Lorena A. Barba

Tingyu Wang, Simon K. Layton, Lorena A. Barba

Boundary element methods produce dense linear systems that can be accelerated via multipole expansions. Solved with Krylov methods, this implies computing the matrix-vector products within each iteration with some error, at an accuracy controlled by the order of the expansion, $p$. We take advantage of a unique property of Krylov iterations that allow lower accuracy of the matrix-vector products as convergence proceeds, and propose a relaxation strategy based on progressively decreasing $p$. In extensive numerical tests of the relaxed Krylov iterations, we obtained speed-ups of between $2.1\times$ and $3.3\times$ for Laplace problems and between $1.7\times$ and $4.0\times$ for Stokes problems. We include an application to Stokes flow around red blood cells, computing with up to 64 cells and problem size up to 131k boundary elements and nearly 400k unknowns. The study was done with an in-house multi-threaded C++ code, on a hexa-core CPU. The code is available on its version-control repository, \href{https://github.com/barbagroup/fmm-bem-relaxed}{https://github.com/barbagroup/fmm-bem-relaxed}.

Pi-Yueh Chuang, Lorena A. Barba

Though PINNs (physics-informed neural networks) are now deemed as a complement to traditional CFD (computational fluid dynamics) solvers rather than a replacement, their ability to solve the Navier-Stokes equations without given data is still of great interest. This report presents our not-so-successful experiments of solving the Navier-Stokes equations with PINN as a replacement for traditional solvers. We aim to, with our experiments, prepare readers for the challenges they may face if they are interested in data-free PINN. In this work, we used two standard flow problems: 2D Taylor-Green vortex at Re=100 and 2D cylinder flow at Re=200. The PINN method solved the 2D Taylor-Green vortex problem with acceptable results, and we used this flow as an accuracy and performance benchmark. About 32 hours of training were required for the PINN method's accuracy to match the accuracy of a 16x16 finite-difference simulation, which took less than 20 seconds. The 2D cylinder flow, on the other hand, did not produce a physical solution. The PINN method behaved like a steady-flow solver and did not capture the vortex shedding phenomenon. By sharing our experience, we would like to emphasize that the PINN method is still a work-in-progress, especially in terms of solving flow problems without any given data. More work is needed to make PINN feasible for real-world problems in such applications.

Rio Yokota, Lorena A. Barba

With the current hybridization of treecodes and FMMs, combined with auto-tuning capabilities on heterogeneous architectures, the flexibility of fast N-body methods has been greatly enhanced. These features are a requirement to developing a black-box software library for fast N-body algorithms on heterogeneous systems, which is our immediate goal.

Lorena A. Barba, Laura Stegner

Traditional assessment methods collapse when students use generative AI to complete work without genuine engagement, creating an illusion of competence where they believe they're learning but aren't. This paper presents the conversational exam -- a scalable oral examination format that restores assessment validity by having students code live while explaining their reasoning. Drawing on human-computer interaction principles, we examined 58 students in small groups across just two days, demonstrating that oral exams can scale to typical class sizes. The format combines authentic practice (students work with documentation and supervised AI access) with inherent validity (real-time performance cannot be faked). We provide detailed implementation guidance to help instructors adapt this approach, offering a practical path forward when many educators feel paralyzed between banning AI entirely or accepting that valid assessment is impossible.

Pi-Yueh Chuang, Lorena A. Barba

The recent surge of interest in physics-informed neural network (PINN) methods has led to a wave of studies that attest to their potential for solving partial differential equations (PDEs) and predicting the dynamics of physical systems. However, the predictive limitations of PINNs have not been thoroughly investigated. We look at the flow around a 2D cylinder and find that data-free PINNs are unable to predict vortex shedding. Data-driven PINN exhibits vortex shedding only while the training data (from a traditional CFD solver) is available, but reverts to the steady state solution when the data flow stops. We conducted dynamic mode decomposition and analyze the Koopman modes in the solutions obtained with PINNs versus a traditional fluid solver (PetIBM). The distribution of the Koopman eigenvalues on the complex plane suggests that PINN is numerically dispersive and diffusive. The PINN method reverts to the steady solution possibly as a consequence of spectral bias. This case study reaises concerns about the ability of PINNs to predict flows with instabilities, specifically vortex shedding. Our computational study supports the need for more theoretical work to analyze the numerical properties of PINN methods. The results in this paper are transparent and reproducible, with all data and code available in public repositories and persistent archives; links are provided in the paper repository at \url{https://github.com/barbagroup/jcs_paper_pinn}, and a Reproducibility Statement within the paper.

John D. Jakeman, Lorena A. Barba, Joaquim R. R. A. Martins et al.

Scientific machine learning (SciML) models are transforming many scientific disciplines. However, the development of good modeling practices to increase the trustworthiness of SciML has lagged behind its application, limiting its potential impact. The goal of this paper is to start a discussion on establishing consensus-based good practices for predictive SciML. We identify key challenges in applying existing computational science and engineering guidelines, such as verification and validation protocols, and provide recommendations to address these challenges. Our discussion focuses on predictive SciML, which uses machine learning models to learn, improve, and accelerate numerical simulations of physical systems. While centered on predictive applications, our 16 recommendations aim to help researchers conduct and document their modeling processes rigorously across all SciML domains.