The framework of multi-agent learning explores the dynamics of how an agent's

Curvature information -- particularly, the largest eigenvalue of the lossHessian, known as the sharpness -- often forms the basis for learning ratetuners. However, recent work has shown that the curvature information undergoescomplex dynamics during training, going from a phase of increasing sharpness toeventual stabilization. We analyze the closed-loop feedback effect betweenlearning rate tuning and curvature. We find that classical learning rate tunersmay yield greater one-step loss reduction, yet they ultimately underperform inthe long term when compared to constant learning rates in the full batch regime.These models break the stabilization of the sharpness, which we explain using asimplified model of the joint dynamics of the learning rate and the curvature.To further investigate these effects, we introduce a new learning rate tuningmethod, Curvature Dynamics Aware Tuning (CDAT), which prioritizes long termcurvature stabilization over instantaneous progress on the objective. In thefull batch regime, CDAT shows behavior akin to prefixed warm-up schedules on deeplearning objectives, outperforming tuned constant learning rates. In the minibatch regime, we observe that stochasticity introduces confounding effects thatexplain the previous success of some learning rate tuners at appropriate batchsizes. Our findings highlight the critical role of understanding the jointdynamics of the learning rate and curvature, beyond greedy minimization, todiagnose failures and design effective adaptive learning rate tuners.

The problem of orbital stabilization of periodic trajectories has been addressed in a series of publications: [1, 2, 3, 4, 5, 6, 7]. Many of these works, e.g., [1, 2, 4, 7], employ the transverse linearization approach, which approximates the dynamics near a reference periodic orbit by a linear time-varying (LTV) system with periodic coefficients. As shown in [2, 8], a feedback designed to stabilize the trivial solution of this auxiliary LTV system can be used to construct a control law that stabilizes the orbit of the original nonlinear system. Under the mild assumption of controllability of the LTV system over one period, the LQR approach can be used to design the feedback. The practical effectiveness of this method was demonstrated in experiments with real robotic systems in [9, 10, 11]. A substantially different stabilization method for the LTV system was proposed in [5], where the authors developed an alternative scheme combining Floquet theory with sliding-mode control. Following this line of work, we show that a specific feedback linearization of the transverse dynamics yields an LTV system endowed with a stable invariant subspace. In this setting, the control objective reduces to driving all trajectories into the stable subspace, which is achieved via sliding-mode-based control. This method does not require solving the computationally demanding periodic LQR problem.

Animals with foveated vision, including humans, experience microsaccades, small, rapid eye movements that they are not aware of. Inspired by this phenomenon, we develop a method for "Artificial Microsaccade Compensation". It can stabilize video captured by a tailless ornithopter that has resisted attempts to use camera-based sensing because it shakes at 12-20 Hz. Our approach minimizes changes in image intensity by optimizing over 3D rotation represented in SO(3). This results in a stabilized video, computed in real time, suitable for human viewing, and free from distortion. When adapted to hold a fixed viewing orientation, up to occasional saccades, it can dramatically reduce inter-frame motion while also benefiting from an efficient recursive update. When compared to Adobe Premier Pro's warp stabilizer, which is widely regarded as the best commercial video stabilization software available, our method achieves higher quality results while also running in real time.

Long-horizon multivariate time-series forecasting is challenging because realistic predictions must (i) denoise heterogeneous signals, (ii) track time-varying cross-series dependencies, and (iii) remain stable and physically plausible over long rollout horizons. We present PRISM, which couples a score-based diffusion preconditioner with a dynamic, correlation-thresholded graph encoder and a forecast head regularized by generic physics penalties. We prove contraction of the induced horizon dynamics under mild conditions and derive Lipschitz bounds for graph blocks, explaining the model's robustness. On six standard benchmarks , PRISM achieves consistent SOTA with strong MSE and MAE gains.

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.

Spectral GNNs, like ChebyNet, are limited by heterophily and over-smoothing due to their static, low-pass filter design. This work investigates the "Adaptive Orthogonal Polynomial Filter" (AOPF) class as a solution. We introduce two models operating in the [-1, 1] domain: 1) `L-JacobiNet`, the adaptive generalization of `ChebyNet` with learnable alpha, beta shape parameters, and 2) `S-JacobiNet`, a novel baseline representing a LayerNorm-stabilized static `ChebyNet`. Our analysis, comparing these models against AOPFs in the [0, infty) domain (e.g., `LaguerreNet`), reveals critical, previously unknown trade-offs. We find that the [0, infty) domain is superior for modeling heterophily, while the [-1, 1] domain (Jacobi) provides superior numerical stability at high K (K>20). Most significantly, we discover that `ChebyNet`'s main flaw is stabilization, not its static nature. Our static `S-JacobiNet` (ChebyNet+LayerNorm) outperforms the adaptive `L-JacobiNet` on 4 out of 5 benchmark datasets, identifying `S-JacobiNet` as a powerful, overlooked baseline and suggesting that adaptation in the [-1, 1] domain can lead to overfitting.

Many successful methods to learn dynamical systems from data have recently been introduced.

A.1 Proof of Proposition 3.2 First, we consider the solution of Eq. (9) for u( x) = kx . Thus, from the property of the martingale and It ˆ o's isometry formula, it follows that Eη (t) = Eη (0) = 0, Eη (t) So, to satisfy Condition (iii) in Theorem 2.2, we have to set Therefore, the exponential stability of the zero solution is assured. Now, applying Gronwall's inequality, we get E[ x (t) This therefore completes the proof of the whole theorem. A.3.2 Proof of Theorem 4.2 First we prove the estimation for E[ τ Applying It ˆ o's formula to log V (x) yields: log V ( x( t)) = log V (x Then, similar to the procedure for the energy cost in A.3.1, we can get that E[ x (t) Here we explain this term in more detail. The training for ES framework is not as efficient as AS.