GROUP SE­MI­NAR

Title: TBD

Abstract: TBD

Title: Cayley based methods for quantum control systems

Abstract: Optimal control of quantum systems typically requires the use of suitable methods for integrating the underlying Hamiltonian system, in order to preserve the geometric structure of the solution. Although many Lie integrators for solving quantum systems based on the exponential function exist (splitting methods, Magnus development, etc.), in the context of quantum optimal control, they often involve high computational costs due to the optimization, which requires a consequent number of system integrations. In this work, we introduce a “commutator-free” approach based on the Cayley transform for solving quantum optimal control problems governed by Schrödinger's equation.  

Title: Optimal control of second order systems

Asbract: In this talk, we consider the class of optimal control problems, where the control system is given by a controlled second order differential equation. Such control systems naturally appear in mechanics in the form of controlled Euler-Lagrange equations. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem, or alternatively, a variational approach based on an augmented objective for smooth problems. We propose a new Lagrangian approach leading to equivalent optimality conditions in the form of Euler-Lagrange equations and show its geometrical characterization. As a result, the differential geometric structure (similar to classical Lagrangian dynamics) can be exploited in the framework of optimal control problems. In particular, the formulation enables the symplectic discretization of the optimal control problem via variational integrators in a straightforward way.

Title: Neural networks enhanced integrators for systems defined by ordinary differential equations

Abstract: Many applications require numerical solutions to differential equations for a large number of initial conditions and/or system parameters. For example, the analysis of fatigue effects and lifetime prediction of technological systems such as wind energy converters (WECs) often requires a comparison of design site conditions with real site conditions by simulating models of WECs for a large number of different conditions. This contribution evaluates the effectiveness of neural network (NN) enhanced integrators. NNs learn the integration errors, the approximation of which are then used as an additive correction term for the numerical schemes. The resulting integrators are compared with well-established methods in numerical studies, with a particular focus on computational requirements. The analytical properties will be addressed in terms of local errors. Classical Runge-Kutta methods and symplectic integrators are considered.

Title: A new perspective on spectral methods

Abstract: In this talk we describe a new perspective on spectral methods for time-dependent PDEs. Basically, given an equation evolving in a separable Hilbert space, a spectral method is no more than a choice of an orthonormal basis. The choice of such a basis is governed by a raft of considerations: stability, speed of convergence, structure preservation and the ease of numerical algebra. 

Focussing on a single space dimension, we distinguish between two cases: T-systems and W-systems. T-systems are defined on L2(R), can be characterised completely and possess a tridiagonal, skew-Hermitian differentiation matrix: this renders linear algebra very easy. W-systems act on L2(a, b), where (a, b) ⊂ R, and are defined directly from orthogonal polynomials. In their case the differentiation matrix is semi-separable of rank 1, again yielding itself to rapid linear algebra. 

We describe the state of the art with both types of systems, with an emphasis on their approximation-
theoretic features. Time allowing, we will mention a generalisation to Galerkin–Petrov-type methods and to multivariate setting.

Title: Turnpike in Kepler problem

Abstract: The turnpike property in optimal control characterizes the quasi-static behavior of solutions. More precisely, the solutions in this case converge to a steady state or a specific trajectory when the problem is considered on a large time interval. In this talk we show the results of numerical resolution of optimal control problem with the control system given by controlled Hamiltonian equations of Kepler problem. The turnpike in this case is given by a circular orbit. We compare the turnpike obtained with different numerical method to conclude on the influence of the discretization on the turnpike.

Title:  Trajectory optimization in space mechanics with Julia

Abstract: TBA

Title: Higher order fractional variational integrators

Abstract: Fractional dissipation is a powerful tool to study non-local physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. A new approach is proposed by Sina Ober-Blöbaum and ernando Jiménez to deal with dissipative systems in a variational way for both, the continuous and discrete setting. It is based on the doubling of variables and their fractional derivatives. In this contribution, we derive fractional variational integrators of order 2 based on BDF and RK convolution quadratures, then study the numerical properties of those integrators.

Title: Reversible numerical methods in deep learning

Abstract: Deep learning proved to be efficient in different learning tasks ranging from image classification to solving and learning dynamical systems. However, in many real life applications, the use of deep networks is restricted due to the high memory costs. This problem can be solved by considering reversible networks. Reversible networks are based on the neural ODE approach to network architecture. In this case, propagation through layers in deep learning is interpreted as an integration of a dynamical system. This approach permits the use of numerical analysis in order to obtain certain desired properties of the network. The reversible numerical methods have the property that a step forward commutes with the step backward. As a result, computation of gradients does not require saving features during the forward propagation, because the same features can be obtained by the integration backward during the propagation backward phase. This leads to lower memory costs of such networks. However, there is a lack of reversible methods allowing adaptivity. Currently, only second order methods are known. In learning tasks related to dynamical systems, higher order methods are usually more accurate and fast, which makes it important to construct higher order methods. In this work, we show a construction method for higher order reversible methods allowing adaptivity based on composition. We use this construction for reversible network architecture and test it on different examples of learning the dynamical systems. In particular, we show the advantages of the higher order methods in comparison to a known second order reversible network.

Title: Algorithms for Hamiltonian Simulation and Optimal Control

Abstract: The precision and efficiency of quantum simulations and controls are vital for the advancement of quantum technologies. This talk presents three recent developments in algorithms for quantum spin systems and broader Hamiltonian systems. Firstly, we introduce a fourth-order Magnus-based algorithm for simulating many-body systems under the presence of highly-oscillatory time-dependent pulses. These integrators achieve high accuracy despite taking large time-steps, which corresponds to faster computation on classical computers and shorter circuit depths on quantum computers, making our algorithm a suitable candidate on near-term quantum devices.  Secondly, we introduce a method for optimal control of spins utilizing analytical first derivatives and approximation of second derivative using Gauss–Newton method, facilitating efficient quantum gate design. Finally, we discuss a novel iterative linearization approach for solving nonlinear dispersive equations, ensuring the preservation of structural properties such as the L2 norm, momentum, and Hamiltonian energy.

Title: Analysis of 1-particle radial Schrödinger equation with coulomb potential and its application to analyse a two particle quantum system

Abstract: Using the work of Klein and Agmon on Schrödinger operator with long range potential, we investigate the incoming, outgoing and asymptotic solution of Schrödinger equation for Hydrogen Atom and apply this theory to the analysis of Helium-type atoms realized as tensor product of two hydrogen atoms.

Title: Reversible methods in deep learning

Abstract: Deep learning is widely used in practical data analysis. It was shown to be especially successful in such tasks as image recognition and generation, in learning dynamical systems from data and more. Many applications require safety in learning and therefore it is important to develop theoretically based architectures. There has been considerable progress in this direction lately, where one of the approaches is to look at deep learning through the lens of dynamical systems. The propagation through layers in deep learning mimics numerical flow of a control system, where the control must be optimal to fit the data. This permits the use of methods from control theory and numerical analysis to design new architectures. It was recently shown that the combination of reversible discretizations together with the calculation of the gradients using the adjoint method lead to considerable reduction of the memory cost while computing correct gradients. On the other hand, it was shown that higher order methods are more accurate in the learning tasks related to dynamical systems. Only lower order reversible methods applicable to deep learning being known, it is important to also construct higher order methods. In this work we construct two classes of higher order reversible methods based on splitting and show their efficiency in examples.

Title: Commutator-free based on Cayley transform for quantum optimal control problems

Abstract: Controlling complex quantum dynamics is a recurring theme in many different fields of atomic, molecular, and optical physics and physical chemistry. Recent examples include quantum state preparation, imaging or reaction control, Boson Sampling. The central idea of quantum control is to employ external fields to steer the dynamics in a desired way (see for instance C.P Koch et al. 2022). The development of fast, robust and accurate methods for the optimal control of quantum systems comprising interacting particles is one of the most active topics. Although there is a significant set of algorithms for numerical applications in the field of quantum control, the high computational cost remains a challenge for many of them. Indeed, one of the main issues is often how to efficiently integrate controlled dynamics (control necessarily implies a time-dependent Hamiltonian). Methods based on Magnus expansion coupled with splitting methods yield pretty good results in this context, although these involve the computation of commutators (from the Magnus expansion) and several matrix exponential, which are generally costly. Our aim is then to use the Cayley transform coupled with the Magnus expansion and the commutator-free approach introduced in Alvermann & Fehske 2011 to provide high-order methods that avoid both commutator and matrix exponential computations. The use of Cayley transform being possible thanks to the nice quadratic property of quantum operators.

Title: Learning of Lagrangian dynamics from data with uncertainty quantification

Abstract: I will show how to use Gaussian Process regression to learn variational dynamical systems from data. From the statistical framework uncertainty quantification for observables such as the Euler-Lagrange operator and Hamiltonians can be derived. Examples include variational odes and pdes.

Title: Symplectic methods in deep learning

Abstract: Deep Learning is widely used in tasks including image recognition and generation, in learning dynamical systems from data and many more. It is important to construct learning architectures with theoretical guarantees to permit safety in the applications. There has been considerable progress in this direction lately. In particular, symplectic networks were shown to have the non vanishing gradient property, essential for numerical stability. On the other hand, architectures based on higher order numerical methods were shown to be efficient in many tasks where the learned function has an underlying dynamical structure. In this work we construct symplectic networks based on higher order explicit methods with non vanishing gradient property and test their efficiency on various examples.

Title: The inverse problem of variational calculus and machine learning

Abstract: The identification of equations of motions and variational principles for dynamical systems from data is an important task in the context of system identification. I will show a framework to identify variational principles from discrete observations of motion data in an ode and pde context. Indeed, a method based on Lie group theory allows us to identify variational symmetries and conservation laws and correctly predict symmetric solutions, such as travelling waves, even if no travelling waves are part of the training data.

Title: Fractional variational integrators based on convolution quadrature methods.

Abstract: Fractional dissipation is a powerful tool to study non-local physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. A new approach is proposed by F. Jiménez and S. Ober-Blöbaum (Fractional damping through restricted calculus of
variations, 2021)  to deal with dissipative systems including fractionally damped systems in a variational way for both, the continuous and discrete setting. It is based on the doubling of variables and their fractional derivatives. We use convolution quadratures (CQ) for which we derive higher-order fractional variational integrators and then provide numerical schemes that are of order 2 improving a previous result of Jiménez and S. Ober-Blöbaum.

Title: Trim turnpike for control systems with symmetries

Abstract: Nowadays, the turnpike phenomenon is a well-known concept in optimal control, which has been the subject of several studies over the last decade. This notion, first introduced in economics, refers to the particular structure of certain solutions that remain close to a "turnpike" most of the time except at the beginning and at the end. Most of recent works on turnpikes in optimal control focuses on steady-state turnpikes i.e. on problems where the turnpike can be understood as the steady-state equilibrium of infinite-horizon optimal solutions. In this context, one of the most important results is the "exponential turnpike theorem", which gives precision on the exponential convergence of the solution to the steady-state turnpike. However, it is not always possible to have a "static turnpikes", especially for systems with symmetries which we are interested in. The goal of our study in then to analyze the "turnpike phenomena" for symmetric optimal control problems and put the right frame to extend the "exponential turnpike theorem" in this case. We will also discuss on some example the main future of this case. 

Title: Stability in numerical optimal control

Abstract: Optimal control problems are omnipresent in different areas of engineering and were recently discovered to be useful in machine learning. Solutions are usually found using numerical methods. This makes the stability in numerical optimal control essential to ensure safety in applications. In this talk we will see how the implicit Euler method affects the stability in comparison to the explicit Euler method and the Runge Kutta 4 method. In addition, we test the effects of the internal stability of the control system on numerical stability. Our study suggests that the stability of the control system has the major effect on the numerical results.

Title: Curvature and stability of quasi-geostrophic motion.

Abstract: In this talk, first we restate the necessary background about quantomorphism group and central extension of the corresponding Lie algebra. Then we derive the quasi-geostrophic equation as an Euler-Arnold equation of the L^2 metric on this central extension. Afterwards, using the Lie algebra structure constants, we study the curvature and its impact on the stability of geodesics which are solutions of the quasi-geostrophic equation.  

Title: Learning of symmetric models for variational dynamical systems from data.

Abstract: Equations of motions of variational dynamical systems can be derived from an action functional defined by a Lagrangian. When the Lagrangian is not known, it can be identified from dynamical data using machine learning techniques. However, Lagrangians are not uniquely determined by the dynamics. In this talk, I will show a framework to learn symmetric models of Lagrangians. The system’s symmetries and conservation laws do not need to be known a priori but are identified automatically based on a Lie group framework. Learning symmetric over non-symmetric Lagrangians improves qualitative aspects of the model, helps the numerical integration of the data-driven model, and informs the user about important geometric properties of the system.

Title:  Turnpikes in optimal control problems with symmetries.

Abstract:  The turnpike property characterizes a quasi-static behavior of solutions of optimal control problems defined on a large time horizon. In this case, the solutions converge to a neighborhood of a steady state and stay there for a major part of the time interval. In many practical examples in mechanical and biological systems, the convergence is not towards a steady state but some attracting trajectory instead. It was recently shown that for the mechanical systems admitting a symmetry with respect to an Abelian group action, these trajectories correspond to the relative equilibria of the system. In this talk we will show how this property of optimal control for mechanical systems generalizes to non Abelian group actions.

Title: Enforcing Safety in Airborne Wind Energy Systems.

Abstract: 

For Airborne Wind Energy Systems (AWES) to become widely adopted, it is necessary to guarantee continuous autonomous operation. To this end, control systems for airborne wind need to be able to adapt to changing conditions and incorporate safety without greatly affecting power generation. In the past safety control laws have been introduced to flight control systems by means of Hamilton Jacobi reachability [Vertovec 2022a] and in conjunction with power optimal control techniques by means of hybrid control laws [Vertovec 2022b]. However, the necessary switching conditions for hybrid control laws are greatly dependent on accurate and sufficient tuning. An alternative approach to deriving a switching law between a power optimal and a safety controller is to enforce safety by means of a control barrier function (CBF).

CBFs have in the past been successfully deployed to various robotics applications and present an optimal strategy of adapting an existing control input in a minimal fashion so as to enforce safety [Ames 2019]. However, one of the major difficulties of applying control barrier functions lies in finding the necessary Lyapunov-like barrier certificate. Due to the complexity of AWES it has in the past been difficult to derive the necessary control barrier function that ensures safe flight and avoids, e.g., a tether rupture. To this end, we decouple the winch controller from the flight controller and present a barrier certificate that successfully prevents the possibility of the tether rupturing.

In addition to presenting this strategy of safe winch control, we discuss more generally how safety in AWES can be embedded in the control design, and where some of the key outstanding challenges lie.

Title: Perturbation Theory for Pseudo-Laplacians

Title: Turnpike properties in optimal control problems with symmetry

Abstract: In optimal control problems (OCP), the turnpike property refers to the phenomenon in which the solutions of an optimal control problem settled in large time stay in a long-time arc staying close to the optimal steady state solution for the majority of the time. When the OCP, whose dynamical system and the cost function are endowed with an additional structure such as a Lie group symmetry, it may offer enriched insights about the system such as turnpike properties of the OCP. In this talk, I will present the framework of some possible ways to exploit symmetry to study turnpike properties of the OCP that Sofya Maslovkaya, Boris Wembe and I initiated.

Title: State Aggregation for Distributed Value Iteration in Dynamic Programming

Abstract: We propose a distributed algorithm to solve a dynamic programming problem with multiple agents, where each agent has only partial knowledge of the state transition probabilities and costs. We provide consensus proofs for the presented algorithm and derive error bounds of the obtained value function with respect to what is considered as the ”true solution” obtained from conventional value iteration. To minimize communication overhead between agents, state costs are aggregated and shared between agents only when the updated costs are expected to influence the solution of other agents significantly. We demonstrate the efficacy of the proposed distributed aggregation method to a large-scale urban traffic routing problem. Individual agents aim at reaching a common access point and share local congestion information with other agents allowing for fully distributed routing with minimal communication between them.

Title: Embedding Formalism, Noether’s Theorem on Time Scales and Eringen’s Nonlocal Elastica

Abstract: Many fields of applied mathematics deal with the construction of a discrete analogue of a continuous notion, this procedure is typically used in numerical analysis. We define a discrete analogue of Lagrangian systems in the framework of discrete embeddings in order to derive variational integrators. Symmetries play a crucial role in physics and in mathematics and in particular for the study of differential equations. Lagrangian systems take a special place in this setting due to Noether’s theorem first stated in 1918. We establish a time scales version of the Noether theorem relating group of symmetries and conservation laws in the framework of ∆ calculus of variations. Eringen’s nonlocal elastica is a classical equation of mechanics studied first by N. Challamel et al. and suggested to us by N. Challamel from Bretagne Sud University. We construct a discrete version of Eringen’s nonlocal elastica and we study the difference with Challamel’s proposal.

Title: Output reference tracking using funnel pre-compensator

Abstract: We study output reference tracking for systems with high relative degree. The well-known funnel control technique achieves output tracking with transient behaviour of the tracking error, i.e., the tracking error evolves within prescribed bounds. The funnel control law, however, involves higher derivatives of the system’s output, which are not always available. To overcome this, in 2018 Berger and Reis introduced the so-called funnel pre-compensator. It is a dynamical system, placed between the system’s output and the controller. The funnel pre-compensator approximates the system’s output with arbitrary accuracy and its derivatives are known explicitly. Using this combination, output reference tracking is possible for systems with high relative degree without involving higher derivatives of the system’s output.

Title: Learning Hamiltonian Systems and Symmetries

Abstract: During the last years, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning dynamical systems. Hereby, the symplectic system structure is      preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance the HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows to simultaneously learn the symmetry group action and the total energy of the system.

Title: On Machine Learning for Control and Multiobjective Optimization for Machine Learning

Abstract: The control of complex dynamical systems is of great importance in many different engineering applications, for example, in autonomous driving or in the control of combustion processes in power plants. In these areas, model predictive control is often used due to the simple implementation and the high control quality. In recent years, machine learning gained increased attention and found its way also to the area of dynamical systems and control. One possible application is the building of surrogate models used within MPC. This topic will be addressed in the first part of the talk. More specifically, we will discuss how autonomous models, i.e., models that do not get the current control as input, can be used to build approximative models for the full system. In the second part of the talk, we will switch to the training of neural networks. More specifically, we will consider regularized problems with nonsmooth regularization terms as they occur, for instance, in the training of neural networks to avoid overfitting. To this end, we will use techniques from multiobjective optimization.

Title: Funnel-MPC

Abstract: Model Predictive Control (MPC) is nowadays a widely used control technique for linear and nonlinear systems due to its ability to handle multi-input multi-output systems under control and state constraints. However, for the usage of MPC initial and recursive feasibility have to be ensured. Furthermore, it is inherently necessary to have a sufficiently accurate model to predict the system behavior and compute an optimal control signal. Complementary, funnel control is an adaptive high-gain output-error feedback. Under certain structural assumptions this concept guarantees the tracking of a prescribed reference signal within predefined bounds. In this talk we present the new concept Funnel-MPC, which combines both approaches, i.e. Model Predictive control and funnel control. The optimal control problem solved in each iteration of Funnel MPC resembles the basic idea of penalty methods used in optimization. Utilizing a stage cost design which mimics the high-gain idea of funnel control, this new approach allows guaranteed output tracking of smooth reference signals with prescribed performance for multiinput multi-output systems.