@unpublished{59794, abstract = {{The depth of networks plays a crucial role in the effectiveness of deep learning.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.}}, author = {{Maslovskaya, Sofya and Ober-Blöbaum, Sina and Offen, Christian and Singh, Pranav and Wembe Moafo, Boris Edgar}}, title = {{{Adaptive higher order reversible integrators for memory efficient deep learning}}}, year = {{2025}}, } @article{62980, abstract = {{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.}}, author = {{Kopylov, Denis A. and Offen, Christian and Ares, Laura and Wembe Moafo, Boris Edgar and Ober-Blöbaum, Sina and Meier, Torsten and Sharapova, Polina R. and Sperling, Jan}}, issn = {{2643-1564}}, journal = {{Physical Review Research}}, number = {{3}}, publisher = {{American Physical Society (APS)}}, title = {{{Multiphoton, multimode state classification for nonlinear optical circuits}}}, doi = {{10.1103/sv6z-v1gk}}, volume = {{7}}, year = {{2025}}, } @unpublished{62979, abstract = {{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.}}, author = {{Meier, Torsten and Sharapova, Polina R. and Sperling, Jan and Ober-Blöbaum, Sina and Wembe Moafo, Boris Edgar and Offen, Christian}}, title = {{{Multiphoton, multimode state classification for nonlinear optical circuits}}}, year = {{2025}}, } @unpublished{58544, abstract = {{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. }}, author = {{Kopylov, Denis and Offen, Christian and Ares, Laura and Wembe Moafo, Boris Edgar and Ober-Blöbaum, Sina and Meier, Torsten and Sharapova, Polina and Sperling, Jan}}, title = {{{Multiphoton, multimode state classification for nonlinear optical circuits }}}, year = {{2025}}, } @article{53805, abstract = {{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.}}, author = {{Offen, Christian}}, journal = {{Mathematics of Computation}}, publisher = {{American Mathematical Society}}, title = {{{Machine learning of continuous and discrete variational ODEs with convergence guarantee and uncertainty quantification}}}, doi = {{10.1090/mcom/4120}}, year = {{2025}}, } @article{46469, abstract = {{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. }}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, issn = {{1054-1500}}, journal = {{Chaos}}, number = {{1}}, publisher = {{AIP Publishing}}, title = {{{Learning of discrete models of variational PDEs from data}}}, doi = {{10.1063/5.0172287}}, volume = {{34}}, year = {{2024}}, } @unpublished{55159, abstract = {{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.}}, author = {{Offen, Christian}}, keywords = {{System identification, inverse problem of variational calculus, Gaussian process, Lagrangian learning, physics informed machine learning, geometry aware learning}}, pages = {{28}}, title = {{{Machine learning of discrete field theories with guaranteed convergence and uncertainty quantification}}}, year = {{2024}}, } @inproceedings{34135, abstract = {{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.}}, author = {{Lishkova, Yana and Scherer, Paul and Ridderbusch, Steffen and Jamnik, Mateja and Liò, Pietro and Ober-Blöbaum, Sina and Offen, Christian}}, booktitle = {{IFAC-PapersOnLine}}, location = {{ Yokohama, Japan}}, number = {{2}}, pages = {{3203--3210}}, publisher = {{Elsevier}}, title = {{{Discrete Lagrangian Neural Networks with Automatic Symmetry Discovery}}}, doi = {{10.1016/j.ifacol.2023.10.1457}}, volume = {{56}}, year = {{2023}}, } @inproceedings{42163, abstract = {{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.}}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, booktitle = {{Geometric Science of Information}}, editor = {{Nielsen, F and Barbaresco, F}}, keywords = {{System identification, discrete Lagrangians, travelling waves}}, location = {{Saint-Malo, Palais du Grand Large, France}}, pages = {{569--579}}, publisher = {{Springer, Cham.}}, title = {{{Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves}}}, doi = {{10.1007/978-3-031-38271-0_57}}, volume = {{14071}}, year = {{2023}}, } @article{29240, abstract = {{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.}}, author = {{Ober-Blöbaum, Sina and Offen, Christian}}, issn = {{0377-0427}}, journal = {{Journal of Computational and Applied Mathematics}}, keywords = {{Lagrangian learning, variational backward error analysis, modified Lagrangian, variational integrators, physics informed learning}}, pages = {{114780}}, publisher = {{Elsevier}}, title = {{{Variational Learning of Euler–Lagrange Dynamics from Data}}}, doi = {{10.1016/j.cam.2022.114780}}, volume = {{421}}, year = {{2023}}, } @article{29236, abstract = {{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.}}, author = {{McLachlan, Robert and Offen, Christian}}, journal = {{Journal of Geometric Mechanics}}, keywords = {{variational integrators, backward error analysis, Euler--Lagrange equations, multistep methods, conjugate symplectic methods}}, number = {{1}}, pages = {{98--115}}, publisher = {{AIMS Press}}, title = {{{Backward error analysis for conjugate symplectic methods}}}, doi = {{10.3934/jgm.2023005}}, volume = {{15}}, year = {{2023}}, } @article{37654, abstract = {{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.}}, author = {{Dierkes, Eva and Offen, Christian and Ober-Blöbaum, Sina and Flaßkamp, Kathrin}}, issn = {{1054-1500}}, journal = {{Chaos}}, number = {{6}}, publisher = {{AIP Publishing}}, title = {{{Hamiltonian Neural Networks with Automatic Symmetry Detection}}}, doi = {{10.1063/5.0142969}}, volume = {{33}}, year = {{2023}}, } @article{21600, abstract = {{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.}}, author = {{Dellnitz, Michael and Hüllermeier, Eyke and Lücke, Marvin and Ober-Blöbaum, Sina and Offen, Christian and Peitz, Sebastian and Pfannschmidt, Karlson}}, journal = {{SIAM Journal on Scientific Computing}}, number = {{2}}, pages = {{A579--A595}}, title = {{{Efficient time stepping for numerical integration using reinforcement learning}}}, doi = {{10.1137/21M1412682}}, volume = {{45}}, year = {{2023}}, } @article{19941, abstract = {{In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby ‘modified’ equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.}}, author = {{McLachlan, Robert I and Offen, Christian}}, journal = {{Journal of Geometric Mechanics}}, number = {{3}}, pages = {{447 -- 471}}, publisher = {{AIMS}}, title = {{{Backward error analysis for variational discretisations of partial differential equations}}}, doi = {{10.3934/jgm.2022014}}, volume = {{14}}, year = {{2022}}, } @article{23382, abstract = {{Hamiltonian systems are differential equations which describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data needs to be approximated based on the trajectory observations. Moreover, the numerical integrator introduces an additional discretisation error. In this paper, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data avoided. The inverse modified data compensates for the discretisation error such that the discretisation error is eliminated. The technique is developed for Gaussian Processes.}}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, journal = {{Chaos: An Interdisciplinary Journal of Nonlinear Science}}, publisher = {{AIP}}, title = {{{Symplectic integration of learned Hamiltonian systems}}}, doi = {{10.1063/5.0065913}}, volume = {{32(1)}}, year = {{2022}}, } @inproceedings{22894, abstract = {{The first order optimality conditions of optimal control problems (OCPs) can be regarded as boundary value problems for Hamiltonian systems. Variational or symplectic discretisation methods are classically known for their excellent long term behaviour. As boundary value problems are posed on intervals of fixed, moderate length, it is not immediately clear whether methods can profit from structure preservation in this context. When parameters are present, solutions can undergo bifurcations, for instance, two solutions can merge and annihilate one another as parameters are varied. We will show that generic bifurcations of an OCP are preserved under discretisation when the OCP is either directly discretised to a discrete OCP (direct method) or translated into a Hamiltonian boundary value problem using first order necessary conditions of optimality which is then solved using a symplectic integrator (indirect method). Moreover, certain bifurcations break when a non-symplectic scheme is used. The general phenomenon is illustrated on the example of a cut locus of an ellipsoid.}}, author = {{Offen, Christian and Ober-Blöbaum, Sina}}, issn = {{2405-8963}}, keywords = {{optimal control, catastrophe theory, bifurcations, variational methods, symplectic integrators}}, location = {{Berlin, Germany}}, pages = {{334--339}}, title = {{{Bifurcation preserving discretisations of optimal control problems}}}, doi = {{https://doi.org/10.1016/j.ifacol.2021.11.099}}, volume = {{54(19)}}, year = {{2021}}, } @inproceedings{21572, author = {{Ridderbusch, Steffen and Offen, Christian and Ober-Blöbaum, Sina and Goulart, Paul}}, booktitle = {{2021 60th IEEE Conference on Decision and Control (CDC)}}, location = {{Austin, TX, USA}}, pages = {{2896}}, publisher = {{IEEE}}, title = {{{Learning ODE Models with Qualitative Structure Using Gaussian Processes }}}, doi = {{10.1109/CDC45484.2021.9683426}}, year = {{2021}}, } @article{19938, abstract = {{We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic on nonsymplectic integrators, but in some circumstances symplecticity greatly reduces the error. }}, author = {{McLachlan, Robert I and Offen, Christian}}, journal = {{Foundations of Computational Mathematics}}, number = {{6}}, pages = {{1363--1400}}, title = {{{Preservation of Bifurcations of Hamiltonian Boundary Value Problems Under Discretisation}}}, doi = {{10.1007/s10208-020-09454-z}}, volume = {{20}}, year = {{2020}}, } @article{19939, author = {{Kreusser, Lisa Maria and McLachlan, Robert I and Offen, Christian}}, issn = {{0951-7715}}, journal = {{Nonlinearity}}, number = {{5}}, pages = {{2335--2363}}, title = {{{Detection of high codimensional bifurcations in variational PDEs}}}, doi = {{10.1088/1361-6544/ab7293}}, volume = {{33}}, year = {{2020}}, } @phdthesis{19947, abstract = {{Ordinary differential equations (ODEs) and partial differential equations (PDEs) arise in most scientific disciplines that make use of mathematical techniques. As exact solutions are in general not computable, numerical methods are used to obtain approximate solutions. In order to draw valid conclusions from numerical computations, it is crucial to understand which qualitative aspects numerical solutions have in common with the exact solution. Symplecticity is a subtle notion that is related to a rich family of geometric properties of Hamiltonian systems. While the effects of preserving symplecticity under discretisation on long-term behaviour of motions is classically well known, in this thesis (a) the role of symplecticity for the bifurcation behaviour of solutions to Hamiltonian boundary value problems is explained. In parameter dependent systems at a bifurcation point the solution set to a boundary value problem changes qualitatively. Bifurcation problems are systematically translated into the framework of classical catastrophe theory. It is proved that existing classification results in catastrophe theory apply to persistent bifurcations of Hamiltonian boundary value problems. Further results for symmetric settings are derived. (b) It is proved that to preserve generic bifurcations under discretisation it is necessary and sufficient to preserve the symplectic structure of the problem. (c) The catastrophe theory framework for Hamiltonian ODEs is extended to PDEs with variational structure. Recognition equations for A-series singularities for functionals on Banach spaces are derived and used in a numerical example to locate high-codimensional bifurcations. (d) The potential of symplectic integration for infinite-dimensional Lie-Poisson systems (Burgers’ equation, KdV, fluid equations, . . . ) using Clebsch variables is analysed. It is shown that the advantages of symplectic integration can outweigh the disadvantages of integrating over a larger phase space introduced by a Clebsch representation. (e) Finally, the preservation of variational structure of symmetric solutions in multisymplectic PDEs by multisymplectic integrators on the example of (phase-rotating) travelling waves in the nonlinear wave equation is discussed.}}, author = {{Offen, Christian}}, publisher = {{Massey University}}, title = {{{Analysis of Hamiltonian boundary value problems and symplectic integration}}}, year = {{2020}}, }