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In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such…

We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…

We consider dual-primal isogeometric tearing and interconnection (IETI-DP) solvers for multi-patch geometries in Isogeometric Analysis. Recently, the authors have published a convergence analysis for those solvers that is explicit in both…

Federated Learning (FL) enables distributed training on edge devices but faces significant challenges due to resource constraints in edge environments, impacting both communication and computational efficiency. Existing iterative pruning…

It is well known that Newton's method can have trouble converging if the initial guess is too far from the solution. Such a problem particularly occurs when this method is used to solve nonlinear elliptic partial differential equations…

This work addresses the key challenges of applying federated learning to large-scale deep neural networks, particularly the issue of client drift due to data heterogeneity across clients and the high costs of communication, computation, and…

Highly heterogeneous, anisotropic coefficients, e.g. in the simulation of carbon-fibre composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer…

Compared with traditional deep learning techniques, continual learning enables deep neural networks to learn continually and adaptively. Deep neural networks have to learn new tasks and overcome forgetting the knowledge obtained from the…

Recent years have seen the emergence of nonlinear methods for solving partial differential equations (PDEs), such as physics-informed neural networks (PINNs). While these approaches often perform well in practice, their theoretical analysis…

Recent advancements in machine learning (ML), natural language processing (NLP), and foundational models have shown promise for real-life applications in critical, albeit compute-constrainted fields like healthcare. In such areas, combining…

Federated Learning (FL) has emerged as a new paradigm for training machine learning models distributively without sacrificing data security and privacy. Learning models on edge devices such as mobile phones is one of the most common use…

Hyperparameter selection is critical for stable and efficient convergence of heterogeneous federated learning, where clients differ in computational capabilities, and data distributions are non-IID. Tuning hyperparameters is a manual and…

As FMs drive progress toward Artificial General Intelligence (AGI), fine-tuning them under privacy and resource constraints has become increasingly critical particularly when highquality training data resides on distributed edge devices.…

Accurately solving PDEs with localised features requires refined meshes that adapt to the solution. Traditional numerical methods, such as finite elements, are linear in nature and often ineffective for such problems, as the mesh is not…

A multilevel adaptive refinement strategy for solving linear elliptic partial differential equations with random data is recalled in this work. The strategy extends the a posteriori error estimation framework introduced by Guignard and…

The demand for edge AI in vision-language tasks requires models that achieve real-time performance on resource-constrained devices with limited power and memory. This paper proposes two adaptive compression techniques -- Sparse Temporal…

Isogeometric Analysis (IgA) is a versatile method for the discretization of partial differential equations on complex domains, which arise in various applications of science and engineering. Some complex geometries can be better described…

Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…

Scientific machine learning is an emerging field that broadly describes the combination of scientific computing and machine learning to address challenges in science and engineering. Within the context of differential equations, this has…

Sparse signal recovery problems from noisy linear measurements appear in many areas of wireless communications. In recent years, deep learning (DL) based approaches have attracted interests of researchers to solve the sparse linear inverse…