Jacobi operators on certain naturally reductive spaces have constant coefficient ODEs.
problem Understanding geometric properties of naturally reductive spaces.
method Analyzing Jacobi operators and their ODEs on these spaces.
result Jacobi operators on these spaces satisfy constant coefficient ODEs.
Classifies solutions to critical sixth order equations with a singularity.
problem Classifying entire positive singular solutions to critical sixth order equations.
method Integral sliding methods, qualitative analysis of ODEs, topological two-parameter shooting technique.
result Solutions are given by a singular radial factor times a periodic solution to a sixth order IVP with constant coefficients.
Paper introduces LODE-GPs for modeling data following linear ODEs.
problem Modeling data from systems of linear ODEs.
method Symbolic construction of LODE-GPs using Smith normal form algorithms.
result Improves GP modeling of data from systems of linear ODEs.
We give an elementary proof to the asymptotic expansion formula of Rochon-Zhang for the unique complete Kähler-Einstein metric of Cheng-Yau, Kobayashi, Tian-Yau and Bando on quasi-projective manifolds. The main tools are the solution formula for second order ODE's with constant coefficients and spectral theory for Lapl…
Investigates portfolio selection for rank-dependent utilities in incomplete markets.
problem Portfolio selection for agents with rank-dependent utility in incomplete financial markets.
method Characterizes deterministic strict equilibrium strategies for constant-coefficient and time-invariant probability weighting functions. Addresses the issue of selecting an optimal strategy from multiple equilibrium strategies for time-variant probability weighting functions.
result Characterizes deterministic strict equilibrium strategies and identifies optimal strategies from multiple equilibrium strategies.
Study constructs closed curves with constant curvature on cylinders and tori.
problem Creating closed curves with constant curvature.
method ODEs and symmetry arguments, starting with cylinders, then tori, and finally Frenet-Serret equations.
result Closed constant curvature space curves constructed on cylinders and tori.
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…
In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Her…
Constructs metrics on Riemann surfaces with singularities.
problem Creating constant curvature metrics on surfaces with specific singularities.
method Using meromorphic 1-forms and ODEs to construct conformal metrics.
result Classified constant curvature metrics on S2 with two conical singularities. Realizations of stochastic process are often observed temporal data or functional data. There are growing interests in classification of dynamic or functional data. The basic feature of functional data is that the functional data have infinite dimensions and are highly correlated. An essential issue for classifying dyn…
Neural Jump ODEs model Itô processes without adversarial training.
problem Generating samples from Itô processes with irregular data.
method Neural Jump ODEs framework for drift and diffusion approximation.
result NJODEs can recover true parameters of Itô processes in the limit.
General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while s…
Study finds multiple periodic solutions to ODEs related to curvature problems.
problem Finding multiple positive periodic solutions to quasilinear ODEs.
method Global bifurcation techniques applied to second order quasilinear ODEs.
result Bifurcation-theoretic proof of nonuniqueness for conformal metrics with constant scalar curvature.
Study on rotational hypersurfaces with constant Gauss-Kronecker curvature.
problem Exploring hypersurfaces with constant Gauss-Kronecker curvature.
method Solving ODE for generating curves and analyzing geometric properties.
result Discovery of non-compact rotational hypersurfaces with negative Gauss-Kronecker curvature and finite volume.
Enhanced Neural ODEs outperform traditional models in image classification and video prediction.
problem Efficiently modeling time-varying dynamics in neural networks.
method Proposed a novel family of non-autonomous Neural ODEs with time-varying weights.
result Outperformed previous Neural ODE variants in speed and representational capacity.
First-order ODEs linked to flat surfaces, leading to integrability.
problem Integrating first-order ODEs.
method Defined Riemannian metrics on variable spaces, studied surface properties, and established connections between Jacobi fields and Lie point symmetries.
result Flat associated surfaces lead to integrable first-order ODEs.
Unified ODE model explains residual and non-residual networks.
problem Unclear relationship between residual and non-residual networks.
method Introducing a damping term in an ODE model to interpolate between ResNet and CNN architectures.
result Unified framework for understanding residual and non-residual networks.
Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of ga…
Improved neural-ODE for faster convergence and stability.
problem Stability, consistency, and convergence issues in neural-ODE solvers.
method Proposed a first-order Nesterov's accelerated gradient (NAG) based ODE-solver.
result Efficacy demonstrated in three tasks: supervised classification, density estimation, and time-series modelling.
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 …
In this paper we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer shares the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding value/loss functions converge. Our result gives, in the context o…
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.
Study on 4-manifolds for special Kähler metrics with constant Ricci determinant.
problem Existence of complete cohomogeneity one Kähler metrics with zero Ricci determinant.
method Analysis of specific Lie groups and solutions to associated ODE systems.
result Complete classification for SU(2) and existence results for E(2) and nil3. Characterizes differential forms and vector fields with constant coefficients on manifolds.
problem Understanding constant coefficient differential forms and vector fields on manifolds.
method Analyzes differential forms and vector fields of specific degrees, proving obstructions and characterizing solutions to partial differential systems.
result Characterizes differential forms and vector fields with constant coefficients of various degrees on smooth manifolds.
The scalar curvature equation for rotation invariant Kähler metrics on Cn\{0} is reduced to a system of ODEs of order 2. By solving the ODEs, we obtain complete lists of rotation invariant zero or positive csck on Cn\{0} in lower dimensions. We also prove that there doe…
Study proves fluid limits of fragmented limit-order markets.
problem Modeling fragmented limit-order markets with small and frequent orders.
method Proved convergence of discrete system to fluid limit characterized by coupled nonlinear ODEs.
result Fluid system converges to stationary equilibrium state over time.
The generalized Weierstrass representation is used to analyze the asymptotic behavior of a constant mean curvature surface that arises locally from an ordinary differential equation with a regular singularity. We prove that a holomorphic perturbation of an ODE that represents a Delaunay surface generates a constant mea…
NODEs with explicit time dependence can interpolate and generalize like piecewise-constant estimators.
problem Learning from finite datasets with neural ODEs.
method Control-theoretic perspective applied to semi-autonomous NODEs.
result SA-NODEs can interpolate and satisfy SCC, leading to generalization rates similar to histogram and nearest-neighbor estimators.
The fused lasso is analyzed for high-dimensional piecewise-constant regression coefficients.
problem Estimation of high-dimensional piecewise-constant regression coefficients.
method Formulated a restricted isometry condition for the fused lasso estimator and derived estimation bounds.
result The estimation error can be dominated by either the lasso or the fused lasso rate, depending on the number of non-zero coefficients and piece-wise constant segments.
New concept of regular separation for ODEs leads to improved Hardy field results.
problem Understanding solutions of definable ODEs with specific properties.
method Introducing regular separation and proving its implications for ODEs and vector fields.
result The regular separation property leads to improved Hardy field results and non-empty sets of trajectories.
We construct smooth Riemannian metrics with constant scalar curvature on each Hirzebruch surface. These metrics respect the complex structures, fiber bundle structures, and Lie group actions of cohomogeneity one on these manifolds. Our construction is reduced to an ordinary differential equation called Duffing equation…
Curved loxodromes on spheres are explained and their ODE derived.
problem Understanding curved analogues of compass-bearing curves on spheres.
method Explained curved loxodromes and derived the fifth order invariant ODE.
result Derived the fifth order invariant ODE for loxodromes.
New method identifies physical constants from video data alone.
problem Identifying physical constants from video data.
method Proves level-set slope-coverage condition ensures local affine mapping to true physical state, enabling exact parameter recovery.
result Underdamped systems identifiable from a single video clip, other regimes require three diverse trajectories.
New RNN model handles long-term dependencies in irregularly-sampled time series.
problem Handling long-term dependencies in irregularly-sampled time series data.
method Designing ODE-LSTMs that separate memory from continuous-time state.
result ODE-LSTMs outperform other RNN-based models on non-uniformly sampled data with long-term dependencies.
Gradient almost para-Ricci-like solitons have constant coefficients and scalar curvatures.
problem Characterizing gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian Π-manifolds. method Proving constant coefficients and scalar curvatures through analysis of soliton properties.
result Constant coefficients and scalar curvatures for gradient almost para-Ricci-like solitons.
The paper proves an equilibrium in a limited stock market participation model with power utilities.
problem Existence of an equilibrium in a model with limited stock market participation and power utilities.
method Proves existence and uniqueness of a solution to a singular and path-dependent Riccati-type ODE.
result Proves existence of a Radner equilibrium with homogenous power-utility investors.
The Eisenhart lift connects Hamiltonian systems to geodesics in pp-wave spacetimes.
problem Studying the stability and dynamics of Hamiltonian systems.
method Eisenhart lift to pp-wave spacetimes and conformal classes of ODEs.
result Existence of a constant of the motion generalizing conservation of energy.
The paper characterizes surfaces in Heisenberg group with constant p-mean curvature.
problem Characterizing surfaces with constant p-mean curvature in the Heisenberg group. method Using the fundamental theorem of surfaces in H1, the existence of constant p-mean curvature surfaces is linked to solutions of a nonlinear ODE. result Complete set of solutions to the ODE (1.2) or (1.5) divides constant p-mean curvature surfaces into several classes. 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 …
Uniform scaling limits in AdamW-trained transformers converge to ODEs.
problem Understanding the dynamics of large-depth transformers trained with AdamW.
method Modeling transformer dynamics as an interacting particle system coupled through attention, proving convergence to ODEs.
result The joint dynamics of hidden states and backpropagated variables converge uniformly to an ODE system.
Calibration of stochastic local volatility (SLV) models to their underlying local volatility model is often performed by numerically solving a two-dimensional non-linear forward Kolmogorov equation. We propose a novel finite volume (FV) discretization in the numerical solution of general 1D and 2D forward Kolmogorov eq…
We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the s…
Continuous-depth Evoformer reduces protein folding prediction time and resource usage.
problem Efficient protein structure prediction with reduced computational costs.
method Continuous-depth formulation of Evoformer using Neural Ordinary Differential Equations (Neural ODEs).
result The continuous-time Evoformer achieves constant memory cost and improved efficiency.
Study biharmonic conformal immersions into anti-de Sitter space, proving rigidity and local existence.
problem Analyzing biharmonic conformal immersions of surfaces into anti-de Sitter space.
method Using a sign convention, expressing biharmonic equation in terms of induced metric and curvature, deriving cohomogeneity-one analytic system, and solving scalar third-order ODE.
result Local existence and rigidity of biharmonic conformal immersions with nonconstant dilation.
Study optimal portfolios for many players in a market model with random coefficients.
problem Optimal portfolio selection for many players under relative performance criteria in a market model with random coefficients.
method Game theory and stochastic optimal control, focusing on CARA and CRRA risk preferences, and extending to continuum of players.
result Existence of forward Nash equilibrium and mean field equilibrium for the n-agent game and corresponding mean field stochastic optimal control problem.
We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constan…
This study shows why training Neural ODEs is hard and proposes a new method.
problem Training Neural ODEs is challenging, especially in practice.
method Proposed a new stabilization method and provided an analytical convergence analysis.
result Insights and techniques for researchers starting work on Neural ODEs.
The purpose of this article is to give an explicit formula for all curves of constant torsion τ in the unit two-sphere S2(1). These curves and their basic properties have been known since the 1890's, and some of these properties are discussed in the Appendix. Some example curves, computed with a standard ODE packa…