Alexander Mielke

Jia-Jie Zhu, Alexander Mielke

The purpose of this paper is to answer a few open questions in the interface of kernel methods and PDE gradient flows. Motivated by recent advances in machine learning, particularly in generative modeling and sampling, we present a rigorous investigation of Fisher-Rao and Wasserstein type gradient flows concerning their gradient structures, flow equations, and their kernel approximations. Specifically, we focus on the Fisher-Rao (also known as Hellinger) geometry and its various kernel-based approximations, developing a principled theoretical framework using tools from PDE gradient flows and optimal transport theory. We also provide a complete characterization of gradient flows in the maximum-mean discrepancy (MMD) space, with connections to existing learning and inference algorithms. Our analysis reveals precise theoretical insights linking Fisher-Rao flows, Stein flows, kernel discrepancies, and nonparametric regression. We then rigorously prove evolutionary $Γ$-convergence for kernel-approximated Fisher-Rao flows, providing theoretical guarantees beyond pointwise convergence. Finally, we analyze energy dissipation using the Helmholtz-Rayleigh principle, establishing important connections between classical theory in mechanics and modern machine learning practice. Our results provide a unified theoretical foundation for understanding and analyzing approximations of gradient flows in machine learning applications through a rigorous gradient flow and variational method perspective.

Alexander Mielke, Jia-Jie Zhu

We investigate a family of gradient flows of positive and probability measures, focusing on the Hellinger-Kantorovich (HK) geometry, which unifies transport mechanism of Otto-Wasserstein, and the birth-death mechanism of Hellinger (or Fisher-Rao). A central contribution is a complete characterization of global exponential decay behaviors of entropy functionals (e.g. KL, $χ^2$) under Otto-Wasserstein and Hellinger-type gradient flows. In particular, for the more challenging analysis of HK gradient flows on positive measures -- where the typical log-Sobolev arguments fail -- we develop a specialized shape-mass decomposition that enables new analysis results. Our approach also leverages the (Polyak-)Łojasiewicz-type functional inequalities and a careful extension of classical dissipation estimates. These findings provide a unified and complete theoretical framework for gradient flows and underpin applications in computational algorithms for statistical inference, optimization, and machine learning.

Jue Chen, Alexander Mielke, Kaspar Althoefer et al.

The potential of Animal-Robot Interaction (ARI) in welfare applications depends on how much an animal perceives a robotic agent as socially relevant, non-threatening and potentially attractive (acceptance). Here, we present an animal-centered soft robotic affective interface for newly hatched chicks (Gallus gallus). The soft interface provides safe and controllable cues, including warmth, breathing-like rhythmic deformation, and face-like visual stimuli. We evaluated chick acceptance of the interface and chick-robot interactions by measuring spontaneous approach and touch responses during video tracking. Overall, chicks approached and spent increasing time on or near the interface, demonstrating acceptance of the device. Across different layouts, chicks showed strong preference for warm thermal stimulation, which increased over time. Face-like visual cues elicited a swift and stable preference, speeding up the initial approach to the tactile interface. Although the breathing cue did not elicit any preference, neither did it trigger avoidance, paving the way for further exploration. These findings translate affective interface concepts to ARI, demonstrating that appropriate soft, thermal and visual stimuli can sustain early chick-robot interactions. This work establishes a reliable evaluation protocol and a safe baseline for designing multimodal robotic devices for animal welfare and neuroscientific research.