arXiv research

Algorithm reconstructs conserved networks from flow data.

New neural network enforces mass conservation for better ice flow predictions.

This paper analyzes the probability flow in the stock market using the Black-Scholes model.

Data symmetries in neural networks can generate conserved quantities.

Turing complete flow on 4-sphere preserves volume.

Minimal existence time for Willmore flow established for smooth and weak Lipschitz initial data.

Following an approach of the second author for conformally invariant variational problems in two dimensions, we show in four dimensions the existence of a conservation law for fourth order systems, which includes both intrinsic and extrinsic biharmonic maps. With the help of this conservation law we prove the continuit…

The conservation laws of the third order quasilinear scalar evolution equations are considered via differential system and characteristic cohomology. We find a subspace of 2 forms in the infinite prolonged space in which every conservation law has a unique representative. The structure of this subspace naturally gives …

It has been proved that on 2-dimensional orientable compact manifolds of genus $g>1$ there is no integrable geodesic flow with an integral polynomial in momenta. There is a conjecture that all integrable geodesic flows on $T^2$ possess an integral quadratic in momenta. All geodesic flows on $S^2$ and $T^2$ possessing i…

LFlows model fluid densities and velocities using invertible maps that satisfy the continuity equation.

MC-LSTM extends LSTM to conserve mass in neural networks.

We give the following results for Pinkall's central affine curve flow on the plane: (i) a systematic and simple way to construct the known higher commuting curve flows, conservation laws, and a bi-Hamiltonian structure, (ii) Baecklund transformations and a permutability formula, (iii) infinitely many families of explic…

A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equatio…

In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows…

Study Godbillon-Vey invariants in non-Lorentzian spacetimes and fluid dynamics.

We characterize the conjugate linearized Ricci flow and the associated backward heat kernel on closed three--manifolds of bounded geometry. We discuss their properties, and introduce the notion of Ricci flow conjugated constraint sets which characterizes a way of Ricci flow averaging metric dependent geometrical data. …

Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows and general relativity. Recent work by the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existen…

Estimates network structure from node potentials and edge flows under Gaussian injection statistics.

It has been shown that for each Killing-Yano (KY)-form accepted by an $n$-dimensional (pseudo)Riemannian manifold of arbitrary signature, two basic gravitational currents can be defined. Conservation of the currents are explicitly proved by showing co-exactness of the one and co-closedness of the other. Some general ge…

Algorithm reconstructs interaction topology in linear dynamical systems.

Analyzes symmetries in neural networks to predict learning dynamics.

Mathematical model describes how red blood cells return to equilibrium.

We show that the twisted Kähler-Ricci flow on a complex manifold X converges to a flow of moving free boundaries, in a certain scaling limit. This leads to a new phenomenon of singularity formation and topology change which can be seen as a complex generalization of the extensively studied formation of shocks in Hamilt…

New surfaces with special geodesic and horocycle behaviors discovered.

A novel spatio-temporal graph neural network with a learnable Tweedie head improves vessel traffic flow prediction in sparse maritime data.

In this paper we introduce a new geometric flow --- the hyperbolic gradient flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. This kind of flow is new and very natural to understand the geometry of manifolds. We particularly investigate the global existence of the evolution of convex hypers…

Belief Propagation algorithms are instruments used broadly to solve graphical model optimization and statistical inference problems. In the general case of a loopy Graphical Model, Belief Propagation is a heuristic which is quite successful in practice, even though its empirical success, typically, lacks theoretical gu…

We propose the time-dependent generalization of an `ordinary' autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds derived from the extended musculo-skeletal configuration manifold. The …

We address the problem of constructing numerical integrators for nonholonomic Lagrangian systems that enjoy appropriate discrete versions of the geometric properties of the continuous flow, including the preservation of energy. Building on previous work on time-dependent discrete mechanics, our approach is based on a d…

Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according t…

CNFs learn on manifolds using PPD, improving likelihood and sample quality.

We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the …

The paper studies minimal resistance dynamics in radial fields, finding unique solutions for incompressible flows.

Develops a new framework to understand MCMC dynamics as flows on Wasserstein space.

Consider a Riemannian metric on two-torus. We prove that the question of existence of polynomial first integrals leads naturally to a remarkable system of quasi-linear equations which turns out to be a Rich system of conservation laws. This reduces the question of integrability to the question of existence of smooth (q…

The paper connects geodesic flows and limit sets on visibility manifolds.

New algorithm finds important synapses without training data.

Study on massless Vlasov equation on Reissner-Nordström spacetimes, showing decay rates and non-decay phenomena.

Generalizes Hasimoto transformation to arbitrary flows on space curves.

Graph-based method predicts edge flows from partial measurements.

Method learns Dirichlet-to-Neumann maps on graphs using Gaussian processes.

Analyzed geometric and diffusion properties of a coupled system.

Self-test loss functions improve data-driven modeling of weak-form operators and gradient flows.

Theoretical analysis confirms non-conservative algorithms can converge to optimal policies.

Study finds conservation laws for a specific class of parabolic equations.

Study finds conserved quantities for two types of curves on conformal sphere.

We give a proof of Gaussian upper bound for the heat kernel coupled with the Ricci ow. Previous proofs by Lei Ni [5] use Harnack inequality and doubling volume property, also the recent proof by Zhang and Cao [6] uses Sobolev type inequality that is conserved along Ricci ow. We will use a horizontal coupling of curve […

Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as well as the twistor spaces over a compact quaternion-Kahler manifold. Differing f…