Alexander Heinlein

Systems of equations arise in a wide range of scientific and engineering applications. The present work focuses on solvers for systems of general equations, including but not limited to those arising from partial differential equations. These systems can be broadly categorized into linear and nonlinear problems. For large linear systems, iterative solvers are generally preferred over direct methods due to the latter’s superlinear growth of computational costs. Although convergence theory is well-developed under certain assumptions on the coefficient matrix, many classes of systems still pose open challenges. These difficulties become even more severe for systems of nonlinear equations, where nonlinear solvers typically rely on repeated linearization. For example, Newton’s method may even converge quadratically near the solution; it can also converge slowly or diverge when the initial guess is not chosen appropriately. A wide range of solvers with diverse variants and hyperparameter settings exists, and the development of efficient and robust iterative methods remains an active area of research. Recently, machine learning (ML) techniques have been applied to enhance the efficiency of classical iterative methods while preserving their interpretability and reliability. We refer to these ML-enhanced iterative methods as hybrid iterative methods, in the sense that they combine classical iterative methods with ML. This paper provides a comprehensive overview of state-of-the-art approaches to constructing hybrid iterative methods for both systems of linear and nonlinear equations, while also discussing open challenges and outlining potential directions for future research.

Kolmogorov-Arnold networks (KANs) have attracted attention recently as an alternative to multilayer perceptrons (MLPs) for scientific machine learning. However, KANs can be expensive to train, even for relatively small networks. Inspired by finite basis physics-informed neural networks (FBPINNs), in this work, we develop a domain decomposition-based architecture for KANs that allows for several small KANs to be trained in parallel to give accurate solutions for multiscale problems. We denote this new approach as finite basis KANs (FBKANs) and show that it yields accurate results for noisy data as well as physics-informed training.

This work develops a coarse space for overlapping Schwarz solvers using an algebraic multiscale construction. The basis is derived entirely from the system matrix and incorporates heterogeneity at subdomain interfaces through reduced boundary problems. An energy-minimization formulation supports extensions to elasticity and connects the construction to existing theoretical bounds. Numerical experiments examine robustness under coefficient contrast and compare the resulting coarse systems with established Schwarz coarse spaces. For three-dimensional diffusion problems with log-normal coefficients, the method achieves comparable iteration counts with a substantially smaller coarse problem.

Graph neural networks (GNNs) have emerged as a powerful framework for learning from graph-structured data. However, their efficient training remains challenging, particularly in distributed computing environments. This challenge arises from the use of message passing, which couples all graph nodes, leading to expensive optimization steps, high memory requirements, and substantial communication overhead. To alleviate these limitations, we propose a novel domain-decomposition (DD) variant of AG2m, an AdaGrad method enhanced with second-order curvature information and momentum, denoted by DD-AG2m. The proposed DD-AG2m alternates between AG2m optimization on the original (global) graph and AG2m optimization on the partitioned graphs. To incorporate global information at reduced cost, we further introduce a two-level variant (2DD-AG2m) that performs global optimization steps on a coarse graph obtained by randomly subsampling nodes within each subdomain. Numerical experiments spanning graph classification, node-level regression, and spatiotemporal forecasting tasks demonstrate that the proposed DD methods reduce the computational cost required to achieve the same predictive performance by a factor of 4–8. Moreover, for the fixed computational cost, they improve the predictive performance of GNNs by up to 22% compared with the baseline AG2m.

Cardiac digital twin research is moving from subject-specific anatomical replicas toward virtual cohorts that represent clinically relevant population subgroups. Yet access to representative imaging-derived anatomy datasets remains limited by cohort size, subgroup sparsity, and data-sharing constraints. Conditional generative models could help address this gap, but virtual cohorts are useful only if they preserve realistic, metadata-dependent anatomical variability. Existing cardiac anatomy generators largely rely on conditional variational autoencoders (cVAEs), which couple representation learning and metadata conditioning through a shared regularized latent prior. We introduce CAN-FLOW, a two-step Conditional ANatomy generation framework based on normalizing FLOWs that first learns geometry-only latent representations of diffeomorphic cardiac shape momenta and then models their sex-, age-, and body-mass-index-dependent distribution with a conditional normalizing flow. We trained CAN-FLOW on 2,208 healthy UK Biobank subjects and compared it with cVAEs across regularization strengths. CAN-FLOW generated plausible stochastic biventricular anatomies that better reproduced clinical phenotype distributions, metadata-dependent trends, subgroup variability, point-cloud coverage, and high-dimensional shape variability. Together, these results establish CAN-FLOW as a shareable framework for generating realistic, stochastically varying, metadata-conditioned biventricular anatomies for virtual cohort construction and in silico clinical trial workflows.

Developing resilient crops requires models that accurately capture how plant growth dynamics emerge from the continuous interplay between genetic and environmental factors. While high-throughput phenotyping platforms generate data for dynamic growth models, current modeling approaches show limitations: process-based crop growth models (CGMs) capture environmental dynamics but are difficult to scale across many genotypes because they require calibration of many parameters, while quantitative genetic models allow modeling many genotypes but represent environmental effects in a static manner. To address this gap, we propose a data-driven framework to identify parsimonious systems of ordinary differential equations (ODEs) describing crop growth dynamics as functions of environmental variables and a limited number of genotype-specific coefficients (GSCs). Using the Sparse Identification of Nonlinear Dynamics (SINDy) method, we infer model structure directly from data via sparse regression on a physiologically informed library of candidate functions. In this framework, genotype-by-environment (G\timesE) interactions arise naturally, as GSCs represent genotype-specific sensitivities to environmental variables. The resulting models are linear in these GSCs, making them compatible with mixed-model approaches for genomic prediction and selection at low computational cost. We evaluate the approach on an in silico dataset generated with WOFOST for five potato cultivars across twelve growing seasons. Our identified ODEs generalize to unseen environments and achieve R^2 > 0.9 for end-of-season yield prediction when compared to full-scale WOFOST simulations. These results highlight the potential of SINDy-based system identification as a scalable and interpretable approach for modeling crop growth dynamics and G\timesE interactions from phenotyping data.