arXiv research

The abstract discusses families of holomorphic maps to Oka manifolds with approximation theorems.

New methods improve efficiency of sampling algorithms for complex systems.

New deep learning architecture learns martingales efficiently.

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also sho…

New method speeds up SDE inference by matching moments to FPK equation.

The paper develops theory for holomorphic null curves in SL2(C).

The Runge-Kutta-Legendre scheme improves pricing American options and other derivatives.

Study magnetic perturbations in Riemannian and Lorentzian Calderón problems.

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability d…

For any pseudoconvex Runge domain $Ω\subset\mathbb{C}^2$ we prove that every closed discrete subset in $Ω$ is contained in a properly embedded complex curve in $Ω$ with any prescribed topology (possibly infinite).

New probabilistic scheme combines deep learning with Runge-Kutta methods for solving PDEs.

We study a moduli space of ASD connections over $S^3\times \mathbb{R}$. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the …

Novel IMEX scheme solves financial PDEs with mixed derivatives.

We prove a version of the classical Runge and Mergelyan uniform approximation theorems for non-orientable minimal surfaces in Euclidean 3-space R3. Then, we obtain some geometric applications. Among them, we emphasize the following ones: 1. A Gunning-Narasimhan type theorem for non-orientable conformal surfaces. 2. An …

Rex solves the inverse problem for ODE/SDE solvers, improving precision and stability.

Note on instabilities in super-time-stepping methods for Heston model.

We examine spaces of connected tri-/univalent graphs subject to local relations which are motivated by the theory of Vassiliev invariants. It is shown that the behaviour of ladder-like subgraphs is strongly related to the parity of the number of rungs: there are similar relations for ladders of even and odd lengths, re…

Introduces a new stochastic optimization method for deep learning.

New method solves complex financial option pricing with varying time steps.

The study quantifies the information needed for causal queries at different levels of Pearl's hierarchy.

Meta-learning has emerged as an important framework for learning new tasks from just a few examples. The success of any meta-learning model depends on (i) its fast adaptation to new tasks, as well as (ii) having a shared representation across similar tasks. Here we extend the model-agnostic meta-learning (MAML) framewo…

We study gradient-based optimization methods obtained by directly discretizing a second-order ordinary differential equation (ODE) related to the continuous limit of Nesterov's accelerated gradient method. When the function is smooth enough, we show that acceleration can be achieved by a stable discretization of this O…

A new method solves American put options with high accuracy and speed.

New boundary treatment improves accuracy for complex PDEs.

Paper proposes a new method to speed up diffusion models.

Sampling with Markov chain Monte Carlo methods often amounts to discretizing some continuous-time dynamics with numerical integration. In this paper, we establish the convergence rate of sampling algorithms obtained by discretizing smooth Itô diffusions exhibiting fast Wasserstein-$2$ contraction, based on local deviat…

Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.

New causal distances improve evaluation of causal discovery algorithms.

A higher-order Runge-Kutta optimizer performs poorly compared to Adam when evaluated fairly.

A novel symplectic integrator for Hamiltonian equations on $S_2^n \times T^{\ast} \RR^m$ is developed and studied. Partitioned Runge--Kutta methods for Hamiltonian systems on products of Hamiltionian manifolds are studied, specifically, algebraic conditions for their symplecticity are derived.

The paper defines flexible domains for minimal surfaces in Euclidean spaces and explores their properties.

Higher-order ODE solvers improve deep learning performance.

In 1996, Nadirashvili used Runge's theorem to produce a complete minimal disc inside a ball in R^3. In this paper we generalize the techniques used by Nadirashvili to obtain new examples of complete minimal surfaces inside a ball in R^3, with the conformal structure of an annulus.

In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves i…

New definition of patient-specific root causes of disease using counterfactuals.

The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we pr…

We prove Runge-type theorems and universality results for locally univalent holomorphic and meromorphic functions. Refining a result of M. Heins, we also show that there is a universal bounded locally univalent function on the unit disk. These results are used to prove that on any hyperbolic simply connected plane doma…

We derive the explicit formula for the joint Laplace transform of the Wishart process and its time integral which extends the original approach of Bru. We compare our methodology with the alternative results given by the variation of constants method, the linearization of the Matrix Riccati ODE's and the Runge-Kutta al…

Calibrated probabilistic solvers improve accuracy of ODE estimates.

We consider the problem of identifying a unitary Yang-Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the cas…

The paper models star dynamics using Ricci flow and Perelman entropy, revealing chaotic behavior.

Geometric methods integrate Lie systems for optimal control problems.

This paper proposes a new method to learn integration schemes for complex ODEs.

There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al. introduced a sampling-based class of methods that are 'well-calibrated' in a specific sense. But the computational c…

Motivated by numerical integration on manifolds, we relate the algebraic properties of invariant connections to their geometric properties. Using this perspective, we generalize some classical results of Cartan and Nomizu to invariant connections on algebroids. This has fundamental consequences for the theory of numeri…

New samplers reduce NFEs for diffusion models.

The paper studies dynamical systems with evolving geometric structure using numerical methods.

ContinuousNet generalizes ResNets to continuous dynamical systems.