In this paper we analyze the behaviour of the stochastic gradient descent (SGD), a widely used method in supervised learning for optimizing neural network weights via a minimization of non-convex loss functions. Since the pioneering work of E, Li and Tai (2017), the underlying structure of such processes can be understood via parabolic PDEs of Fokker-Planck type, which are at the core of our analysis. Even if Fokker-Planck equations have a long history and a extensive literature, almost nothing is known when the potential is non-convex or when the diffusion matrix is degenerate, and this is the main difficulty that we face in our analysis. We identify two different regimes: in the initial phase of SGD, the loss function drives the weights to concentrate around the nearest local minimum. We refer to this phase as the drift regime and we provide quantitative estimates on this concentration phenomenon. Next, we introduce the diffusion regime, where stochastic fluctuations help the learning process to escape suboptimal local minima. We analyze the Mean Exit Time (MET) and prove upper and lower bounds of the MET. Finally, we address the asymptotic convergence of SGD, for a non-convex cost function and a degenerate diffusion matrix, that do not allow to use the standard approaches, and require new techniques. For this purpose, we exploit two different methods: duality and entropy methods. We provide new results about the dynamics and effectiveness of SGD, offering a deep connection between stochastic optimization and PDE theory, and some answers and insights to basic questions in the Machine Learning processes: How long does SGD take to escape from a bad minimum? Do neural network parameters converge using SGD? How do parameters evolve in the first stage of training with SGD?

Deep operator network (DeepONet) has shown great promise as a surrogate model for systems governed by partial differential equations (PDEs), learning mappings between infinite-dimensional function spaces with high accuracy. However, achieving low generalization errors often requires highly overparameterized networks, posing significant challenges for large-scale, complex systems. To address these challenges, latent DeepONet was proposed, introducing a two-step approach: first, a reduced-order model is used to learn a low-dimensional latent space, followed by operator learning on this latent space. While effective, this method is inherently data-driven, relying on large datasets and making it difficult to incorporate governing physics into the framework. Additionally, the decoupled nature of these steps prevents end-to-end optimization and the ability to handle data scarcity. This work introduces PI-Latent-NO, a physics-informed latent operator learning framework that overcomes these limitations. Our architecture employs two coupled DeepONets in an end-to-end training scheme: the first, termed Latent-DeepONet, identifies and learns the low-dimensional latent space, while the second, Reconstruction-DeepONet, maps the latent representations back to the original physical space. By integrating governing physics directly into the training process, our approach requires significantly fewer data samples while achieving high accuracy. Furthermore, the framework is computationally and memory efficient, exhibiting nearly constant scaling behavior on a single GPU and demonstrating the potential for further efficiency gains with distributed training. We validate the proposed method on high-dimensional parametric PDEs, demonstrating its effectiveness as a proof of concept and its potential scalability for large-scale systems.

The signature kernel is a positive definite kernel for sequential and temporal data that has become increasingly popular in machine learning applications due to powerful theoretical guarantees, strong empirical performance, and recently introduced various scalable variations. In this chapter, we give a short introduction to $\texttt{KSig}$, a $\texttt{Scikit-Learn}$ compatible Python package that implements various GPU-accelerated algorithms for computing signature kernels, and performing downstream learning tasks. We also introduce a new algorithm based on tensor sketches which gives strong performance compared to existing algorithms. The package is available at https://github.com/tgcsaba/ksig.

Multivariate Time Series Classification (MTSC) enables the analysis if complex temporal data, and thus serves as a cornerstone in various real-world applications, ranging from healthcare to finance. Since the relationship among variables in MTS usually contain crucial cues, a large number of graph-based MTSC approaches have been proposed, as the graph topology and edges can explicitly represent relationships among variables (channels), where not only various MTS graph representation learning strategies but also different Graph Neural Networks (GNNs) have been explored. Despite such progresses, there is no comprehensive study that fairly benchmarks and investigates the performances of existing widely-used graph representation learning strategies/GNN classifiers in the application of different MTSC tasks. In this paper, we present the first benchmark which systematically investigates the effectiveness of the widely-used three node feature definition strategies, four edge feature learning strategies and five GNN architecture, resulting in 60 different variants for graph-based MTSC. These variants are developed and evaluated with a standardized data pipeline and training/validation/testing strategy on 26 widely-used suspensor MTSC datasets. Our experiments highlight that node features significantly influence MTSC performance, while the visualization of edge features illustrates why adaptive edge learning outperforms other edge feature learning methods. The code of the proposed benchmark is publicly available at \url{https://github.com/CVI-yangwn/Benchmark-GNN-for-Multivariate-Time-Series-Classification}.

Venezuelan banks have historically made credit card limit adjustment decisions manually through committees. However, since the number of credit card holders in Venezuela is expected to increase in the upcoming months due to economic improvements, manual decisions are starting to become unfeasible. In this project, a machine learning model that uses cost-sensitive learning is proposed to automate the task of handing out credit card limit increases. To accomplish this, several neural network and XGBoost models are trained and compared, leveraging Venezolano de Credito's data and using grid search with 10-fold cross-validation. The proposed model is ultimately chosen due to its superior balance of accuracy, cost-effectiveness, and interpretability. The model's performance is evaluated against the committee's decisions using Cohen's kappa coefficient, showing an almost perfect agreement.

Whenever data-based systems are employed in engineering applications, defining an optimal statistical representation is subject to the problem of model selection. This paper focusses on how well models can generalise in Structural Health Monitoring (SHM). Although statistical model validation in this field is often performed heuristically, it is possible to estimate generalisation more rigorously using the bounds provided by Statistical Learning Theory (SLT). Therefore, this paper explores the selection process of a kernel smoother for modelling the impulse response of a linear oscillator from the perspective of SLT. It is demonstrated that incorporating domain knowledge into the regression problem yields a lower guaranteed risk, thereby enhancing generalisation.

We introduce a new representation for 3D molecules based on their continuous atomic density fields. Using this representation, we propose a new model based on walk-jump sampling for unconditional 3D molecule generation in the continuous space using neural fields. Our model, FuncMol, encodes molecular fields into latent codes using a conditional neural field, samples noisy codes from a Gaussian-smoothed distribution with Langevin MCMC (walk), denoises these samples in a single step (jump), and finally decodes them into molecular fields. FuncMol performs all-atom generation of 3D molecules without assumptions on the molecular structure and scales well with the size of molecules, unlike most approaches. Our method achieves competitive results on drug-like molecules and easily scales to macro-cyclic peptides, with at least one order of magnitude faster sampling. The code is available at https://github.com/prescient-design/funcmol.

The COVID-19 pandemic has profoundly impacted billions globally. It challenges public health and healthcare systems due to its rapid spread and severe respiratory effects. An effective strategy to mitigate the COVID-19 pandemic involves integrating testing to identify infected individuals. While RT-PCR is considered the gold standard for diagnosing COVID-19, it has some limitations such as the risk of false negatives. To address this problem, this paper introduces a novel Deep Learning Diagnosis System that integrates pre-trained Deep Convolutional Neural Networks (DCNNs) within an ensemble learning framework to achieve precise identification of COVID-19 cases from Chest X-ray (CXR) images. We combine feature vectors from the final hidden layers of pre-trained DCNNs using the Choquet integral to capture interactions between different DCNNs that a linear approach cannot. We employed Sugeno-$\lambda$ measure theory to derive fuzzy measures for subsets of networks to enable aggregation. We utilized Differential Evolution to estimate fuzzy densities. We developed a TensorFlow-based layer for Choquet operation to facilitate efficient aggregation, due to the intricacies involved in aggregating feature vectors. Experimental results on the COVIDx dataset show that our ensemble model achieved 98\% accuracy in three-class classification and 99.50\% in binary classification, outperforming its components-DenseNet-201 (97\% for three-class, 98.75\% for binary), Inception-v3 (96.25\% for three-class, 98.50\% for binary), and Xception (94.50\% for three-class, 98\% for binary)-and surpassing many previous methods.

Aviation safety is a global concern, requiring detailed investigations into incidents to understand contributing factors comprehensively. This study uses the National Transportation Safety Board (NTSB) dataset. It applies advanced natural language processing (NLP) techniques, including Latent Dirichlet Allocation (LDA), Non-Negative Matrix Factorization (NMF), Latent Semantic Analysis (LSA), Probabilistic Latent Semantic Analysis (pLSA), and K-means clustering. The main objectives are identifying latent themes, exploring semantic relationships, assessing probabilistic connections, and cluster incidents based on shared characteristics. This research contributes to aviation safety by providing insights into incident narratives and demonstrating the versatility of NLP and topic modelling techniques in extracting valuable information from complex datasets. The results, including topics identified from various techniques, provide an understanding of recurring themes. Comparative analysis reveals that LDA performed best with a coherence value of 0.597, pLSA of 0.583, LSA of 0.542, and NMF of 0.437. K-means clustering further reveals commonalities and unique insights into incident narratives. In conclusion, this study uncovers latent patterns and thematic structures within incident narratives, offering a comparative analysis of multiple-topic modelling techniques. Future research avenues include exploring temporal patterns, incorporating additional datasets, and developing predictive models for early identification of safety issues. This research lays the groundwork for enhancing the understanding and improvement of aviation safety by utilising the wealth of information embedded in incident narratives.

Drug combination therapies have shown promising therapeutic efficacy in complex diseases and have demonstrated the potential to reduce drug resistance. However, the huge number of possible drug combinations makes it difficult to screen them all in traditional experiments. In this study, we proposed MD-Syn, a computational framework, which is based on the multidimensional feature fusion method and multi-head attention mechanisms. Given drug pair-cell line triplets, MD-Syn considers one-dimensional and two-dimensional feature spaces simultaneously. It consists of a one-dimensional feature embedding module (1D-FEM), a two-dimensional feature embedding module (2D-FEM), and a deep neural network-based classifier for synergistic drug combination prediction. MD-Syn achieved the AUROC of 0.919 in 5-fold cross-validation, outperforming the state-of-the-art methods. Further, MD-Syn showed comparable results over two independent datasets. In addition, the multi-head attention mechanisms not only learn embeddings from different feature aspects but also focus on essential interactive feature elements, improving the interpretability of MD-Syn. In summary, MD-Syn is an interpretable framework to prioritize synergistic drug combination pairs with chemicals and cancer cell line gene expression profiles. To facilitate broader community access to this model, we have developed a web portal (https://labyeh104-2.life.nthu.edu.tw/) that enables customized predictions of drug combination synergy effects based on user-specified compounds.