Wil H. A. Schilders

Reduced rank extrapolation (RRE) is an acceleration method typically used to accelerate the iterative solution of nonlinear systems of equations using a fixed-point process. In this context, the iterates are vectors generated from a fixed-point mapping function. However, when considering the iterative solution of large-scale matrix equations, the iterates are low-rank matrices generated from a fixed-point process for which, generally, the mapping function changes in each iteration. To enable acceleration of the iterative solution for these problems, we propose two novel generalizations of RRE. First, we show how to effectively compute RRE for sequences of low-rank matrices. Second, we derive a formulation of RRE that is suitable for fixed-point processes for which the mapping function changes each iteration. We demonstrate the potential of the methods on several numerical examples involving the iterative solution of large-scale algebraic Lyapunov and Riccati matrix equations.

Scientific Machine Learning (SciML) is an emerging interdisciplinary field that integrates the strengths of scientific computing and machine learning to address complex modeling, simulation, and inference tasks in the natural and engineering sciences. This paper provides a concise overview of the foundational paradigms underlying SciML, where we highlight the complementary roles of physics-based models and data-driven methods. We discuss the core challenges, including data scarcity, physical consistency, high dimensionality, and computational cost and then introduce key methodological advances such as physics-informed neural networks, operator learning, hybrid modeling, and probabilistic approaches, each designed to address specific limitations of traditional methods. The techniques are illustrated through applications in fluid dynamics and turbulence modeling, automation and control, energy management, and sustainable mobility, demonstrating how SciML enables more accurate, efficient, and interpretable solutions. Finally, we outline open problems and future directions, and emphasize the need for theoretical understanding, scalable algorithms, and interdisciplinary collaboration.

Discovering a suitable neural network architecture for modeling complex dynamical systems poses a formidable challenge, often involving extensive trial and error and navigation through a high-dimensional hyper-parameter space. In this paper, we discuss a systematic approach to constructing neural architectures for modeling a subclass of dynamical systems, namely, Linear Time-Invariant (LTI) systems. We use a variant of continuous-time neural networks in which the output of each neuron evolves continuously as a solution of a first-order or second-order Ordinary Differential Equation (ODE). Instead of deriving the network architecture and parameters from data, we propose a gradient-free algorithm to compute sparse architecture and network parameters directly from the given LTI system, leveraging its properties. We bring forth a novel neural architecture paradigm featuring horizontal hidden layers and provide insights into why employing conventional neural architectures with vertical hidden layers may not be favorable. We also provide an upper bound on the numerical errors of our neural networks. Finally, we demonstrate the high accuracy of our constructed networks on three numerical examples.