51,490 research outputs found
Higher Order Variational Integrators: a polynomial approach
We reconsider the variational derivation of symplectic partitioned
Runge-Kutta schemes. Such type of variational integrators are of great
importance since they integrate mechanical systems with high order accuracy
while preserving the structural properties of these systems, like the
symplectic form, the evolution of the momentum maps or the energy behaviour.
Also they are easily applicable to optimal control problems based on mechanical
systems as proposed in Ober-Bl\"obaum et al. [2011].
Following the same approach, we develop a family of variational integrators
to which we refer as symplectic Galerkin schemes in contrast to symplectic
partitioned Runge-Kutta. These two families of integrators are, in principle
and by construction, different one from the other. Furthermore, the symplectic
Galerkin family can as easily be applied in optimal control problems, for which
Campos et al. [2012b] is a particular case.Comment: 12 pages, 1 table, 23rd Congress on Differential Equations and
Applications, CEDYA 201
Infinite Horizon Mean-Field Linear Quadratic Optimal Control Problems with Jumps and the related Hamiltonian Systems
In this work, we focus on an infinite horizon mean-field linear-quadratic
stochastic control problem with jumps. Firstly, the infinite horizon linear
mean-field stochastic differential equations and backward stochastic
differential equations with jumps are studied to support the research of the
control problem. The global integrability properties of their solution
processes are studied by introducing a kind of so-called dissipation conditions
suitable for the systems involving the mean-field terms and jumps. For the
control problem, we conclude a sufficient and necessary condition of open-loop
optimal control by the variational approach. Besides, a kind of infinite
horizon fully coupled linear mean-field forward-backward stochastic
differential equations with jumps is studied by using the method of
continuation. Such a research makes the characterization of the open-loop
optimal controls more straightforward and complete.Comment: 27page
Variational Inference for SDEs Driven by Fractional Noise
We present a novel variational framework for performing inference in (neural)
stochastic differential equations (SDEs) driven by Markov-approximate
fractional Brownian motion (fBM). SDEs offer a versatile tool for modeling
real-world continuous-time dynamic systems with inherent noise and randomness.
Combining SDEs with the powerful inference capabilities of variational methods,
enables the learning of representative function distributions through
stochastic gradient descent. However, conventional SDEs typically assume the
underlying noise to follow a Brownian motion (BM), which hinders their ability
to capture long-term dependencies. In contrast, fractional Brownian motion
(fBM) extends BM to encompass non-Markovian dynamics, but existing methods for
inferring fBM parameters are either computationally demanding or statistically
inefficient. In this paper, building upon the Markov approximation of fBM, we
derive the evidence lower bound essential for efficient variational inference
of posterior path measures, drawing from the well-established field of
stochastic analysis. Additionally, we provide a closed-form expression to
determine optimal approximation coefficients. Furthermore, we propose the use
of neural networks to learn the drift, diffusion and control terms within our
variational posterior, leading to the variational training of neural-SDEs. In
this framework, we also optimize the Hurst index, governing the nature of our
fractional noise. Beyond validation on synthetic data, we contribute a novel
architecture for variational latent video prediction,-an approach that, to the
best of our knowledge, enables the first variational neural-SDE application to
video perception.Comment: 24 pages, under revie
A Variational Approach to Parameter Estimation in Ordinary Differential Equations
Ordinary differential equations are widely-used in the field of systems
biology and chemical engineering to model chemical reaction networks. Numerous
techniques have been developed to estimate parameters like rate constants,
initial conditions or steady state concentrations from time-resolved data. In
contrast to this countable set of parameters, the estimation of entire courses
of network components corresponds to an innumerable set of parameters. The
approach presented in this work is able to deal with course estimation for
extrinsic system inputs or intrinsic reactants, both not being constrained by
the reaction network itself. Our method is based on variational calculus which
is carried out analytically to derive an augmented system of differential
equations including the unconstrained components as ordinary state variables.
Finally, conventional parameter estimation is applied to the augmented system
resulting in a combined estimation of courses and parameters. The combined
estimation approach takes the uncertainty in input courses correctly into
account. This leads to precise parameter estimates and correct confidence
intervals. In particular this implies that small motifs of large reaction
networks can be analysed independently of the rest. By the use of variational
methods, elements from control theory and statistics are combined allowing for
future transfer of methods between the two fields
Optimal Feedback Control of Thermal Networks
An improved approach to the mathematical modeling of feedback control of thermal networks has been devised. Heretofore software for feedback control of thermal networks has been developed by time-consuming trial-and-error methods that depend on engineers expertise. In contrast, the present approach is a systematic means of developing algorithms for feedback control that is optimal in the sense that it combines performance with low cost of implementation. An additional advantage of the present approach is that a thermal engineer need not be expert in control theory. Thermal networks are lumped-parameter approximations used to represent complex thermal systems. Thermal networks are closely related to electrical networks commonly represented by lumped-parameter circuit diagrams. Like such electrical circuits, thermal networks are mathematically modeled by systems of differential-algebraic equations (DAEs) that is, ordinary differential equations subject to a set of algebraic constraints. In the present approach, emphasis is placed on applications in which thermal networks are subject to constant disturbances and, therefore, integral control action is necessary to obtain steady-state responses. The mathematical development of the present approach begins with the derivation of optimal integral-control laws via minimization of an appropriate cost functional that involves augmented state vectors. Subsequently, classical variational arguments provide optimality conditions in the form of the Hamiltonian equations for the standard linear-quadratic-regulator (LQR) problem. These equations are reduced to an algebraic Riccati equation (ARE) with respect to the augmented state vector. The solution of the ARE leads to the direct computation of the optimal proportional- and integral-feedback control gains. In cases of very complex networks, large numbers of state variables make it difficult to implement optimal controllers in the manner described in the preceding paragraph
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