arXiv research

Minimal surfaces in third-order ODEs identified for linear second-order ODEs.

The paper explores solving inverse problems for ODEs with and without constraints.

We apply the Cartan equivalence method to the study of real analytic second order ODEs under the local real analytic diffeomorphism of $\C^2$ which are area-preserving. This enables us to give a characterization of the second order ODEs which are equivalent to $y^{\prime\prime}=0$ under such transformations. Moreover w…

We present Ordinary Differential Equation Variational Auto-Encoder (ODE$^2$VAE), a latent second order ODE model for high-dimensional sequential data. Leveraging the advances in deep generative models, ODE$^2$VAE can simultaneously learn the embedding of high dimensional trajectories and infer arbitrarily complex conti…

It is demonstrated that point symmetry algebras of general analytic second order ODEs, not necessary of principal type, can have all dimensions between 0 and 8 except for 7. For the symmetry dimension 8 the ODE must be locally trivializable.

SONODEs and ANODEs improve learning of second order dynamics.

In the present paper we consider the problem of local equivalence of second order ODEs which are cubic in second derivative under the action of the pseudogroup of contact transformations. We show how it may be reduced to the equivalence problem of 2-webs in $\mathbb{R}^3$ under the action of finite-dimensional group, a…

We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also show that, modulo Cartan duality and point transformations, the Painlevé-I equati…

We use the solution space of a pair of ODEs of at least second order to construct a smooth surface in Euclidean space. We describe when this surface is a proper embedding which is geodesically complete with finite total Gauss curvature. If the associated roots of the ODEs are real and distinct, we give a universal uppe…

The equivalence problem for second order ODEs given modulo point transformations is solved in full analogy with the equivalence problem of nondegenerate 3-dimensional CR structures. This approach enables an analog of the Feffereman metrics to be defined. The conformal class of these (split signature) metrics is well de…

We solve the local equivalence problem for second order (smooth or analytic) ordinary differential equations. We do so by presenting a {\em complete convergent normal form} for this class of ODEs. The normal form is optimal in the sense that it is defined up to the automorphism group of the model (flat) ODE $y"=0$. For…

This is an review on the point classification of second order ODE's by Ruslan Sharipov. His works were published in 1997-1998 at the Electronic Archive at LANL and undeservedly forgotten. Last chapter is an application of this classification to the investigation of Painleve equations.

These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting f…

The class of second order ODE's cubic with respect to the first order derivative is considered. Using geometric structures associated with these equations, the subclasses of umbilical equations, zero mean curvature equations, and zero Gaussian curvature equations are defined. Zero mean curvature equations are studied w…

New method for linear connections in ODEs with constraints.

This work improves likelihood of score-based diffusion ODEs using high-order denoising score matching.

This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler-L…

The paper improves ODE solvers by integrating diverse information types.

A systematic algorithm for building integrating factors of the form mu(x,y') or mu(y,y') for non-linear second order ODEs is presented. When such an integrating factor exists, the algorithm determines it without solving any differential equations. Examples of ODEs not having point symmetries are shown to be solvable us…

Study finds multiple periodic solutions to ODEs related to curvature problems.

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…

Optimizes curves on Riemannian manifolds to minimize curvature.

We provide five examples of conformal geometries which are naturally associated with ordinary differential equations (ODEs). The first example describes a one-to-one correspondence between the Wuenschmann class of 3rd order ODEs considered modulo contact transformations of variables and (local) 3-dimensional conformal …

We find the complete set of fundamental invariants for systems of ordinary differential equations of order $\ge 4$ under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. It turns out that starting fr…

In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew ($S$) and mean ($H$) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that $H$ and $S$ can be interpr…

New symmetries found for scalar and vector ODEs of arbitrary dimensions.

New optimality conditions for sub-Riemannian geodesics derived.

New method identifies physical constants from video data alone.

We derive a second-order ordinary differential equation (ODE) which is the limit of Nesterov's accelerated gradient method. This ODE exhibits approximate equivalence to Nesterov's scheme and thus can serve as a tool for analysis. We show that the continuous time ODE allows for a better understanding of Nesterov's schem…

Portable, Wearable and Wireless electrocardiogram (ECG) Systems have the potential to be used as point-of-care for cardiovascular disease diagnostic systems. Such wearable and wireless ECG systems require automatic detection of cardiovascular disease. Even in the primary care, automation of ECG diagnostic systems will …

We study complex analytic (possibly singular) projective connections on the plane. We characterize some of them in terms of their families of integral curves. We also give a beginning of classification of second order odes polynomial in the first and second derivatives, and with holomorphic coefficients.

Method studies equivalence of second order ODEs under specific transformations.

In the present paper we establish the necessary and sufficient conditions for two ordinary differential equations of the form $y"{}^2+A(x,y,y') y"+B(x,y,y')=0$ to be equivalent under the action of the pseudogroup of contact transformations. These conditions are formulated in terms of integrals of some one-dimensional d…

Classifies solutions to critical sixth order equations with a singularity.

New symmetry dimensions for higher order ODEs are identified.

Paper proves higher-order flow matching preserves optimality in generative modeling.

We compute the characteristic Cartan connection associated with a system of third order ODEs. Our connection is different from Tanaka normal one, but still is uniquely associated with the system of third order ODEs. This allows us to find all fundamental invariants of a system of third order ODEs and, in particular, de…

First-order ODEs linked to flat surfaces, leading to integrability.

We prove that there is a correspondence between projective structures defined by torsion-free connections with skew-symmetric Ricci tensor and Veronese webs on a plane. The correspondence is used to characterise the projective structures in terms of second order ODEs.

New vector fields integrate first-order ODEs.

WENDy now estimates nonlinear ODEs with noisy data.

Study compares 5 ODE solvers on 3 case studies, finding varying accuracy.

The aim of this paper is to construct a Riemann-Lagrange geometry on 1-jet spaces, in the sense of d-connections, d-torsions, d-curvatures, electromagnetic d-field and geometric electromagnetic Yang-Mills energy, starting from a given linear ODEs system or a given superior order ODE. The case of a non-homogenous linear…

Soft-constrained PINN solves ODEs with minimal data, improving efficiency and robustness.

In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g.~the volume standing at the top of the book). We study the fluctuations of the …

We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature $(2, 2)$. We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in…

We develop a second-order model for limit order books in a single scaling regime.

We show how the tangent bundle decomposition generated by a system of ordinary differential equations may be generalized to the case of a system of second order PDEs `of connection type'. Whereas for ODEs the decomposition is intrinsic, for PDEs it is necessary to specify a closed 1-form on the manifold of independent …