Paper classifies minimal graph transformations into new families of surfaces.
problem Classifying minimal graph transformations into new families of surfaces.
method Formulated and solved a coupled system of partial differential equations, reduced to solving an ordinary differential equation.
result Established rigorous equivalence to a modified problem for a harmonic function, yielding new families of minimal surfaces.
We introduce the framework of continuous--depth graph neural networks (GNNs). Graph neural ordinary differential equations (GDEs) are formalized as the counterpart to GNNs where the input-output relationship is determined by a continuum of GNN layers, blending discrete topological structures and differential equations.…
The paper classifies shapes of translating solitons from isoparametric graphs.
problem Understanding shapes of translating solitons from isoparametric graphs.
method Analyzing ordinary differential equations and isoparametric functions.
result Classification of shapes of translating solitons.
The paper classifies shapes of translating solitons for a specific flow.
problem Understanding the shapes of translating solitons in a specific flow.
method Analyzing functions on a unit sphere and solving an ODE.
result Classification of the shapes of translating solitons.
New methods prove existence of rotating shapes moving in space.
problem Existence of rotating shapes moving in space.
method Different methods to prove existence based on singular ordinary differential equation.
result Existence of rotationally symmetric translating solutions proven without partial differential equations.
Neural ODEs control graph dynamics with low energy feedback.
problem Controlling complex dynamical systems on graphs.
method Neural Ordinary Differential Equation Control (NODEC) framework.
result NODEC learns low-energy control signals for graph dynamical systems.
New model predicts neural network performance from early training epochs, incorporating architecture impact.
problem Predicting neural network performance from early training epochs, neglecting architecture impact.
method Architecture-aware graph ordinary differential equation model.
result Model outperforms state-of-the-art methods for MLP and CNN learning curves.
Proposes SDE framework for uncertainty quantification in graph neural networks.
problem Lack of uncertainty quantification in graph neural networks.
method Introduces Latent Graph Neural Stochastic Differential Equations (LGNSDE) with Bayesian prior-posterior mechanism and Brownian motion.
result LGNSDEs provide theoretically sensible guarantees for uncertainty estimates and are robust to perturbations.
Study shows neural ODEs generalize well on synthetic graphs but struggle with degree heterogeneity and clustering.
problem Understanding neural ODEs on complex networks, especially with varying graph sizes and structures.
method Synthetic data from five dynamical systems on graphs, using Barabási-Barzel form vector fields.
result Degree heterogeneity and dynamical system type are primary factors affecting neural ODEs' generalization.
The theory of Lie remarkable equations, i.e. differential equations characterized by their Lie point symmetries, is reviewed and applied to ordinary differential equations. In particular, we consider some relevant Lie algebras of vector fields on Rk and characterize Lie remarkable equations admitted by the …
We use E. Cartan's method to solve the problem of equivalence of the second order ordinary differential equations with respect to the pseudogroup of point transformations.
The systems of complex analytic second order ordinary differential equations whose solutions close up to become rational curves (after analytic continuation) are characterized by the vanishing of an explicit differential invariant, and turn out to provide an infinite dimensional family of integrable systems.
To a system of second order ordinary differential equations (SODE) one can assign a canonical nonlinear connection that describes the geometry of the system. In this work we develop a geometric setting that allows us to assign a canonical nonlinear connection also to a system of higher order ordinary differential equat…
Paper solves a class of differential equations with specific solutions.
problem Identifying solutions to a class of nonlinear ODEs.
method Solves using a proposed side condition involving a third-order linear ODE.
result New closed and integral-form solutions for the Tzitzeica curve equation.
We show that classical Wilczynski--Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship…
We shall study the equivalence problem for ordinary differential equations with respect to the affine transformations group.
Logic approach finds real singularities in differential equations.
problem Finding geometric singularities of implicit ODEs over the reals.
method Vessiot theory, parametric Gaussian elimination, heuristic simplification, real quantifier elimination.
result Effective computation of geometric singularities using logic methods.
Paper develops a consistent algorithm for learning graph structure from continuous-time stochastic differential equations.
problem Learning structure from continuous-time stochastic differential equations.
method Score-based structure learning using Neural Ordinary Differential Equations with adaptive regularization.
result The method consistently recovers directed graphs of local independencies in systems of stochastic differential equations.
We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman dia…
Introduces geometric control theory for students.
problem No specific problem addressed in the abstract.
method Expository presentation of geometric control theory.
result Suitable for advanced students with solid math background.
Novel method for solving ODEs on k-polysymplectic manifolds.
problem Solving ordinary differential equations on k-polysymplectic manifolds.
method k-polysymplectic energy-momentum method.
result Novel stability analysis techniques applied to Hamiltonian systems.
A Gaussian Process Ordinary Differential Equation framework for large continuous dynamical systems
problem Forecasting complex dynamical systems
method Kernel autonomous ODE approach based on Gaussian Processes and Quadratic Order Model Reduction
result Full model outperforms ROM methods in terms of accuracy or computational costs
Characterizes Bonnet surfaces using analytic conditions.
problem Characterizing Bonnet surfaces by geometric conditions.
method Derives Bonnet surfaces using two analytic conditions: mean curvature reduction to an ODE and the Painlevé property.
result Bonnet surfaces characterized by analytic conditions.
Differential invariants and equivalence of ODEs y′′=a3(x,y)y′3+a2(x,y)y′2+a1(x,y)y′+a0(x,y)math.DG The paper solves the equivalence problem for a specific class of ODEs.
problem Solving the equivalence problem for a specific class of ordinary differential equations.
method Construction of the algebra of differential invariants for point transformations.
result The equivalence problem is solved for the given class of ODEs.
The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable equations and even on systems of equations. However, little has been done in the…
The goal of the present paper is to propose an enhanced ordinary differential equations solver by exploitation of the powerful equivalence method of Élie Cartan. This solver returns a target equation equivalent to the equation to be solved and the transformation realizing the equivalence. The target ODE is a member of …
Reconstructing signature features from randomized vector fields in differential equations.
problem Reconstructing signature features from controlled differential equations with random vector fields.
method Using controlled ordinary differential equations driven by continuous bounded variation curves, the study explores the extent to which signature features can be reconstructed from the non-linear flow of these equations.
result The number of signature features that can be reconstructed from the non-linear flow of controlled ordinary differential equations with random vector fields is exponential in the hidden dimension, under certain conditions.
We express the first jet bundle of curves in Euclidean space as homogeneous spaces associated to a Galilean-type group. Certain Cartan connections on a manifold with values in the Lie algebra of the Galilean group are characterized as geometries associated to systems of second order ordinary differential equations. We …
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the other to gradient matching. All methods have greater accuracy than previously sugge…
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.
Let dxi/dt=fi(x1,⋯,xn), (i=1,⋯,n) be a system of n first order autonomous ordinary differential equations. We use E. Cartan's equivalence method to study the invariants of this system under diffeomorphisms of the form Φ(t,x1,⋯,xn)=(φ0(t),φ1(x1),⋯,φ1(x1)).
GrADE uses graph neural networks and Neural ODE for solving time-dependent nonlinear PDEs efficiently.
problem Solving time-dependent nonlinear PDEs is computationally challenging and time-consuming.
method GrADE combines graph neural networks for spatial modeling and Neural ODE for temporal modeling, using attention mechanisms.
result GrADE efficiently solves PDEs, demonstrating scalability and better accuracy compared to existing methods.
Steerable neural ODEs on homogeneous spaces for equivariant feature dynamics.
problem Learning continuous-time equivariant dynamics of vector-valued features on homogeneous spaces.
method Introduces steerable neural ordinary differential equations on homogeneous spaces, interpreting features as sections of associated vector bundles over M. result Steerable NODEs are G-equivariant when the flow and connection are G-invariant, and they incorporate existing models. New approach to k-dimensional torus differential equations.
problem Generalizing Poincaré's results for higher-order equations on a torus.
method Non-Hamiltonian approach using continuous vector functions.
result New results even in the k=1 case. New method for linear connections in ODEs with constraints.
problem Constructing linear connections for ODEs with and without constraints.
method Novel method using submodule covariant derivatives.
result Closed form expressions for Massa-Pagani connection and its extension.
Paper derives explicit expression of Alekseev-Meinrenken diffeomorphism.
problem Understanding the Alekseev-Meinrenken diffeomorphism.
method Via the Stokes phenomenon of meromorphic linear systems of ODEs with Poncaré rank 1.
result Explicit expression of the Alekseev-Meinrenken diffeomorphism.
Modeling interacting objects with latent Gaussian process ODEs.
problem Time uncertainty-aware modeling of continuous-time dynamics of interacting objects.
method A new model using latent Gaussian process ordinary differential equations to infer independent dynamics and interactions.
result Our model improves long-term predictions and successfully encapsulates independent dynamics and interactions.
Neural controlled DEs model irregular time series by adjusting based on observations.
problem Modeling irregularly sampled multivariate time series with memory-efficient adjoint-based backpropagation.
method Neural controlled differential equations (CDEs) that adjust based on subsequent observations.
result Achieves state-of-the-art performance on various datasets.
Most deep neural networks use simple, fixed activation functions, such as sigmoids or rectified linear units, regardless of domain or network structure. We introduce differential equation units (DEUs), an improvement to modern neural networks, which enables each neuron to learn a particular nonlinear activation functio…
Surveying connections between algebraic geometry and surface topology.
problem Non-abelian analogues of standard conjectures on cohomology.
method Study mapping class group actions on character varieties and isomonodromy differential equations.
result Open questions and conjectures on these topics.
NP-ODE models FEA simulations with uncertainty, improving accuracy and efficiency.
problem Limitations of FEA in terms of computational cost and uncertainty quantification.
method Physics-informed neural process aided ordinary differential equations (NP-ODE).
result NP-ODE outperforms benchmark methods in uncertainty quantification and prediction accuracy.
In 1883, as an early result, Sophus Lie established an explicit necessary and sufficient condition for an analytic second order ordinary differential equation y_xx = F(x,y,y_x) to be equivalent, through a point transformation (x,y) --> (X(x,y), Y(x,y)), to the Newtonian free particle equation Y_XX = 0. This result, pre…
ICODEN models survival data with interval-censored times using neural networks and ODEs.
problem Predicting time-to-event outcomes with interval-censored data, especially when models require strong assumptions or cannot handle high-dimensional predictors.
method ICODEN uses ordinary differential equations and deep neural networks to model the hazard function and cumulative hazard without proportional hazards assumption.
result ICODEN achieves satisfactory predictive accuracy across various simulation and real-world applications, handling high-dimensional predictors robustly.
By the classical Martingale Representation Theorem, replication of random vectors can be achieved via stochastic integrals or solutions of stochastic differential equations. We introduce a new approach to replication of random vectors via adapted differentiable processes generated by a controlled ordinary differential …
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…
A variational proof is provided of the existence and uniqueness of evolutions of regular Lagrangian systems.
Solutions to a differential equation link to contact structures.
problem Linking solutions of a specific differential equation to contact structures.
method Established a correspondence between solutions of Noth's equation and diffeomorphisms of contact structures.
result Established a correspondence between solutions of Noth's equation and diffeomorphisms of contact structures of type G2. In this paper we propose the use of continuous residual modules for graph kernels in Graph Neural Networks. We show how both discrete and continuous residual layers allow for more robust training, being that continuous residual layers are those which are applied by integrating through an Ordinary Differential Equation …