In the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics. Standing in the middle of a field, we can easily forget that we live on a round planet. We're so small in comparison to the Earth that from our point of view, it looks flat. The world is full of such shapes--ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds.

This article investigates the modeling and control of Lagrangian systems involving non-conservative forces using a hybrid method that does not require acceleration calculations. It focuses in particular on the derivation and identification of physically consistent models, which are essential for model-based control synthesis. Lagrangian or Hamiltonian neural networks provide useful structural guarantees but the learning of such models often leads to inconsistent models, especially on real physical systems where training data are limited, partial and noisy. Motivated by this observation and the objective to exploit these models for model-based nonlinear control, a learning algorithm relying on an original loss function is proposed to improve the physical consistency of Lagrangian systems. A comparative analysis of different learning-based modeling approaches with the proposed solution shows significant improvements in terms of physical consistency of the learned models, on both simulated and experimental systems. The model's consistency is then exploited to demonstrate, on an experimental benchmark, the practical relevance of the proposed methodology for feedback linearization and energy-based control techniques.

System identification in control theory aims to approximate dynamical systems from trajectory data. While neural networks have demonstrated strong predictive accuracy, they often fail to preserve critical physical properties such as stability and typically assume stationary dynamics, limiting their applicability under distribution shifts. Existing approaches generally address either stability or adaptability in isolation, lacking a unified framework that ensures both. We propose LILAD (Learning In-Context Lyapunov-stable Adaptive Dynamics), a novel framework for system identification that jointly guarantees adaptability and stability. LILAD simultaneously learns a dynamics model and a Lyapunov function through in-context learning (ICL), explicitly accounting for parametric uncertainty. Trained across a diverse set of tasks, LILAD produces a stability-aware, adaptive dynamics model alongside an adaptive Lyapunov certificate. At test time, both components adapt to a new system instance using a short trajectory prompt, which enables fast generalization. To rigorously ensure stability, LILAD also computes a state-dependent attenuator that enforces a sufficient decrease condition on the Lyapunov function for any state in the new system instance. This mechanism extends stability guarantees even under out-of-distribution and out-of-task scenarios. We evaluate LILAD on benchmark autonomous systems and demonstrate that it outperforms adaptive, robust, and non-adaptive baselines in predictive accuracy.

Abstract--Electromagnetic navigation systems (eMNS) enable a number of magnetically guided surgical procedures. A challenge in magnetically manipulating surgical tools is that the effective workspace of an eMNS is often severely constrained by power and thermal limits. We show that system-level control design significantly expands this workspace by reducing the currents needed to achieve a desired motion. We identified five key system approaches that enable this expansion: (i) motion-centric torque/force objectives, (ii) energy-optimal current allocation, (iii) real-time pose estimation, (iv) dynamic feedback, and (v) high-bandwidth eMNS components. As a result, we stabilize a 3D inverted pendulum on an eight-coil OctoMag eMNS with significantly lower currents (0.1-0.2 We generalize to multi-agent control by simultaneously stabilizing two inverted pendulums within a shared workspace, exploiting magnetic-field nonlinearity and coil redundancy for independent actuation. A structured analysis compares the electromagnetic workspaces of both paradigms and examines current-allocation strategies that map motion objectives to coil currents. Cross-platform evaluation of the clinically oriented Navion eMNS further demonstrates substantial workspace expansion by maintaining stable balancing at distances up to 50 cm from the coils. The results demonstrate that feedback is a practical path to scalable, efficient, and clinically relevant magnetic manipulation. A video presenting our approach is available at https://youtu.be/PQeAKPL_iS0. Magnetic navigation systems are rapidly emerging as a key technology in medical robotics, enabling breakthroughs from precision drug delivery to sophisticated endoscopic procedures [1]-[3]. These systems act on nanometer to centimeter scales and encompass both soft and hard magnetomagnetic materials [4], [5]. Michael Muehlebach is with the Learning and Dynamical Systems Group, Max Planck Institute for Intelligent Systems, 72076 T ubingen, Germany (email: michael.muehlebach@tuebingen.mpg.de). We balance two 3D inverted pendulums simultaneously within the same magnetic workspace, leveraging the magnetic field created by the OctoMag eMNS. Because both pendulums are identical, independent actuation under a global field requires exploiting the nonlinearity of the magnetic field. This setup is used as an experimental platform to compare different strategies for multi-agent control. Each inverted pendulum system includes an arm driven by the external magnetic field and a non-magnetic pendulum. Balancing two inverted pendulums within the same magnetic workspace is challenging due to coupling effects not only between each coil and the permanent magnets, but also between the magnets themselves.

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We thank the reviewers for their close reading of the paper and helpful feedback. We are also excited to apply ideas from DualDICE to the policy improvement problem, as mentioned by the reviewers. We are exploring several potential approaches to this problem. How are these assumptions handled practically, e.g. What are the x-axes in figure 2 and figure 4? We will remedy this in the final draft.

Data-driven discovery of governing equations from data remains a fundamental challenge in nonlinear dynamics. Although sparse regression techniques have advanced system identification, they struggle with rational functions and noise sensitivity in complex mechanical systems. The Lagrangian formalism offers a promising alternative, as it typically avoids rational expressions and provides a more concise representation of system dynamics. However, existing Lagrangian identification methods are significantly affected by measurement noise and limited data availability. This paper presents a novel differentiable sparse identification framework that addresses these limitations through three key contributions: (1) the first integration of cubic B-Spline approximation into Lagrangian system identification, enabling accurate representation of complex nonlinearities, (2) a robust equation discovery mechanism that effectively utilizes measurements while incorporating known physical constraints, (3) a recursive derivative computation scheme based on B-spline basis functions, effectively constraining higher-order derivatives and reducing noise sensitivity on second-order dynamical systems. The proposed method demonstrates superior performance and enables more accurate and reliable extraction of physical laws from noisy data, particularly in complex mechanical systems compared to baseline methods.

The only learnable parameters (117,963 in all) in our model are those of the autoencoder. The decoder is a symmetric copy of the encoder. Models without updates take 2.5 hours to train on a Tesla V100-SXM2 GPU, and models with updates Measurements (i.e, images) are taken every Our model does not exhibit such limitations. Figure 3 shows how the baseline model is unable to predict future frames correctly, for even a single step in the future (first frame of the left block), when it is trained on a dataset with multiple pendulums. In the case of this simple system, our model without updates is enough. 2 Figure 2: The first row shows ground truth (GT) images.

Identifying an effective model of a dynamical system from sensory data and using it for future state prediction and control is challenging.

The approach works by jointly learning a dynamics model and Lyapunov function that guarantees non-expansiveness of the dynamics under the learned Lyapunov function.