Philipp Knechtges

Fabian Hoppe, Melven Röhrig-Zöllner, Philipp Knechtges

We consider LLM-based algorithm development through a case study on contractionorder optimisation for tensor networks with OpenEvolve. We pay particular attention to the choice of the LLM as well as design choices such as evaluation metric and test instances. Our results highlight both the promise of verifier-guided evolutionary coding agents for algorithm development/improvement and the continuing importance of evaluation, validation, and interpretation -- and corresponding challenges -- by the human scientist.

Florian Zwicke, Philipp Knechtges, Marek Behr et al.

In an effort to increase the versatility of finite element codes, we explore the possibility of automatically creating the Jacobian matrix necessary for the gradient-based solution of nonlinear systems of equations. Particularly, we aim to assess the feasibility of employing the automatic differentiation tool TAPENADE for this purpose on a large Fortran codebase that is the result of many years of continuous development. As a starting point we will describe the special structure of finite element codes and the implications that this code design carries for an efficient calculation of the Jacobian matrix. We will also propose a first approach towards improving the efficiency of such a method. Finally, we will present a functioning method for the automatic implementation of the Jacobian calculation in a finite element software, but will also point out important shortcomings that will have to be addressed in the future.

Franziska Griese, Fabian Hoppe, Alexander Rüttgers et al.

The numerical simulation and optimization of technical systems described by partial differential equations is expensive, especially in multi-query scenarios in which the underlying equations have to be solved for different parameters. A comparatively new approach in this context is to combine the good approximation properties of neural networks (for parameter dependence) with the classical finite element method (for discretization). However, instead of considering the solution mapping of the PDE from the parameter space into the FEM-discretized solution space as a purely data-driven regression problem, so-called physically informed regression problems have proven to be useful. In these, the equation residual is minimized during the training of the neural network, i.e., the neural network "learns" the physics underlying the problem. In this paper, we extend this approach to saddle-point and non-linear fluid dynamics problems, respectively, namely stationary Stokes and stationary Navier-Stokes equations. In particular, we propose a modification of the existing approach: Instead of minimizing the plain vanilla equation residual during training, we minimize the equation residual modified by a preconditioner. By analogy with the linear case, this also improves the condition in the present non-linear case. Our numerical examples demonstrate that this approach significantly reduces the training effort and greatly increases accuracy and generalizability. Finally, we show the application of the resulting parameterized model to a related inverse problem.

Felix Terhag, Philipp Knechtges, Achim Basermann et al.

Cardiac real-time magnetic resonance imaging (MRI) is an emerging technology that images the heart at up to 50 frames per second, offering insight into the respiratory effects on the heartbeat. However, this method significantly increases the number of images that must be segmented to derive critical health indicators. Although neural networks perform well on inner slices, predictions on outer slices are often unreliable. This work proposes sparse Bayesian learning (SBL) to predict the ventricular volume on outer slices with minimal manual labeling to address this challenge. The ventricular volume over time is assumed to be dominated by sparse frequencies corresponding to the heart and respiratory rates. Moreover, SBL identifies these sparse frequencies on well-segmented inner slices by optimizing hyperparameters via type -II likelihood, automatically pruning irrelevant components. The identified sparse frequencies guide the selection of outer slice images for labeling, minimizing posterior variance. This work provides performance guarantees for the greedy algorithm. Testing on patient data demonstrates that only a few labeled images are necessary for accurate volume prediction. The labeling procedure effectively avoids selecting inefficient images. Furthermore, the Bayesian approach provides uncertainty estimates, highlighting unreliable predictions (e.g., when choosing suboptimal labels).

Markus Götz, Daniel Coquelin, Charlotte Debus et al.

To cope with the rapid growth in available data, the efficiency of data analysis and machine learning libraries has recently received increased attention. Although great advancements have been made in traditional array-based computations, most are limited by the resources available on a single computation node. Consequently, novel approaches must be made to exploit distributed resources, e.g. distributed memory architectures. To this end, we introduce HeAT, an array-based numerical programming framework for large-scale parallel processing with an easy-to-use NumPy-like API. HeAT utilizes PyTorch as a node-local eager execution engine and distributes the workload on arbitrarily large high-performance computing systems via MPI. It provides both low-level array computations, as well as assorted higher-level algorithms. With HeAT, it is possible for a NumPy user to take full advantage of their available resources, significantly lowering the barrier to distributed data analysis. When compared to similar frameworks, HeAT achieves speedups of up to two orders of magnitude.

Brendan Keith, Philipp Knechtges, Nathan V. Roberts et al.

We explore a vexing benchmark problem for viscoelastic fluid flows with the discontinuous Petrov-Galerkin (DPG) finite element method of Demkowicz and Gopalakrishnan [1,2]. In our analysis, we develop an intrinsic a posteriori error indicator which we use for adaptive mesh generation. The DPG method is useful for the problem we consider because the method is inherently stable---requiring no stabilization of the linearized discretization in order to handle the advective terms in the model. Because stabilization is a pressing issue in these models, this happens to become a very useful property of the method which simplifies our analysis. This built-in stability at all length scales and the a posteriori error indicator additionally allows for the generation of parameter-specific meshes starting from a common coarse initial mesh. A DPG discretization always produces a symmetric positive definite stiffness matrix. This feature allows us to use the most efficient direct solvers for all of our computations. We use the Camellia finite element software package [3,4] for all of our analysis.