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.
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48 results for “ordinary differential 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…
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
Let , () be a system of 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 .
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.
Signature tensors uniquely identify ODE solutions.
problem Identifying ODE solutions from signature tensors.
method Geometric theory of nonlinear systems of ODEs.
result Necessary and sufficient algebraic conditions for signature tensors to represent ODE solutions.
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…
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…
In this paper we consider an alternative approach to "un-reduction". This is the process where one associates to a Lagrangian system on a manifold a dynamical system on a principal bundle over that manifold, in such a way that solutions project. We show that, when written in terms of second-order ordinary differential …
This work introduces a polynomial kernel method for inferring ODE models.
problem Estimating future behavior of dynamical systems from observations.
method Parametric polynomial kernel regression using Backpropagation and Stochastic Gradient Descent.
result Successfully tracks future behavior of chaotic dynamical systems over long time periods.
NeuPDE uses neural networks to model time-dependent data using differential equations.
problem Modeling time-dependent data from dynamic datasets.
method Neural network approach with both shallow multilayer perceptrons and nonlinear differential terms.
result Demonstrated on various dynamical systems, NeuPDE outperforms other methods.
Symmetries of variational problems are considered as symmetries of vector bundle valued exterior differential systems. This approach is then applied to third order ordinary variational equations of motion of the semi-classical spinning particle.
A variational proof is provided of the existence and uniqueness of evolutions of regular Lagrangian systems.
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 …
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.
New approach to -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 case.
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…
FSNN learns causal relationships in complex systems using neural nets.
problem Inferring causality in directed cyclic graphs.
method Constructs a non-linear system of ODEs using feed forward neural nets.
result Accurately models complex, non-linear systems with causal relationships.
Paper studies rigidity phenomena for elliptic systems on Riemannian manifolds.
problem Local radial rigidity of elliptic systems on Riemannian manifolds.
method Reduction to singular ordinary differential equations of Euler type.
result Local uniqueness and existence results for solutions with prescribed initial jets.
A Lie system is a system of first-order ordinary differential equations describing the integral curves of a -dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields: a so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the $k…
The paper defines conjugate points for systems of second-order ODEs and discusses their properties.
problem Defining and analyzing conjugate points for systems of second-order ordinary differential equations.
method Using Jacobi fields and eigendistributions of the Jacobi endomorphism, the paper extends the concept of conjugate points to a generalization of locally symmetric spaces and Lagrangian systems with symmetry.
result The paper provides a comprehensive analysis of conjugate points for various types of systems of second-order ordinary differential equations.
We consider a problem of equivalence of generic pairs on a manifold , where is a distribution of rank and is a distribution of rank one. We construct a canonical bundle with a canonical frame. We prove that two pairs are equivalent if and only if the corresponding frames are diffeomorphic. As a p…
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.
New method improves nonlinear filtering accuracy with reduced computation.
problem Complex nonlinear filtering with small system noise.
method Asymptotic expansion with ordinary differential equations and Edgeworth-type correction.
result Significantly lower computational cost with improved accuracy.
A Logic Based Approach to Finding Real Singularities of Implicit Ordinary Differential Equationsmath.LO
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.
Quasi-Lie schemes for PDEsmath.AP
The theory of quasi-Lie systems, i.e. systems of first order ordinary differential equations which can be related via a generalised flow to Lie systems, is extended to systems of partial differential equations and its applications to obtaining -dependent superposition rules and integrability conditions are analysed.…
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.
The -symplectic structures appear in the geometric study of the partial differential equations of classical field theories. Meanwhile, we present a new application of the -symplectic structures to investigate a type of systems of first-order ordinary differential equations, the -symplectic Lie systems. In part…
New approach predicts microfinance borrower risk to manage portfolios.
problem Non-repayment problem in microfinance due to asymmetric information.
method Modeling and simulation of ordinary differential systems.
result Prediction of solvent and insolvent borrowers over time.
We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are …
New methods solve inverse problem for Lagrangians in calculus of variations.
problem Determining Lagrangians for systems of differential equations.
method Exterior differential systems theory (EDS) and generalisation of Jesse Douglas's solution.
result Generalised solutions in arbitrary dimension .
We consider gradient Ricci solitons conformal to a -dimensional pseudo-Euclidean space and we completely describe the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary differential equations. As a consequence, the gradient Ricci solitons that arise from t…
In this paper we analyze the tangential symmetries of Darboux integrable decomposable exterior differential systems. The decomposable systems generalize the notion of a hyperbolic exterior differential system and include the classic notion of Darboux integrability for first order systems and second order scalar equatio…
New sampling scheme improves ML accuracy in physics simulations.
problem Improving accuracy of ML models in physics simulations.
method Taylor-based data sampling scheme for DNNs.
result Reduces error in DNN solutions of ODE systems.
In two recent papers necessary and sufficient conditions for a given system of second-order ordinary differential equations to be of Lagrangian form with additional dissipative forces were derived. We point out that these conditions are not independent and prove a stronger result accordingly.
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 .
result Steerable NODEs are -equivariant when the flow and connection are -invariant, and they incorporate existing models.
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.
NDS learns dynamical models with prior knowledge, improving accuracy and efficiency.
problem Learning accurate dynamical models with limited data and varying dynamics.
method Neural Dynamical Systems (NDS) integrates prior knowledge in ODEs with neural networks to estimate parameters and predict states.
result NDS achieves higher accuracy and uses fewer samples compared to other methods.
For the implicit systems of first order ordinary differential equations on the plane there is presented the complete local classification of generic singularities of family of its phase curves up to smooth orbital equivalence. Besides the well known singularities of generic vector fields on the plane and the singularit…
Superposition rules form a class of functions that describe general solutions of systems of first-order ordinary differential equations in terms of generic families of particular solutions and certain constants. In this work we extend this notion and other related ones to systems of higher-order differential equations …
Modeling Atmospheric Data and Identifying Dynamics: Temporal Data-Driven Modeling of Air Pollutantsstat.AP
Modeling air pollutants using data-driven techniques and sparse identification of nonlinear dynamics.
problem Predicting concentrations of air pollutants using hidden physical laws.
method Sparse identification of nonlinear dynamics (SINDy) for parsimonious systems of ordinary differential equations.
result More than half of the critical points are saddle points, indicating system instability.
New method finds global Lagrangians for variational systems.
problem Constructing global variational principles for variational systems.
method Analyzing Lepage 2-forms and finding global Lagrangians for systems defined by homogeneous functions of degree \(c
eq 0, 1\).
result Locally variational systems defined by homogeneous functions of degree \(c
eq 0, 1\) are globally variational.
We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructe…
About Some Applications of Kolmogorov Equations to the Simulation of Financial Institutions Activityq-fin.GN
The goal of this article is to describe the concepts of system dynamics and its applications to the simulation modeling of financial institutions daily activity. The hybrid method of the re-engineering of banking business processes based upon combination of system dynamics, queuing theory and tools of ordinary differen…
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.
Graph neural networks are extended to continuous-depth models using differential equations.
problem Improving graph neural networks for static and dynamic graph data.
method Formalizing GNNs as GDEs, blending discrete structures with differential equations.
result GDEs offer computational advantages in static settings and improved performance in dynamic settings.
DyNODE uses neural ODEs to model system dynamics in continuous control tasks.
problem Modeling the dynamics of systems in continuous control tasks.
method Neural Ordinary Differential Equations (ODEs) combined with actor-critic RL.
result DyNODE outperforms standard neural networks in sample efficiency and predictive performance.