On the variational discretisation of optimal control problems for unconstrained Lagrangian dynamics - Konopik, Michael; Leyendecker, Sigrid; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Sato Martín de Almagro, Rodrigo T. (Multibody System Dynamics)
We discretise a recently proposed new Lagrangian approach to optimal control problems with dynamics described by force-controlled Euler-Lagrange equations (Konopik et al., in Nonlinearity 38:11, 2025). The resulting discretisations are in the form of discrete Lagrangians. We show that the discrete necessary conditions for optimality obtained provide variational integrators for the continuous problem, akin to Karush-Kuhn-Tucker (KKT) conditions for standard direct approaches. This approach paves the way for the use of variational error analysis to derive the order of convergence of the resulting numerical schemes for both state and costate variables and to apply discrete Noether’s theorem to compute conserved quantities, distinguishing itself from existing geometric approaches. We show for a family of low-order discretisations that the resulting numerical schemes are ‘doubly-symplectic’, meaning they yield forced symplectic integrators for the underlying controlled mechanical system and overall symplectic integrators in the state-adjoint space. Multi-body dynamics examples are solved numerically using the new approach. In addition, the new approach is compared to standard direct approaches in terms of computational performance and error convergence. The results highlight the advantages of the new approach, namely, better performance and convergence behaviour of state and costate variables consistent with variational error analysis and automatic preservation of certain first integrals.

Trim turnpikes for optimal control problems with symmetries - Flaßkamp, Kathrin; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Wembe Moafo, Boris Edgar (Mathematics of Control, Signals, and Systems)
Abstract Motivated by mechanical systems with symmetries, we focus on optimal control problems possessing certain symmetries. Following recent works (Faulwasser in Math Control Signals Syst 34:759–788 2022; Trélat in Math Control Signals Syst 35:685–739 2023), which generalized the classical concept of static turnpike to manifold turnpike we extend the exponential turnpike property to the exponential trim turnpike for control systems with symmetries induced by abelian or non-abelian groups. Our analysis is mainly based on the geometric reduction of control systems with symmetries. More concretely, we first reduce the control system on the quotient space and state the turnpike theorem for the reduced problem. Then we use the group properties to obtain the trim turnpike theorem for the full problem. Finally, we illustrate our results on the Kepler problem and the rigid body problem.

Adaptive higher order reversible integrators for memory efficient deep learning - Maslovskaya, Sofya; Ober-Blöbaum, Sina; Offen, Christian; Singh, Pranav; Wembe Moafo, Boris Edgar
The depth of networks plays a crucial role in the effectiveness of deep learning. However, the memory requirement for backpropagation scales linearly with the number of layers, which leads to memory bottlenecks during training. Moreover, deep networks are often unable to handle time-series data appearing at irregular intervals. These issues can be resolved by considering continuous-depth networks based on the neural ODE framework in combination with reversible integration methods that allow for variable time-steps. Reversibility of the method ensures that the memory requirement for training is independent of network depth, while variable time-steps are required for assimilating time-series data on irregular intervals. However, at present, there are no known higher-order reversible methods with this property. High-order methods are especially important when a high level of accuracy in learning is required or when small time-steps are necessary due to large errors in time integration of neural ODEs, for instance in context of complex dynamical systems such as Kepler systems and molecular dynamics. The requirement of small time-steps when using a low-order method can significantly increase the computational cost of training as well as inference. In this work, we present an approach for constructing high-order reversible methods that allow adaptive time-stepping. Our numerical tests show the advantages in computational speed when applied to the task of learning dynamical systems.

Multiphoton, multimode state classification for nonlinear optical circuits - Kopylov, Denis A.; Offen, Christian; Ares, Laura; Wembe Moafo, Boris Edgar; Ober-Blöbaum, Sina; Meier, Torsten; Sharapova, Polina R.; Sperling, Jan (Physical Review Research)
We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which is in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes.

Second order optimality conditions in a new Lagrangian formulation for optimal control problems - Konopik, Michael; Leyendecker, Sigrid; Maslovskaya, Sofya; Sina Ober-Blöbaum, Sina Ober-Blöbaum; Almagro, Rodrigo T. Sato Martín de (arXiv:2507.06024)
It has been shown recently that optimal control problems with the dynamical constraint given by a second order system admit a regular Lagrangian formulation. This implies that the optimality conditions can be obtained in a new form based on the variational approach. In this paper we extend the first order necessary optimality conditions obtained previously to second order optimality conditions. This results in a complete characterization of the optimality conditions in a new Lagrangian form.

Variational integrators for a new Lagrangian approach to control affine systems with a quadratic Lagrange term - Konopik, Michael; T. Sato Martín de Almagro, Rodrigo; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Leyendecker, Sigrid (Journal of Nonlinear Science)

A new Lagrangian approach to optimal control of second-order systems - Konopik, Michael; Leyendecker, Sigrid; Maslovskaya, Sofya; Ober-Blöbaum, Sina; T. Sato Martín de Almagro, Rodrigo (Nonlinearity)

Multiphoton, multimode state classification for nonlinear optical circuits - Kopylov, Denis; Offen, Christian; Ares, Laura; Wembe Moafo, Boris Edgar; Ober-Blöbaum, Sina; Meier, Torsten; Sharapova, Polina; Sperling, Jan
We introduce a new classification of multimode states with a fixed number of photons. This classification is based on the factorizability of homogeneous multivariate polynomials and is invariant under unitary transformations. The classes physically correspond to field excitations in terms of single and multiple photons, each of which being in an arbitrary irreducible superposition of quantized modes. We further show how the transitions between classes are rendered possible by photon addition, photon subtraction, and photon-projection nonlinearities. We explicitly put forward a design for a multilayer interferometer in which the states for different classes can be generated with state-of-the-art experimental techniques. Limitations of the proposed designs are analyzed using the introduced classification, providing a benchmark for the robustness of certain states and classes.

Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification - Offen, Christian (Mathematics of Computation)
The article introduces a method to learn dynamical systems that are governed by Euler–Lagrange equations from data. The method is based on Gaussian process regression and identifies continuous or discrete Lagrangians and is, therefore, structure preserving by design. A rigorous proof of convergence as the distance between observation data points converges to zero and lower bounds for convergence rates are provided. Next to convergence guarantees, the method allows for quantification of model uncertainty, which can provide a basis of adaptive sampling techniques. We provide efficient uncertainty quantification of any observable that is linear in the Lagrangian, including of Hamiltonian functions (energy) and symplectic structures, which is of interest in the context of system identification. The article overcomes major practical and theoretical difficulties related to the ill-posedness of the identification task of (discrete) Lagrangians through a careful design of geometric regularisation strategies and through an exploit of a relation to convex minimisation problems in reproducing kernel Hilbert spaces.

A new Lagrangian approach to control affine systems with a quadratic Lagrange term - Leyendecker, Sigrid; Maslovskaya, Sofya; Ober-Blöbaum, Sina; Almagro, Rodrigo T. Sato Martín de; Szemenyei, Flóra Orsolya (Journal of Computational Dynamics)
In this work, we consider optimal control problems for mechanical systems with fixed initial and free final state and a quadratic Lagrange term. Specifically, the dynamics is described by a second order ODE containing an affine control term. Classically, Pontryagin's maximum principle gives necessary optimality conditions for the optimal control problem. For smooth problems, alternatively, a variational approach based on an augmented objective can be followed. Here, we propose a new Lagrangian approach leading to equivalent necessary optimality conditions in the form of Euler-Lagrange equations. Thus, 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 discretisation of the optimal control problem via variational integrators in a straightforward way.
Keywords: Optimal control problem; Lagrangian system; Hamiltonian system; Variations; Pontryagin's maximum principle.

Learning of discrete models of variational PDEs from data - Offen, Christian; Ober-Blöbaum, Sina (Chaos)
We show how to learn discrete field theories from observational data of fields on a space-time lattice. For this, we train a neural network model of a discrete Lagrangian density such that the discrete Euler--Lagrange equations are consistent with the given training data. We, thus, obtain a structure-preserving machine learning architecture. Lagrangian densities are not uniquely defined by the solutions of a field theory. We introduce a technique to derive regularisers for the training process which optimise numerical regularity of the discrete field theory. Minimisation of the regularisers guarantees that close to the training data the discrete field theory behaves robust and efficient when used in numerical simulations. Further, we show how to identify structurally simple solutions of the underlying continuous field theory such as travelling waves. This is possible even when travelling waves are not present in the training data. This is compared to data-driven model order reduction based approaches, which struggle to identify suitable latent spaces containing structurally simple solutions when these are not present in the training data. Ideas are demonstrated on examples based on the wave equation and the Schrödinger equation.

Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification - Offen, Christian
We introduce a method based on Gaussian process regression to identify discrete variational principles from observed solutions of a field theory. The method is based on the data-based identification of a discrete Lagrangian density. It is a geometric machine learning technique in the sense that the variational structure of the true field theory is reflected in the data-driven model by design. We provide a rigorous convergence statement of the method. The proof circumvents challenges posed by the ambiguity of discrete Lagrangian densities in the inverse problem of variational calculus. Moreover, our method can be used to quantify model uncertainty in the equations of motions and any linear observable of the discrete field theory. This is illustrated on the example of the discrete wave equation and Schrödinger equation. The article constitutes an extension of our previous article arXiv:2404.19626 for the data-driven identification of (discrete) Lagrangians for variational dynamics from an ode setting to the setting of discrete pdes.
Keywords: System identification; inverse problem of variational calculus; Gaussian process; Lagrangian learning; physics informed machine learning; geometry aware learning

Symplectic Methods in Deep Learning - Maslovskaya, Sofya; Ober-Blöbaum, Sina (IFAC-PapersOnLine)

Inverse optimal control problem in the non autonomous linear-quadratic case - Jean, Frédéric; Maslovskaya, Sofya

Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery - Lishkova, Yana; Scherer, Paul; Ridderbusch, Steffen; Jamnik, Mateja; Liò, Pietro; Ober-Blöbaum, Sina; Offen, Christian (IFAC-PapersOnLine)
By one of the most fundamental principles in physics, a dynamical system will exhibit those motions which extremise an action functional. This leads to the formation of the Euler-Lagrange equations, which serve as a model of how the system will behave in time. If the dynamics exhibit additional symmetries, then the motion fulfils additional conservation laws, such as conservation of energy (time invariance), momentum (translation invariance), or angular momentum (rotational invariance). To learn a system representation, one could learn the discrete Euler-Lagrange equations, or alternatively, learn the discrete Lagrangian function Ld which defines them. Based on ideas from Lie group theory, in this work we introduce a framework to learn a discrete Lagrangian along with its symmetry group from discrete observations of motions and, therefore, identify conserved quantities. The learning process does not restrict the form of the Lagrangian, does not require velocity or momentum observations or predictions and incorporates a cost term which safeguards against unwanted solutions and against potential numerical issues in forward simulations. The learnt discrete quantities are related to their continuous analogues using variational backward error analysis and numerical results demonstrate the improvement such models can have both qualitatively and quantitatively even in the presence of noise.

Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves - Offen, Christian; Ober-Blöbaum, Sina (Geometric Science of Information)
The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.
Keywords: System identification; discrete Lagrangians; travelling waves

Variational Learning of Euler–Lagrange Dynamics from Data - Ober-Blöbaum, Sina; Offen, Christian (Journal of Computational and Applied Mathematics)
The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do not need to be observed or approximated and no prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.
Keywords: Lagrangian learning; variational backward error analysis; modified Lagrangian; variational integrators; physics informed learning

Backward error analysis for conjugate symplectic methods - McLachlan, Robert; Offen, Christian (Journal of Geometric Mechanics)
The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.
Keywords: variational integrators; backward error analysis; Euler--Lagrange equations; multistep methods; conjugate symplectic methods

Hamiltonian Neural Networks with Automatic Symmetry Detection - Dierkes, Eva; Offen, Christian; Ober-Blöbaum, Sina; Flaßkamp, Kathrin (Chaos)
Recently, Hamiltonian neural networks (HNN) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian 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. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered.

Efficient time stepping for numerical integration using reinforcement learning - Dellnitz, Michael; Hüllermeier, Eyke; Lücke, Marvin; Ober-Blöbaum, Sina; Offen, Christian; Peitz, Sebastian; Pfannschmidt, Karlson (SIAM Journal on Scientific Computing)
Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it results in prohibitively large errors or computational effort. To this end, adaptive schemes, such as solvers based on Runge–Kutta pairs, have been developed which adapt the step size based on local error estimations at each step. While the classical schemes apply very generally and are highly efficient on regular systems, they can behave suboptimally when an inefficient step rejection mechanism is triggered by structurally complex systems such as chaotic systems. To overcome these issues, we propose a method to tailor numerical schemes to the problem class at hand. This is achieved by combining simple, classical quadrature rules or ODE solvers with data-driven time-stepping controllers. Compared with learning solution operators to ODEs directly, it generalizes better to unseen initial data as our approach employs classical numerical schemes as base methods. At the same time it can make use of identified structures of a problem class and, therefore, outperforms state-of-the-art adaptive schemes. Several examples demonstrate superior efficiency. Source code is available at https://github.com/lueckem/quadrature-ML.