Variational reduction simplifies Lagrangian systems with scaling symmetries.
problem Simplifying Lagrangian systems with scaling symmetries.
method Defining a variational reduction procedure for homogenous Lagrangian systems.
result Reconstructing trajectories from critical points of reduced variational principle.
Paper variates Navier-Stokes-Fourier system for thermodynamic consistency.
problem Modeling compressible fluid dynamics with thermodynamic constraints.
method Variational discretization with discrete exterior calculus.
result Derives a nonholonomic variational integrator for NSF system.
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.
Study nonholonomic systems with collisions using variational principles.
problem Variational problems on nonholonomic systems with collisions.
method Extended variational principle, introduced connection on principal bundles, applied Lagrange–Poincaré–Pontryagin reduction.
result Implicit Lagrange–d'Alembert–Pontryagin equations for nonholonomic systems with collisions.
We develop variational integrators from discrete Hamiltonian systems with external forces.
problem Creating accurate discrete models of continuous Hamiltonian systems.
method Constructing discrete Hamiltonian systems with external forces, analyzing symplectic structure, and combining methods to build variational integrators.
result We derive variational integrators that approximate continuous Hamiltonian systems with high accuracy.
Systems of ordinary differential equations (or dynamical forms in Lagrangian mechanics), induced by embeddings of smooth fibered manifolds over one-dimensional basis, are considered in the class of variational equations. For a given non-variational system, conditions assuring variationality (the Helmholtz conditions) o…
We discuss a recently proposed variational principle for deriving the variational equations associated to any Lagrangian system. The principle gives simultaneously the Lagrange and the variational equations of the system. We define a new Lagrangian in an extended configuration space ---which we call D'Alambert's--- com…
A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and sufficient algebraic and differential conditions for the existence of a variational…
The article concerns the problem if a~given system of differential equations is identical with the Euler--Lagrange system of an~appropriate variational integral. Elementary approach is applied. The main results involve the determination of the first--order variational integrals related to the second--order Euler--Lagra…
New variational principles found for conformal geodesics.
problem Challenges in Lagrangian formulation for conformal geodesics.
method Enlarging the class of variations leads to a variational formulation with a third-order conformally invariant Lagrangian.
result Some integral curves of the fourth-order ODE system are spirals.
Derives equations for systems with external forces using variational methods.
problem Deriving equations for systems subjected to external forces.
method Variational derivation of Euler-Poincaré equations using Poisson groupoid geometry.
result Derives variational error for numerical integrators of forced systems.
Paper analyzes dynamics of nonholonomic systems with collisions using variational techniques.
problem Analyzing the dynamics of nonholonomic mechanical systems with impacts.
method Variational techniques extended to nonsmooth context for collisions.
result Variational formulation for implicit nonholonomic mechanical systems with energy-momentum preserving collisions.
Geometric integrator preserves coadjoint orbits in dissipative systems.
problem Preserving coadjoint orbits in dissipative mechanical systems.
method Adapted discrete variational integrators for forced Euler-Poincaré and Lie-Poisson systems.
result Preserves coadjoint orbits exactly, improving over general-purpose methods.
We are able to derive the equations of motion for forced mechanical systems in a purely variational setting, both in the context of Lagrangian or Hamiltonian mechanics, by duplicating the variables of the system as introduced by Galley [2013], Galley, Tsang, and Stein [2014]. We show that this construction is useful to…
Extends exterior diff. sys. to Lie algebroids with examples.
problem Invariant inverse problem of the calculus of variations
method Extends exterior differential systems to Lie algebroids, defines integral manifolds.
result Defines integral manifolds for exterior diff. systems on Lie algebroids.
A variational principle is proposed for obtaining the Jacobi equations in systems admitting a Lagrangian description. The variational principle gives simultaneously the Lagrange equations of motion and the Jacobi variational equations for the system. The approach can be of help in finding constants of motion in the Jac…
Develops VAEs for learning complex physical systems from data.
problem Learning low-dimensional representations of nonlinear physical systems.
method Variational Autoencoders with manifold latent spaces.
result Effective in learning nonlinear Burgers equation and constrained mechanical systems.
New algorithm for continuous-time switching systems using variational inference.
problem Inference in time-series data with continuous-time switching systems.
method Developed a variational inference algorithm combining Gaussian process approximation and posterior inference for Markov jump processes.
result Bayesian latent state estimates and point estimates of unknown parameters for arbitrary points on the real axis.
The paper analyzes errors in mechanical systems with external forces.
problem Error analysis of mechanical systems with external forces.
method Analysis of variational integrators with contact order r for discrete mechanical systems. result The contact order of the integrator is the same as the contact order of the original systems.
New approach to Lagrangian systems using intrinsic geometry.
problem Developing a new framework for Lagrangian systems.
method Direct reformulation of Hamiltonian formalism, introduction of spatial equation and spatial-gauge symmetry.
result Covariant and non-covariant canonical variational principles demonstrated for Maxwell equations.
We use a method, inspired by Pohozeav's work, to study asymptotic behaviors of non-variational elliptic systems in dimension n greater than two. The results apply to changing sign solutions.
Develops theory of contact systems with nonholonomic constraints.
problem Nonholonomic constraints in contact systems.
method Variational principle and projection of Hamiltonian vector field.
result Nonholonomic dynamics as projection of unconstrained dynamics.
Estimates parameters of interconnected linear systems using total variation penalization.
problem Joint estimation of parameters in interconnected linear dynamical systems.
method Total variation penalized least-squares estimator.
result The MSE goes to zero as the number of systems increases, even with constant trajectory length.
In this paper, we consider a generalization of variational calculus which allows us to consider in the same framework different cases of mechanical systems, for instance, Lagrangian mechanics, Hamiltonian mechanics, systems subjected to constraints, optimal control theory and so on. This generalized variational calculu…
Switching linear dynamics improves model-based reinforcement learning and system identification.
problem Complex and nonlinear systems can be approximated by linear dynamical systems.
method Bayesian inference, Variational Autoencoders, Concrete relaxations.
result Improved accuracy in learning dynamics from partial and high-dimensional observations.
The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangi…
Paper proposes variational inference for piecewise-linear systems.
problem Intractability of switching dynamical systems.
method Variational approximation and expectation-maximization algorithms.
result Parameters can be estimated off-line, including the number of linear modes.
New method calibrates predictions in chaotic systems using variational inference.
problem Uncertainty in data assimilation for chaotic systems.
method Variational inference applied to multivariate Gaussian distribution.
result Nearly perfectly calibrated predictions in chaotic Lorenz-96 dynamics.
The elastica is a curve in R3 that is stationary under variations of the integral of the square of the curvature. Elastica is viewed as a dynamical system that arises from the second order calculus of variations, and its quantization is discussed.
Method converts sparse systems to dense ones for statistical mechanics problems.
problem Statistical mechanics on sparse graphs
method Extracts a Feedback Vertex Set, learns variational distribution, estimates free energy.
result More accurate and faster than existing methods for sparse systems.
Variational calculus on a vector bundle E equipped with a structure of a general algebroid is developed, together with the corresponding analogs of Euler-Lagrange equations. Constrained systems are introduced in the variational and in the geometrical setting. The constrained Euler-Lagrange equations are derived for ana…
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.
Lagrangian formalism on graded manifolds is phrased in terms of the Grassmann-graded variational bicomplex, generalizing the familiar variational bicomplex for even Lagrangian systems on fiber bundles.
This paper proposes using variational autoencoders to model opponents in multi-agent systems.
problem Understanding and interacting with opponents in multi-agent systems.
method Variational autoencoders for opponent modeling, with a modification to use local information.
result Our opponent modeling methods achieve equal or greater episodic returns.
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 n. Efficiently infers switching nonlinear systems with collapsed amortized variational inference.
problem Inference in switching nonlinear dynamical systems with discrete latent variables.
method Learn an inference network as a proposal for continuous latent variables, performing exact marginalization of discrete variables.
result Successfully segments time series data into meaningful regimes using piece-wise nonlinear dynamics.
A new method learns complex dynamical systems from data efficiently.
problem Learning complex dynamical systems from large-scale data efficiently.
method Low-rank structured variational autoencoding framework for nonlinear Gaussian state-space models.
result Consistently demonstrates better predictive capabilities compared to other models.
Diversifies reply suggestions for IM systems using M-CVAE.
problem Improving diversity of automated reply suggestions in instant messaging systems.
method Formulated a generative latent variable model with Conditional Variational Auto-Encoder (M-CVAE) to diversify responses.
result Increased diversity by ~30-40% without significant impact on relevance.
DVE models dynamic changes in feature embeddings for better sequence-aware applications.
problem Dynamic characterization of feature variations in time-varying data.
method Dynamic Variational Embedding (DVE) using recurrent neural networks.
result DVE models intrinsic nature and temporal variation of nodes effectively.
Alternative discrete Dirac mechanics using Dirac structures.
problem Developing a new framework for discrete mechanics.
method Introducing 'continuous Dirac system' and proposing a definition of 'discrete Dirac system'.
result It is possible to recover discrete Lagrangian and Hamiltonian systems.
The Duffing oscillator's parameters are identified online using variational message passing.
problem Estimating parameters of a nonlinear Duffing oscillator in real-time.
method Variational message passing on a factor graph of the Duffing oscillator's generative model.
result The online inference procedure performs as well as offline methods.
New algorithm learns unstable, partially observable systems.
problem Learning in unstable and partially observable Gaussian Process State-Space Models.
method Structured variational inference with efficient forward-backward pass and modified conditioning step.
result Good test performance in stable and unstable real systems with hidden states.
Variationality of conformal geodesics fails in higher dimensions.
problem The variationality of conformal geodesics in higher dimensions.
method Analysis of conformal geodesics in three and higher dimensions.
result Variationality fails in both parametrized and un-parametrized conformal geodesics in higher dimensions.
We focus on variational inference in dynamical systems where the discrete time transition function (or evolution rule) is modelled by a Gaussian process. The dominant approach so far has been to use a factorised posterior distribution, decoupling the transition function from the system states. This is not exact in gene…
In this work we show that the systems of balance equations (balance systems) of continuum thermodynamics occupy a natural place in the variational bicomplex formalism. We apply the vertical homotopy decomposition to get a local splitting (in a convenient domain) of a general balance system as the sum of a Lagrangian pa…
Optimizes variational autoencoder for detecting missing data in Mars rover transmissions.
problem Detecting missing data in Mars rover transmissions to prevent volume loss and corruption.
method Applies derivative-free optimization to tune variational autoencoder.
result Improves variational autoencoder's ability to detect missing data, aiding GDSA team.
Automates VI divergence selection for efficient few-shot learning.
problem Efficiently selecting divergence measures for VI to improve performance.
method Meta-learning algorithm to learn optimal divergence metric and variational parameter initialization.
result Meta-learning approach outperforms standard VI methods across various tasks.
This paper extends, to a class of systems of semi-linear hyperbolic second order PDEs in three variables, the geometric study of a single nonlinear hyperbolic PDE in the plane as presented in [Anderson I.M., Kamran N., Duke Math. J. 87 (1997), 265-319]. The constrained variational bi-complex is introduced and used to d…