Large Language Models (LLMs) have emerged as transformative tools in natural language processing, demonstrating remarkable capabilities in tasks ranging from text generation and translation to question answering and code completion. At the heart of these models lies a sophisticated mechanism for representing text: high-dimensional vector embeddings. These embeddings map words, phrases, and even entire documents into a continuous semantic space, where geometric relationships reflect semantic similarities. For instance, words with related meanings are positioned closer together, while dissimilar concepts are further apart. While these embedding spaces are often treated as continuous for practical purposes, a fundamental aspect of LLMs hints at an underlying discreteness: their reliance on a finite vocabulary of tokens. This discrete foundation suggests that the seemingly continuous semantic space might, in fact, possess a quantized structure, analogous to the discrete energy levels observed in quantum systems. This inherent quantization prompts a compelling question: can we leverage the powerful theoretical frameworks of mathematical physics and tools of quantum mechanics to gain a deeper understanding of the organization and dynamics of these semantic spaces? Furthermore, if this quantization is valid, could quantum computing, for example, offer new approaches to training or exploiting these models, potentially unlocking significant performance gains?

In this work, we explore advanced machine learning techniques for minimizing Gibbs free energy in full 3D micromagnetic simulations. Building on Brown's bounds for magnetostatic self-energy, we revisit their application in the context of variational formulations of the transmission problems for the scalar and vector potential. To overcome the computational challenges posed by whole-space integrals, we reformulate these bounds on a finite domain, making the method more efficient and scalable for numerical simulation. Our approach utilizes an alternating optimization scheme for joint minimization of Brown's energy bounds and the Gibbs free energy. The Cayley transform is employed to rigorously enforce the unit norm constraint, while R-functions are used to impose essential boundary conditions in the computation of magnetostatic fields. Our results highlight the potential of mesh-free Physics-Informed Neural Networks (PINNs) and Extreme Learning Machines (ELMs) when integrated with hard constraints, providing highly accurate approximations. These methods exhibit competitive performance compared to traditional numerical approaches, showing significant promise in computing magnetostatic fields and the application for energy minimization, such as the computation of hysteresis curves. This work opens the path for future directions of research on more complex geometries, such as grain structure models, and the application to large scale problem settings which are intractable with traditional numerical methods.

The modeling of multi-scale and multi-physics complex systems typically involves the use of scientific software that can optimally leverage extreme scale computing. Despite major developments in recent years, these simulations continue to be computationally intensive and time consuming. Here we explore the use of AI to accelerate the modeling of complex systems at a fraction of the computational cost of classical methods, and present the first application of physics informed neural operators to model 2D incompressible magnetohydrodynamics simulations. Our AI models incorporate tensor Fourier neural operators as their backbone, which we implemented with the TensorLY package. Our results indicate that physics informed neural operators can accurately capture the physics of magnetohydrodynamics simulations that describe laminar flows with Reynolds numbers $Re\leq250$. We also explore the applicability of our AI surrogates for turbulent flows, and discuss a variety of methodologies that may be incorporated in future work to create AI models that provide a computationally efficient and high fidelity description of magnetohydrodynamics simulations for a broad range of Reynolds numbers. The scientific software developed in this project is released with this manuscript.

Fast and stable fluid simulations are an essential prerequisite for applications ranging from computer aided aerodynamic design of automobiles or airplanes to simulations of physical effects in CGI to research in meteorology. Recent differentiable fluid simulations allow gradient based methods to optimize e.g. fluid control systems in an informed manner. Solving the partial differential equations governed by the dynamics of the underlying physical systems, however, is a challenging task and current numerical approximation schemes still come at high computational costs. In this work, we propose an unsupervised framework that allows powerful deep neural networks to learn the dynamics of incompressible fluids end to end on a grid-based representation. For this purpose, we introduce a loss function that penalizes residuals of the incompressible Navier Stokes equations. After training, the framework yields models that are capable of fast and differentiable fluid simulations and can handle various fluid phenomena such as the Magnus effect and K\'arm\'an vortex streets. Besides demonstrating its real-time capability on a GPU, we exploit our approach in a control optimization scenario.