Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machine learning. Training neural networks for singularly perturbed problems is particularly challenging due to certain parameters in the PDEs that introduce near-singularities in the loss function. In this study, we overcome this challenge by introducing a novel method based on homotopy dynamics to effectively manipulate these parameters. From a theoretical perspective, we analyze the effects of these parameters on training difficulty in these singularly perturbed problems and establish the convergence of the proposed homotopy dynamics method. Experimentally, we demonstrate that our approach significantly accelerates convergence and improves the accuracy of these singularly perturbed problems. These findings present an efficient optimization strategy leveraging homotopy dynamics, offering a robust framework to extend the applicability of neural networks for solving singularly perturbed differential equations.

Solving nonlinear partial differential equations (PDEs) with multiple solutions is essential in various fields, including physics, biology, and engineering. However, traditional numerical methods, such as finite element and finite difference methods, often face challenges when dealing with nonlinear solvers, particularly in the presence of multiple solutions. These methods can become computationally expensive, especially when relying on solvers like Newton's method, which may struggle with ill-posedness near bifurcation points. In this paper, we propose a novel approach, the Newton Informed Neural Operator, which learns the Newton solver for nonlinear PDEs. Our method integrates traditional numerical techniques with the Newton nonlinear solver, efficiently learning the nonlinear mapping at each iteration. This approach allows us to compute multiple solutions in a single learning process while requiring fewer supervised data points than existing neural network methods.

With the recent approval by the FDA of the first disease-modifying drug for Alzheimer's Disease (AD), personalized medicine will be increasingly important for appropriate management and counseling of patients with AD and those at risk. The growing availability of clinical biomarker data and data-driven computational modeling techniques provide an opportunity for new approaches to individualized AD therapeutic planning. In this paper, we develop a new mathematical model, based on AD cognitive, cerebrospinal fluid (CSF) and MRI biomarkers, to provide a personalized optimal treatment plan for individuals. This model is parameterized by biomarker data from the AD Neuroimaging Initiative (ADNI) cohort, a large multi-institutional database monitoring the natural history of subjects with AD and mild cognitive impairment (MCI). Optimal control theory is used to incorporate time-varying treatment controls and side-effects into the model, based on recent clinical trial data, to provide a personalized treatment regimen with anti-amyloid-beta therapy. In-silico treatment studies were conducted on the approved treatment, aducanumab, as well as on another promising anti-amyloid-beta therapy under evaluation, donanemab. Clinical trial simulations were conducted over both short-term (78 weeks) and long-term (10 years) periods with low-dose (6 mg/kg) and high-dose (10 mg/kg) regimens for aducanumab, and a single-dose regimen (1400 mg) for donanemab. Results confirm those of actual clinical trials showing a large and sustained effect of both aducanumab and donanemab on amyloid beta clearance. The effect on slowing cognitive decline was modest for both treatments, but greater for donanemab. This optimal treatment computational modeling framework can be applied to other single and combination treatments for both prediction and optimization, as well as incorporate new clinical trial data as it becomes available.

Bertini_real is a compiled command line program for numerically decomposing the real portion of a positive-dimensional complex component of an algebraic set. The software uses homotopy continuation to solve a series of systems via regeneration from a witness set to compute a cell decomposition. The implemented decomposition algorithms are similar to the well-known cylindrical algebraic decomposition (CAD) first established by Collins in that they produce a set of connected cells. In contrast to the CAD, Bertini_real produces cells with midpoints connected to boundary points by homotopies, which can easily be numerically tracked. Furthermore, the implemented decomposition for surfaces naturally yields a triangulation. This CAD-like decomposition captures the topological information and permits further computation on the real sets, such as sampling, visualization, and three-dimensional printing.

A Matlab implementation, multiplicity, of a numerical algorithm for computing the multiplicity structure of a nonlinear system at an isolated zero is presented. The software incorporates a newly developed equation-by-equation strategy that significantly improves the efficiency of the closedness subspace algorithm and substantially reduces the storage requirement. The equation-by-equation strategy is actually based on a variable-by-variable closedness subspace approach. As a result, the algorithm and software can handle much larger nonlinear systems and higher multiplicities than their predecessors, as shown in computational experiments on the included test suite of benchmark problems.