arXiv subjects

Alexander Mielke

Publications and source records attributed to Alexander Mielke.

At least 19 recordsLinked to original sources

arXiv2026

From diffusion to transmission via EDP-convergence: a paradigmatic multiscale limit

We consider nonlinear diffusion equations on an interval where the diffusion coefficient in a small region near the center is scaled such that it approximates a transmission condition at a membrane. While the limiting behavior of the solutions is well-understood, we study the convergence in the sense of the energy-dissipation principle (EDP) of associated gradient structures given in terms of a free energy and a dissipation potential. EDP-convergence provides a uniquely specified limiting gradient structure that reformulates the transmission condition in terms of an effective kinetic relation for the membrane, which relates the jump of the chemical potential and the flux through the membrane. We show how \AAA properties of the chosen free energy and the mobility of small-scale diffusion migrate to the effective kinetic relation. A surprising result is that starting from the linear Onsager relation of Otto's gradient structure for the linear diffusion equation, one obtains an exponentially growing kinetic relation, the so-called Marcelin-De Donder kinetic.

math.AP↗

arXiv2026

Degenerate Diffusions on Continuum Percolation and Hamilton-Jacobi-Bellman Equations

We study degenerate controlled diffusions and Hamilton--Jacobi--Bellman equations posed on genuine continuum percolation clusters. The diffusion is constrained to evolve inside the infinite cluster and degenerates according to the distance to the irregular random boundary. Under suitable regularity and degeneracy assumptions on the diffusion matrix, we prove that the corresponding controlled diffusion never reaches the boundary and therefore admits a unique global strong solution. Using this result, we establish a stochastic control representation formula for viscosity solutions of degenerate Hamilton--Jacobi--Bellman equations posed on the random cluster, without imposing boundary conditions. We further verify that the structural assumptions introduced in this work, including quantitative integrability properties of the distance-to-the-boundary function and the associated degeneracy, hold for concrete continuum percolation models such as the Boolean model. In particular, although the diffusion never reaches the boundary, the boundary geometry remains a fundamental ingredient of the theory through the admissible degeneracy regime. The results establish a quenched framework for stochastic control and degenerate partial differential equations on genuine continuum percolation geometries, and identify a link between the analytic structure of the diffusion and the stochastic geometry of the underlying cluster. They also provide the foundational framework for the homogenization theory developed in the companion article [BMM26].

math.PR↗

arXiv2026

Gradient-flow characterizations of one-dimensional quasistatic viscoelasticity with Bhattacharya-like viscosity

We study the equation of one-dimensional quasistatic nonlinear viscoelasticity with Dirichlet boundary conditions, in the particular case that the underlying dissipation geometry (provided by the viscosity) is comparable to the Bhattacharya metric on probability densities. We establish a global existence result for weak solutions, with an approach based on a spatial discretization allowing us to work directly with the Riemannian metric associated to the viscosity. Strong convergence of spatially discrete solutions is shown directly - this is possible thanks to Lipschitz estimates achieved locally on energy sublevels enabled by an explicit derivation of the stretching of tangent vectors under the flow in the discrete setting and the relationship to the Bhattacharya metric. We furthermore prove gradient-flow representations for the solutions: they are curves of maximal slope and, under a global convexity hypothesis on the energy sublevels, we prove they satisfy a metric evolutionary variational inequality.

math.AP↗

arXiv2026

A Soft Robotic Interface for Chick-Robot Affective Interactions

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.

cs.RO↗

arXiv2025

Derivation of the fourth-order DLSS equation with nonlinear mobility via chemical reactions

We provide a derivation of the one-dimensional fourth-order DLSS equation based on an interpretation as a chemical reaction network. We consider the rate equation on the discretized circle for a process in which pairs of particles occupying the same site simultaneously jump to the two neighboring sites; the reverse process involves pairs of particles at adjacent sites simultaneously jumping back to the site located between them. Depending on the rates, in the vanishing-mesh-size limit we obtain either the classical DLSS equation or a variant with nonlinear mobility of power type. Via EDP convergence, we identify the limiting gradient structure to be driven by entropy with respect to a generalization of diffusive transport with nonlinear mobility. Interestingly, the DLSS equation with power-type mobility shares qualitative similarities with the fast diffusion and porous medium equation, since we find traveling wave solutions with algebraic tails or compactly supported polynomials, respectively.

math.AP↗

arXiv2025

Some notes on the Hellinger distance and various Fisher-Rao distances

These expository notes introduce the Hellinger distance on the set of all measures and the induced Fisher-Rao distances for subsets of measures, such as probability measures or Gaussian measures. The historical background is highlighted and the relations and the distinct features of the two distances are discussed. Moreover, we provide a dynamic characterization of absolutely continuous curves in the Hellinger spaces in terms of the growth equation, which replaces the continuity equation in the theory of optimal transport.

math-ph↗

arXiv2025

Chemical Space of Molecular Nanomotors: Optimizing Photochemical Properties for One- and Two-photon Applications

Light-driven molecular nanomotors hold promise for applications in material science and biomedicine. Significant efforts have focused on improving their efficiency, often targeting single candidate molecules. Here, we present a systematic data-driven approach to design nanomotors with high isomerization quantum yields for one- and two-photon applications, the latter being critical for biomedical applications requiring near-infrared light. We analyze the excited state properties of a dataset of 2016 nanomotors substituted with electron-donating and electron-withdrawing (push-pull) groups. Among the the top candidates, we achieved an increase in two-photon absorption strengths of up to two orders of magnitude compared to existing nanomotors. To ensure that the pi-pi*-character of the excited state is preserved, which is necessary to achieve the required photoisomerization, we introduce a photoreactivity score, that gauges the excited state character based on the transition. Furthermore, we benchmark three machine learning (ML) models Kernel Ridge Regression, XGBoost, and a Neural Network using physical and connectivity-based molecular descriptors. The excellent accuracy of our ML predictions holds promise to replace computationally costly quantum chemistry calculations in chemical space explorations.

physics.chem-ph↗

arXiv2025

Evolution of Gaussians in the Hellinger-Kantorovich-Boltzmann gradient flow

This study leverages the basic insight that the gradient-flow equation associated with the relative Boltzmann entropy, in relation to a Gaussian reference measure within the Hellinger-Kantorovich (HK) geometry, preserves the class of Gaussian measures. This invariance serves as the foundation for constructing a reduced gradient structure on the parameter space characterizing Gaussian densities. We derive explicit ordinary differential equations that govern the evolution of mean, covariance, and mass under the HK-Boltzmann gradient flow. The reduced structure retains the additive form of the HK metric, facilitating a comprehensive analysis of the dynamics involved. We explore the geodesic convexity of the reduced system, revealing that global convexity is confined to the pure transport scenario, while a variant of sublevel semi-convexity is observed in the general case. Furthermore, we demonstrate exponential convergence to equilibrium through Polyak-Lojasiewicz-type inequalities, applicable both globally and on sublevel sets. By monitoring the evolution of covariance eigenvalues, we refine the decay rates associated with convergence. Additionally, we extend our analysis to non-Gaussian targets exhibiting strong log-lambda-concavity, corroborating our theoretical results with numerical experiments that encompass a Gaussian-target gradient flow and a Bayesian logistic regression application.

math.AP↗

arXiv2025

Discrete-to-continuum limit for nonlinear reaction-diffusion systems via EDP convergence for gradient systems

We investigate the convergence of spatial discretizations for reaction-diffusion systems with mass-action law satisfying a detailed balance condition. Considering systems on the d-dimensional torus, we construct appropriate space-discrete processes and show convergence not only on the level of solutions, but also on the level of the gradient systems governing the evolutions. As an important step, we prove chain rule inequalities for the reaction-diffusion systems as well as their discretizations, featuring a non-convex dissipation functional. The convergence is then obtained with variational methods by building on the recently introduced notion of gradient systems in continuity equation format.

math.AP↗

arXiv2025

An Eulerian formulation for dissipative materials using Lie derivatives and GENERIC

We recall the systematic formulation of Eulerian mechanics in terms of Lie derivatives along the vector field of the material points. Using the abstract properties of Lie derivatives we show that the transport via Lie derivatives generates in a natural way a Poisson structure on the chosen phase space. The evolution equations for thermo-viscoelastic-viscoplastic materials in the Eulerian setting is formulated in the abstract framework of GENERIC (General Equations for Non-Equilibrium Reversible Irreversible Coupling). The equations may not be new, but the systematic splitting between reversible Hamiltonian and dissipative effects allows us to see the equations in a new light that is especially useful for future generalizing of the system, e.g., for adding new effects like reactive species.

math-ph↗

arXiv2025

Hellinger-Kantorovich Gradient Flows: Global Exponential Decay of Entropy Functionals

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, $\chi^2$) 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-)\L{}ojasiewicz-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.

math.AP↗

arXiv2024

Kernel Approximation of Fisher-Rao Gradient Flows

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 $\Gamma$-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.

stat.ML↗

arXiv2024

On De Giorgi's lemma for variational interpolants in metric and Banach spaces

Variational interpolants are an indispensable tool for the construction of gradient-flow solutions via the Minimizing Movement Scheme. The De Giorgi lemma provides the associated discrete energy-dissipation inequality. It was originally developed for metric gradient systems. Drawing from this theory we study the case of generalized gradient systems in Banach spaces, where a refined theory allows us to extend the validity of the discrete energy-dissipation inequality and to establish it as an equality. For the latter we have to impose the condition of radial differentiability of the dissipation potential. Several examples are discussed to show how sharp the results are.

math.AP↗

arXiv2024

Interaction-Force Transport Gradient Flows

This paper presents a new gradient flow dissipation geometry over non-negative and probability measures. This is motivated by a principled construction that combines the unbalanced optimal transport and interaction forces modeled by reproducing kernels. Using a precise connection between the Hellinger geometry and the maximum mean discrepancy (MMD), we propose the interaction-force transport (IFT) gradient flows and its spherical variant via an infimal convolution of the Wasserstein and spherical MMD tensors. We then develop a particle-based optimization algorithm based on the JKO-splitting scheme of the mass-preserving spherical IFT gradient flows. Finally, we provide both theoretical global exponential convergence guarantees and improved empirical simulation results for applying the IFT gradient flows to the sampling task of MMD-minimization. Furthermore, we prove that the spherical IFT gradient flow enjoys the best of both worlds by providing the global exponential convergence guarantee for both the MMD and KL energy.

cs.LG↗

arXiv2024

On the equilibrium solutions of electro-energy-reaction-diffusion systems

Electro-energy-reaction-diffusion systems are thermodynamically consistent continuum models for reaction-diffusion processes that account for temperature and electrostatic effects in a way that total charge and energy are conserved. The question of the long-time asymptotic behavior of electro-energy-reaction-diffusion systems motivates the characterization of their equilibrium solutions, which leads to a maximization problem of the entropy on the manifold of states with fixed values for the linear charge and the nonlinear convex energy functional. As the main result, we establish the existence, uniqueness, and regularity of solutions to this constrained optimization problem. We give two conceptually different proofs, which are related to different perspectives on the constrained maximization problem. The first one is based on the method of Lagrange multipliers, while the second one employs the direct method of the calculus of variations.

math.AP↗

arXiv2024

A general thermodynamical model for finitely-strained continuum with inelasticity and diffusion, its GENERIC derivation in Eulerian formulation, and some application

A thermodynamically consistent visco-elastodynamical model at finite strains is derived that also allows for inelasticity (like plasticity or creep), thermal coupling, and poroelasticity with diffusion. The theory is developed in the Eulerian framework and is shown to be consistent with the thermodynamic framework given by General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). For the latter we use that the transport terms are given in terms of Lie derivatives. Application is illustrated by two examples, namely volumetric phase transitions with dehydration in rocks and martensitic phase transitions in shape-memory alloys. A strategy towards a rigorous mathematical analysis is only very briefly outlined.

math-ph↗

arXiv2024

Deriving a GENERIC system from a Hamiltonian system

We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets). We start from a general finite-dimensional Hamiltonian system that is coupled linearly to an infinite-dimensional heat bath with linear dynamics. The latter is assumed to admit a compression to a finite-dimensional dissipative semigroup (i.e., the heat bath is a dilation of the semigroup) describing the dissipative evolution of new macroscopic variables. Already in the finite-energy case (zero-temperature heat bath) we obtain the so-called GENERIC structure (General Equations for Non-Equilibrium Reversible Irreversibe Coupling), with conserved energy, nondecreasing entropy, a new Poisson structure, and an Onsager operator describing the dissipation. However, their origin is not obvious at this stage. After extending the system in a natural way to the case of positive temperature, giving a heat bath with infinite energy, the compression property leads to an exact multivariate Ornstein-Uhlenbeck process that drives the rest of the system. Thus, we are able to identify a conserved energy, an entropy, and an Onsager operator (involving the Green-Kubo formalism) which indeed provide a GENERIC structure for the macroscopic system.

math-ph↗

arXiv2023

Evolutionary Variational Inequalities on the Hellinger-Kantorovich and Spherical Hellinger-Kantorovich spaces

We study the minimizing movement scheme for families of geodesically semi-convex functionals defined on either the Hellinger-Kantorovich or the Spherical Hellinger-Kantorovich space. By exploiting some of the finer geometric properties of those spaces, we prove that the sequence of curves, which are produced by geodesically interpolating the points generated by the minimizing movement scheme, converges to curves that satisfy the Evolutionary Variational Inequality (EVI), when the time step goes to 0.

math.AP↗