AI Driven Near Real-time Locational Marginal Pricing Method: A Feasibility and Robustness Study

Accurate price predictions are essential for market participants in order to optimize their operational schedules and bidding strategies, especially in the current context where electricity prices become more volatile and less predictable using classical approaches. Locational Marginal Pricing (LMP) pricing mechanism is used in many modern power markets, where the traditional approach utilizes optimal power flow (OPF) solvers. However, for large electricity grids this process becomes prohibitively time-consuming and computationally intensive. Machine learning solutions could provide an efficient tool for LMP prediction, especially in energy markets with intermittent sources like renewable energy. The study evaluates the performance of popular machine learning and deep learning models in predicting LMP on multiple electricity grids. The accuracy and robustness of these models in predicting LMP is assessed considering multiple scenarios. The results show that machine learning models can predict LMP 4-5 orders of magnitude faster than traditional OPF solvers with 5-6\% error rate, highlighting the potential of machine learning models in LMP prediction for large-scale power models with the help of hardware solutions like multi-core CPUs and GPUs in modern HPC clusters.

Evolving Generalizable Multigrid-Based Helmholtz Preconditioners with Grammar-Guided Genetic Programming

Solving the indefinite Helmholtz equation is not only crucial for the understanding of many physical phenomena but also represents an outstandingly-difficult benchmark problem for the successful application of numerical methods. Here we introduce a new approach for evolving efficient preconditioned iterative solvers for Helmholtz problems with multi-objective grammar-guided genetic programming. Our approach is based on a novel context-free grammar, which enables the construction of multigrid preconditioners that employ a tailored sequence of operations on each discretization level. To find solvers that generalize well over the given domain, we propose a custom method of successive problem difficulty adaption, in which we evaluate a preconditioner's efficiency on increasingly ill-conditioned problem instances. We demonstrate our approach's effectiveness by evolving multigrid-based preconditioners for a two-dimensional indefinite Helmholtz problem that outperform several human-designed methods for different wavenumbers up to systems of linear equations with more than a million unknowns.

Known Operator Learning and Hybrid Machine Learning in Medical Imaging --- A Review of the Past, the Present, and the Future

In this article, we perform a review of the state-of-the-art of hybrid machine learning in medical imaging. We start with a short summary of the general developments of the past in machine learning and how general and specialized approaches have been in competition in the past decades. A particular focus will be the theoretical and experimental evidence pro and contra hybrid modelling. Next, we inspect several new developments regarding hybrid machine learning with a particular focus on so-called known operator learning and how hybrid approaches gain more and more momentum across essentially all applications in medical imaging and medical image analysis. As we will point out by numerous examples, hybrid models are taking over in image reconstruction and analysis. Even domains such as physical simulation and scanner and acquisition design are being addressed using machine learning grey box modelling approaches. Towards the end of the article, we will investigate a few future directions and point out relevant areas in which hybrid modelling, meta learning, and other domains will likely be able to drive the state-of-the-art ahead.

Optimizing Geometric Multigrid Methods with Evolutionary Computation

For many linear and nonlinear systems that arise from the discretization of partial differential equations the construction of an efficient multigrid solver is a challenging task. Here we present a novel approach for the optimization of geometric multigrid methods that is based on evolutionary computation, a generic program optimization technique inspired by the principle of natural evolution. A multigrid solver is represented as a tree of mathematical expressions which we generate based on a tailored grammar. The quality of each solver is evaluated in terms of convergence and compute performance using automated local Fourier analysis (LFA) and roofline performance modeling, respectively. Based on these objectives a multi-objective optimization is performed using strongly typed genetic programming with a non-dominated sorting based selection. To evaluate the model-based prediction and to target concrete applications, scalable implementations of an evolved solver can be automatically generated with the ExaStencils framework. We demonstrate our approach by constructing multigrid solvers for the steady-state heat equation with constant and variable coefficients that consistently perform better than common V- and W-cycles.