S. Becker

Victor Boussange, Sebastian Becker, Arnulf Jentzen et al.

Nonlinear partial differential equations (PDEs) are used to model dynamical processes in a large number of scientific fields, ranging from finance to biology. In many applications standard local models are not sufficient to accurately account for certain non-local phenomena such as, e.g., interactions at a distance. In order to properly capture these phenomena non-local nonlinear PDE models are frequently employed in the literature. In this article we propose two numerical methods based on machine learning and on Picard iterations, respectively, to approximately solve non-local nonlinear PDEs. The proposed machine learning-based method is an extended variant of a deep learning-based splitting-up type approximation method previously introduced in the literature and utilizes neural networks to provide approximate solutions on a subset of the spatial domain of the solution. The Picard iterations-based method is an extended variant of the so-called full history recursive multilevel Picard approximation scheme previously introduced in the literature and provides an approximate solution for a single point of the domain. Both methods are mesh-free and allow non-local nonlinear PDEs with Neumann boundary conditions to be solved in high dimensions. In the two methods, the numerical difficulties arising due to the dimensionality of the PDEs are avoided by (i) using the correspondence between the expected trajectory of reflected stochastic processes and the solution of PDEs (given by the Feynman-Kac formula) and by (ii) using a plain vanilla Monte Carlo integration to handle the non-local term. We evaluate the performance of the two methods on five different PDEs arising in physics and biology. In all cases, the methods yield good results in up to 10 dimensions with short run times. Our work extends recently developed methods to overcome the curse of dimensionality in solving PDEs.

Julia Lademann, Jannik Henze, Nadine Honke et al.

The rapid integration of artificial intelligence (AI) in education requires teachers to develop AI competencies while preparing students for a society influenced by AI. This study evaluates the impact of an online teacher training program on German in-service teachers' AI literacy, usage behaviors, and attitudes toward AI. A pre-post design study was conducted with teachers (N1 = 291 for AI literacy, N2 = 436 for attitude assessment) participating in the course. The program combined synchronous and asynchronous learning formats, including webinars, self-paced modules, and practical projects. The participants exhibited notable improvements across all domains: AI literacy scores increased significantly, and all attitude items regarding AI usage and integration demonstrated significant positive changes. Teachers reported increased confidence in AI integration. Structured teacher training programs effectively enhance AI literacy and foster positive attitudes toward AI in education.

Jonas Kirch, Sebastian Becker, Tiago Koketsu Rodrigues et al.

Clustered Federated Learning has emerged as an effective approach for handling heterogeneous data across clients by partitioning them into clusters with similar or identical data distributions. However, most existing methods, including the Iterative Federated Clustering Algorithm (IFCA), rely on a central server to coordinate model updates, which creates a bottleneck and a single point of failure, limiting their applicability in more realistic decentralized learning settings. In this work, we introduce DFCA, a fully decentralized clustered FL algorithm that enables clients to collaboratively train cluster-specific models without central coordination. DFCA uses a sequential running average to aggregate models from neighbors as updates arrive, providing a communication-efficient alternative to batch aggregation while maintaining clustering performance. Our experiments on various datasets demonstrate that DFCA outperforms other decentralized algorithms and performs comparably to centralized IFCA, even under sparse connectivity, highlighting its robustness and practicality for dynamic real-world decentralized networks.

Joe O'Brien, Jeremy Dolan, Jay Kim et al.

Our survey of 53 specialists across 105 AI reliability and security research areas identifies the most promising research prospects to guide strategic AI R&D investment. As companies are seeking to develop AI systems with broadly human-level capabilities, research on reliability and security is urgently needed to ensure AI's benefits can be safely and broadly realized and prevent severe harms. This study is the first to quantify expert priorities across a comprehensive taxonomy of AI safety and security research directions and to produce a data-driven ranking of their potential impact. These rankings may support evidence-based decisions about how to effectively deploy resources toward AI reliability and security research.

Christian Beck, Sebastian Becker, Patrick Cheridito et al.

In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.

Christian Beck, Sebastian Becker, Patrick Cheridito et al.

In this paper we introduce a numerical method for nonlinear parabolic PDEs that combines operator splitting with deep learning. It divides the PDE approximation problem into a sequence of separate learning problems. Since the computational graph for each of the subproblems is comparatively small, the approach can handle extremely high-dimensional PDEs. We test the method on different examples from physics, stochastic control and mathematical finance. In all cases, it yields very good results in up to 10,000 dimensions with short run times.

Christian Beck, Sebastian Becker, Philipp Grohs et al.

Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black-Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.