transformations that map from high-dimensional input spaces to low-dimensional embeddings without providing mechanisms for inverse mapping. This fundamental asymmetry severely constrains the applicability of NLDR techniques in generative modelling, data synthesis, and interactive exploration scenarios where bidirectional transformations are essential. Unlike autoen-coders, which explicitly incorporate decoder architectures during training, classical manifold learning approaches such as t-SNE, Uniform Manifold Approximation and Projection (UMAP) [8], and diffusion maps optimize embeddings through eigen decomposition, neighbourhood preservation, or probabilistic formulations that do not naturally yield invertible mappings. Consequently, despite their superior performance in preserving local neighbourhood structures and global topological properties, these methods remain confined to analysis and visualization tasks. This work addresses the reconstruction gap in NLDR methods by developing specialized decoder architectures that enable bidirectional mapping between high-dimensional data and learned manifold representations.

Solving inverse problems in cardiovascular modeling is particularly challenging due to the high computational cost of running high-fidelity simulations. In this work, we focus on Bayesian parameter estimation and explore different methods to reduce the computational cost of sampling from the posterior distribution by leveraging low-fidelity approximations. A common approach is to construct a surrogate model for the high-fidelity simulation itself. Another is to build a surrogate for the discrepancy between high- and low-fidelity models. This discrepancy, which is often easier to approximate, is modeled with either a fully connected neural network or a nonlinear dimensionality reduction technique that enables surrogate construction in a lower-dimensional space. A third possible approach is to treat the discrepancy between the high-fidelity and surrogate models as random noise and estimate its distribution using normalizing flows. This allows us to incorporate the approximation error into the Bayesian inverse problem by modifying the likelihood function. We validate five different methods which are variations of the above on analytical test cases by comparing them to posterior distributions derived solely from high-fidelity models, assessing both accuracy and computational cost. Finally, we demonstrate our approaches on two cardiovascular examples of increasing complexity: a lumped-parameter Windkessel model and a patient-specific three-dimensional anatomy.

We consider the problem of propagating the uncertainty from a possibly large number of random inputs through a computationally expensive model. Stratified sampling is a well-known variance reduction strategy, but its application, thus far, has focused on models with a limited number of inputs due to the challenges of creating uniform partitions in high dimensions. To overcome these challenges, we perform stratification with respect to the uniform distribution defined over the unit interval, and then derive the corresponding strata in the original space using nonlinear dimensionality reduction. We show that our approach is effective in high dimensions and can be used to further reduce the variance of multifidelity Monte Carlo estimators.

We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the model output variability, plus a simultaneously learnt surrogate model with inputs on this manifold. The proposed dimensionality reduction framework can then be applied to perform outer loop many-query tasks, like sensitivity analysis and uncertainty propagation. In particular, we prove, both theoretically under idealized conditions, and numerically in challenging test cases, how NeurAM can be used to obtain multifidelity sampling estimators with reduced variance by sampling the models on the discovered low-dimensional and shared manifold among models. Several numerical examples illustrate the main features of the proposed dimensionality reduction strategy, and highlight its advantages with respect to existing approaches in the literature.

A valuable step in the modeling of multiscale dynamical systems in fields such as computational chemistry, biology, materials science and more, is the representative sampling of the phase space over long timescales of interest; this task is not, however, without challenges. For example, the long term behavior of a system with many degrees of freedom often cannot be efficiently computationally explored by direct dynamical simulation; such systems can often become trapped in local free energy minima. In the study of physics-based multi-time-scale dynamical systems, techniques have been developed for enhancing sampling in order to accelerate exploration beyond free energy barriers. On the other hand, in the field of Machine Learning, a generic goal of generative models is to sample from a target density, after training on empirical samples from this density. Score based generative models (SGMs) have demonstrated state-of-the-art capabilities in generating plausible data from target training distributions. Conditional implementations of such generative models have been shown to exhibit significant parallels with long-established -- and physics based -- solutions to enhanced sampling. These physics-based methods can then be enhanced through coupling with the ML generative models, complementing the strengths and mitigating the weaknesses of each technique. In this work, we show that that SGMs can be used in such a coupling framework to improve sampling in multiscale dynamical systems.

Recently proposed algorithms for nonlinear dimensionality reduction fall broadly into two categories which have different advantages and disad- vantages: global (Isomap [1]), and local (Locally Linear Embedding [2], Laplacian Eigenmaps [3]). We present two variants of Isomap which combine the advantages of the global approach with what have previ- ously been exclusive advantages of local methods: computational spar- sity and the ability to invert conformal maps.

Nonlinear dimensionality reduction methods are ubiquitously applied for visualization and preprocessing highdimensional data in machine learning [1, 2, 3, 4, 5, 6, 7, 8]. These methods assume that the intrinsic dimension of the underlying manifold is much lower than the ambient dimension of the real-world data [9, 10, 11]. Based on approximating the manifold by k nearest neighbour (kNN) graph, nonlinear dimensionality reduction projects data from high to low-dimensional space and retains the topological structure of original data. While nonlinear dimensionality reduction is effective for visualizing high-dimensional data, one major weakness is lacking interpretability of the reduced-dimension results [8]. The reduced dimensions of nonlinear dimensionality reduction have no specific meaning, compared with linear methods like Principal Component Analysis (PCA) where the dimensions of the embedding space represent the directions of the largest variance of original data. Particularly, nonlinear dimensionality reduction focuses on preserving distance between observations and thereby loses source feature information in the embedding space, resulting in failing to illustrate feature loadings that linear methods such as PCA can provide to explain the feature contribution in each dimension. In this paper, we seek to improve the interpretability of nonlinear dimensionality reduction. In addition to preserving the local topological structure between observations in the embedding space, we aim to incorporate the source features to devise an interpretable nonlinear dimensionality reduction method. The feature information is encoded in the column space of data, and we use the tangent space to locally depict the column space [12, 13].

Artificial intelligence (AI) has become a part of everyday conversation and our lives. It is considered as the new electricity that is revolutionizing the world. AI is heavily invested in both industry and academy. However, there is also a lot of hype in the current AI debate. AI based on so-called deep learning has achieved impressive results in many problems, but its limits are already visible. AI has been under research since the 1940s, and the industry has seen many ups and downs due to over-expectations and related disappointments that have followed. The purpose of this book is to give a realistic picture of AI, its history, its potential and limitations. We believe that AI is a helper, not a ruler of humans. We begin by describing what AI is and how it has evolved over the decades. After fundamentals, we explain the importance of massive data for the current mainstream of artificial intelligence. The most common representations for AI, methods, and machine learning are covered. In addition, the main application areas are introduced. Computer vision has been central to the development of AI. The book provides a general introduction to computer vision, and includes an exposure to the results and applications of our own research. Emotions are central to human intelligence, but little use has been made in AI. We present the basics of emotional intelligence and our own research on the topic. We discuss super-intelligence that transcends human understanding, explaining why such achievement seems impossible on the basis of present knowledge,and how AI could be improved. Finally, a summary is made of the current state of AI and what to do in the future. In the appendix, we look at the development of AI education, especially from the perspective of contents at our own university.

Deep neural networks are vulnerable to adversarial examples, i.e., carefully-perturbed inputs aimed to mislead classification. This work proposes a detection method based on combining non-linear dimensionality reduction and density estimation techniques. Our empirical findings show that the proposed approach is able to effectively detect adversarial examples crafted by non-adaptive attackers, i.e., not specifically tuned to bypass the detection method. Given our promising results, we plan to extend our analysis to adaptive attackers in future work.

We present a novel view of nonlinear manifold learning using derivative-free optimization techniques. Specifically, we propose an extension of the classical multi-dimensional scaling (MDS) method, where instead of performing gradient descent, we sample and evaluate possible "moves" in a sphere of fixed radius for each point in the embedded space. A fixed-point convergence guarantee can be shown by formulating the proposed algorithm as an instance of General Pattern Search (GPS) framework. Evaluation on both clean and noisy synthetic datasets shows that pattern search MDS can accurately infer the intrinsic geometry of manifolds embedded in high-dimensional spaces. Additionally, experiments on real data, even under noisy conditions, demonstrate that the proposed pattern search MDS yields state-of-the-art results.