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We develop theory and applications of forward characteristic processes in discrete time following a seminal paper of Jan Kallsen and Paul Kr\"uhner. Particular emphasis is placed on the dynamics of volatility surfaces which can be easily…

Predicting dynamic behaviors is one of the goals of science in general as well as essential to many specific applications of human knowledge to real world systems. Here we introduce an analytic approach using the sigmoid growth curve to…

A discretisation scheme that preserves topological features of a physical problem is extended so that differential geometric structures can be approximated in a consistent way thus giving access to the study of physical systems which are…

We study a general framework of distributional computational graphs: computational graphs whose inputs are probability distributions rather than point values. We analyze the discretization error that arises when these graphs are evaluated…

We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy…

Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make…

Derivation of the probability density evolution provides invaluable insight into the behavior of many stochastic systems and their performance. However, for most real-time applica-tions, numerical determination of the probability density…

Coordinate-based neural networks parameterizing implicit surfaces have emerged as efficient representations of geometry. They effectively act as parametric level sets with the zero-level set defining the surface of interest. We present a…

Stochastic differential equations (SDEs) are established tools to model physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE intrinsic randomness of a system around its drift can be identified…

This paper presents an end-to-end differentiable algorithm for robust and detail-preserving surface normal estimation on unstructured point-clouds. We utilize graph neural networks to iteratively parameterize an adaptive anisotropic kernel…

A method is developed within an adaptive framework to solve quasilinear diffusion problems with internal and possibly boundary layers starting from a coarse mesh. The solution process is assumed to start on a mesh where the problem is badly…

We study a discrete denoising diffusion framework that integrates a sample-efficient estimator of single-site conditionals with round-robin noising and denoising dynamics for generative modeling over discrete state spaces. Rather than…

We propose a novel approach for deformation-aware neural networks that learn the weighting and synthesis of dense volumetric deformation fields. Our method specifically targets the space-time representation of physical surfaces from liquid…

We consider a distributionally robust formulation of stochastic optimization problems arising in statistical learning, where robustness is with respect to uncertainty in the underlying data distribution. Our formulation builds on…

Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of…

Mathematical models of biological populations commonly use discrete structure classes to capture trait variation among individuals (e.g. age, size, phenotype, intracellular state). Upscaling these discrete models into continuum descriptions…

In this paper, we investigate the dynamics of surface growth resulting from sedimentation of spherical granular particles in a fluid environment, using experiments and simulations. In the experimental part, spherical polystyrene particles…

We develop a discrete differential geometry for surfaces of non-constant negative curvature, which can be used to model various phenomena from the growth of flower petals to marine invertebrate swimming. Specifically, we derive and…

We propose a numerical method for fluid deformable surfaces governed by surface Stokes flow and Helfrich bending energy under active growth, aiming to model shape evolution of the epithelium in developmental processes. To prevent…

We introduce an interface growth model exhibiting a nonequilibrium roughening transition (NRT). In the model, particles consist of two species, and deposit or evaporate on one dimensional substrate according to a given dynamic rule. When…