Benjamin Berkels

The comparison of protein structures is necessary to determine molecular functions and evolutionary relationships and can also facilitate drug design and the prediction of separation options. The number of available protein structures is rapidly increasing, driven by experimental determination but also prediction methods such as AlphaFold. In this context, traditional structure comparison based on atomic superposition becomes computationally inefficient. To enable scalable analysis and surface shape comparison, alternative approaches represent protein structures using compact, fixed-length vectors known as descriptors. Current vectors often contain hundreds of entries, such as three-dimensional Zernike descriptors composed of 121 entries, or combinations of molecular and geometric properties. Here, we propose to use the Laplace-Beltrami spectrum, a mathematical representation of object surfaces derived from the eigenvalues of the Laplace-Beltrami operator, as an efficient option to capture protein structural properties and facilitate structural classification. Specifically, protein surfaces are encoded using only the first 10 non-zero eigenvalues of this spectrum. We used the SHREC 2025 dataset as a benchmark and achieved 85.8% accuracy on the original 97 protein structure classes and 97.7% accuracy on the homology-grouped 45 classes. Our method is fast, requiring ~75 min computation time for the 11,555 protein surface meshes of the SHREC 2025 dataset on a conventional AMD Ryzen 7 5700X 8-Core processor with 32 GB RAM. The approach can easily be applied to large protein structure databases because it is readily parallelizable and allows the incorporation of other descriptors, including surface features such as charge. This rapid screening tool can therefore be used to identify proteins with related surface shapes independent of sequence homology.

In this paper, a posteriori error estimates are derived for the approximation error of minimizers of functionals on the space of functions with bounded variation with a nonconvex lower-order term. To this end, the calibration method by Alberti, Bouchitté, and Dal Maso [ Calc. Var. Partial Differential Equations, 16 (2003), pp. 299–333] allows the problem to be reformulated as a uniformly convex variational problem over characteristic functions of subgraphs in one dimension higher. A primal-dual approach is formulated where the duality of divergence and gradient properly incorporates boundary conditions for the primal variable. Based on this, a posteriori error estimates can be derived first for the relaxed problem in the [Formula: see text]-norm. A cut-out argument allows converting this into an [Formula: see text]-error estimate for the characteristic subgraph functions apart from the jump interface, whereas the area of the interfacial region is estimated separately. To apply the estimate, we consider as one possible discretization a conforming finite element space for the primal variable and a nonconforming space for the dual variable. Finally, we validate the a posteriori error estimates in numerical experiments for a prototypical nonconvex functional in one and two dimensions as well as depth estimation in stereo imaging, a classical computer vision problem.

Optimizing manufacturing processes to reduce scrap and enhance process stability presents significant challenges, particularly when multiple conflicting objectives must be addressed concurrently. As the number of objectives increases, the complexity of the optimization task escalates. This difficulty is further intensified in online optimization scenarios, where optimal parameter settings must be delivered in real time within active production environments. In this work, we propose a reinforcement learning-based framework for the multi-objective optimization of manufacturing parameters, demonstrated through a case study on pinion gear manufacturing. The framework utilizes the Multi-Objective Maximum a Posteriori Optimization (MO-MPO) algorithm to train a reinforcement learning agent. A high-fidelity simulation of the pinion manufacturing process is constructed in Simufact, serving both data generation and validation purposes. The agent’s performance is assessed using a hold-out test set along with additional simulations of the physical process. To ensure the generalizability of the approach, further validation is performed using open-source manufacturing datasets and synthetically generated data. The results demonstrate the feasibility of the proposed method for real-time industrial deployment. Moreover, Pareto-optimality is verified via half-space analysis, emphasizing the framework’s effectiveness in managing trade-offs among competing objectives.

Phase transformations and crystallographic defects are two essential tools to drive innovations in materials. Bulk materials design via tuning chemical compositions is systematized using phase diagrams. It is shown here that the same thermodynamic concept can be applied to manipulate the chemistry at defects. Grain boundaries in Mg–Ga system are chosen as a model system, because Ga segregates to the boundaries, while simultaneously improving the strength and ductility of Mg alloys. To reveal the role of grain boundaries, correlated atomic‐scale characterization and simulation to scope and build phase diagrams for defects are presented. The discovery is enabled by triggering phase transformations of individual grain boundaries through local alloying, and sequentially imaging the structural and chemical changes using atomic‐resolution scanning transmission electron microscopy. Ab initio simulations determined the thermodynamic stability of grain boundary phases, and found out that increasing Ga content enhances grain boundary cohesion, relating to improved ductility. The methodology to trigger, trace, and simulate defect transformation at atomic resolution enables a systematic development of defect phase diagrams, providing a valuable tool to utilize chemical complexity and phase transformations at defects.

Phase transformations and crystallographic defects are two essential tools to drive innovations in materials. Bulk materials design via tuning chemical compositions has been systematized using phase diagrams. We show here that the same thermodynamic concept can be applied to understand the chemistry at defects. We present a combined experimental and modelling approach to scope and build phase diagrams for defects. The discovery was enabled by triggering phase transformations of individual defects through local alloying, and sequentially imaging the structural and chemical changes using atomic-resolution scanning transmission electron microscopy. By observing atomic-scale phase transformations of a Mg grain boundary through Ga alloying, we exemplified the method to construct a grain boundary phase diagram using ab initio simulations and thermodynamic principles. The methodology enables a systematic development of defect phase diagrams to propel a new paradigm for materials design utilizing chemical complexity and phase transformations at defects.