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Optimal Transport has received much attention in Machine Learning as it allows to compare probability distributions by exploiting the geometry of the underlying space. However, in its original formulation, solving this problem suffers from…
In this paper we study a model for traffic flow on networks based on a hyperbolic system of conservation laws with discontinuous flux. Each equation describes the density evolution of vehicles having a common path along the network. In this…
The aim of this paper is to derive macroscopic equations for processes on large co-evolving networks, examples being opinion polarization with the emergence of filter bubbles or other social processes such as norm development. This leads to…
In this paper we propose an accurate, and computationally efficient method for incorporating adaptive spatial resolution into weakly-compressible Smoothed Particle Hydrodynamics (SPH) schemes. Particles are adaptively split and merged in an…
We study stochastic model reduction for evolution equations in infinite dimensional Hilbert spaces, and show the convergence to the reduced equations via abstract results of Wong-Zakai type for stochastic equations driven by a scaled…
Flow models are a cornerstone of modern machine learning. They are generative models that progressively transform probability distributions according to learned dynamics. Specifically, they learn a continuous-time Markov process that…
Wasserstein gradient flows on probability measures have found a host of applications in various optimization problems. They typically arise as the continuum limit of exchangeable particle systems evolving by some mean-field interaction…
We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures $m_0,m_1$. The effect of the additional entropy functional results into an elliptic…
In finite-dimensions, minimal surfaces that fill in the space delineated by closed curves and have minimal area arose naturally in classical physics in several contexts. No such concept seems readily available in infinite dimensions. The…
Given a family of probability measures in P(X), the space of probability measures on a Hilbert space X, our goal in this paper is to highlight one ore more curves in P(X) that summarize efficiently that family. We propose to study this…
We derive general depth-integrated model equations for overland flows featuring the evolution of suspended sediment that may be eroded from or deposited onto the underlying topography ('morphodynamics'). The resulting equations include…
A series of novel filters for probabilistic inference that propose an alternative way of performing Bayesian updates, called particle flow filters, have been attracting recent interest. These filters provide approximate solutions to…
We show that the average trajectories of relativistic quantum particles in Schwarzschild spacetime, obtained via quantum mechanical weak measurements of momentum and energy, are equivalent to the predicted flow lines of probability current…
We study Benamou's domain decomposition algorithm for optimal transport in the entropy regularized setting. The key observation is that the regularized variant converges to the globally optimal solution under very mild assumptions. We prove…
We study a mathematical model of a compressible viscous fluid driven by stochastic forces under slip boundary conditions of friction type. We introduce a notion of a weak solution that is analytically and probabilistically consistent with…
Classical optimal transport problem seeks a transportation map that preserves the total mass betwenn two probability distributions, requiring their mass to be the same. This may be too restrictive in certain applications such as color or…
In this article, we formulate topology optimization problems concerning the mass distribution as minimization problems for functionals on the Wasserstein space. We relax optimization problems regarding non-convex objective functions on the…
Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, Fokker-Planck equations, which model the diffusion of probability measures, can be understood as gradient descent over…
In this article dedicated to the memory of Igor D. Chueshov, I first summarize in a few words the joint results that we obtained over a period of six years regarding the long-time behavior of solutions to a class of semilinear stochastic…
This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparameterized two-layer neural networks. In particular, we consider the minimax optimization problem stemming from estimating linear…