Oscillatory Dynamics

Robots are moving out of the structured factory floor and into unstructured environments such as disaster sites, planetary surfaces, and agricultural fields, for which the right robot often does not yet exist. We present GLAMDRING, a framework that synthesizes the optimal robot for a locomotion task and, jointly, learns the controller that drives it. For the given specifications of forward-velocity bounds, a per-actuator power budget, an actuator library, and a payload requirement, GLAMDRING returns a matched quadruped morphology (link geometry and per-joint actuators) and a Hopf-oscillator Central Pattern Generator (CPG) gait policy. We rank feasible designs against a target design objective, viz., maximum speed, minimum Cost of Transport (CoT), or max Payload Margin. Because body and locomotion are coupled, the optimal morphology dictates how a robot is driven, while optimal gait depends on the physical body. We train a small number of CPG policies by reinforcement learning across the space of candidate morphologies, co-learning the gait with the underlying robot hardware. Link lengths and actuators are then resolved post-hoc from the policy's logged operating envelope, reducing synthesis cost to a small, fixed number of reinforcement-learning runs instead of one per candidate. Our experiments show three key findings: co-designing body and gait is necessary to satisfy locomotion constraints; actuator-envelope feasibility, rather than locomotion success alone, determines realizable payload capacity; and canonical animal gaits emerge naturally in most designs from morphology and constraints alone. A real-world demonstration further highlights the efficacy of our work.

Learned quadruped locomotion policies commonly map observations directly to joint-level actions, leaving the periodic structure of locomotion implicit in the policy. We investigate an alternative representation in which each joint trajectory is expressed as a command-conditioned Fourier series and modified online using feedback from the robot state. A context network generates the Fourier coefficients and the weights of a per-step feedback network, whose outputs adjust joint offsets, harmonic gains, frequency, and phase during execution. In simulation, we examine this explicit frequency structure alongside the hidden activations of an MLP policy that directly outputs joint targets. The harmonic waveforms change frequency and shape with commanded speed. Dynamic mode decomposition of selected MLP rollouts reveals dominant activation modes near the foot-height oscillation frequency and its second harmonic, showing periodic structure without an explicit Fourier generator. On a Unitree Go2, the simulation-trained harmonic controller records a provisional onboard-estimated peak speed of 3.67 meter per second and carries added loads up to 5.883 kilogram in separate trials.

Animals coordinate their movements through distributed neural circuits, but soft robots still typically depend on external, centralized electronics for control. Building soft robots that operate without centralized electronic controllers while remaining responsive to their environment remains a frontier challenge in soft robotics. In this work we introduce a soft-robot control architecture inspired by leaky integrate-and-fire models of biological neural circuits. The Pneumatic neuron (Pneu-ron) is a soft actuator that unifies energy conversion, logic, and actuation in one component. Each module combines a low-boiling-point fluid (LBF), a heater, and a mechanical switch into a self-excitable unit. Boiling the LBF inflates the module and triggers excitation and inhibition of adjacent modules in a process we call "inflate-and-fire". When interconnected into excitatory-inhibitory rings, Pneu-rons generate stable, sequential oscillations whose frequency emerges from the material dynamics and environmental conditions. By harnessing the inflation of Pneu-rons for actuation these networks can drive oscillatory locomotion of soft robots. Pneu-ron networks sustain oscillation under mechanical load and thermal variations, adapting through material physics rather than computation. Dynamical modeling of these networks reveals a dimensionless bifurcation diagram that dictates the network's oscillatory behavior. Encoding logic and actuation into material-level modules presents a new avenue for adaptive, electronics controller-free, soft robots.

Smectic liquid crystals are layered liquid-crystalline phases characterized by orientational order and periodic density modulation. Although their structures can be modeled using continuum theories, computing stable configurations remains challenging in complex geometries, particularly when the high-frequency density modulations associated with smectic layering should be resolved. We propose a deep variational framework (DVF) for computing these configurations within the modified Landau--de Gennes model, in which the coupled orientational and positional order parameters are represented on a regular reference domain while physical confinement is incorporated through coordinate mappings. A warmup penalty mitigates the spectral bias of neural networks toward smooth, nonlayered fields, enabling robust recovery of oscillatory smectic states. Comparisons with a neural-network baseline and finite-difference relaxation demonstrate the essential role of this penalty and the numerical stability of the resulting layered states. The DVF reproduces experimentally established smectic-A defect structures and layer morphologies across diverse confinement geometries and further predicts a chevron-like smectic-C state in a tangent-anchored sphere. Together, these results demonstrate the applicability of the DVF to computing stable smectic configurations across experimentally relevant confinement geometries and anchoring conditions.

Reconstructing oscillatory wave fields from scattered sensors is a severely underdetermined inverse problem. Beyond the challenges of general physical-field reconstruction, wave responses are complex-valued, frequency-sensitive, and highly oscillatory, while costly simulation and sensing often leave only extreme-sparse observations. Existing low-rank, operator, and diffusion approaches are largely designed for real-valued, smoother fields; dense pixel-space diffusion is particularly inefficient for oscillatory complex fields and difficult to scale to 3D. We propose HarmoCore, which places a generative prior in a compact, continuous, and structured wave-field latent. HarmoCore represents joint real--imaginary channels with Functional Tucker cores over shared continuous spatial bases, learns a frequency-conditioned core diffusion prior, and performs Diffusion Posterior Sampling directly in core space. At fixed sensor coordinates, the multilinear decoder induces an explicit likelihood guidance operator, avoiding dense pixel-space correction. Optional target-equation residual guidance further promotes physical consistency. Experiments on 2D Helmholtz, 2D synthetic wave fields, and 3D Helmholtz show substantial gains under 1%--2% sensing while remaining practical in three dimensions.

Animals' body morphology shapes the gait patterns they can perform, where mechanical resonance reduces the need for active control. By tuning posture and muscle stiffness, they leverage their embodied intelligence to achieve effective gaits for different speeds. In contrast, most quadruped robots are not specifically designed to exploit mechanical resonance due to the complexity of nonlinear dynamics and require dedicated locomotion controllers. To provide an alternative, we present a proof of concept framework making the nonlinear dynamics of a robot predictable in the design process and show how this knowledge can be leveraged such that multi-gait locomotion can emerge from nonlinear resonances, shaped by gravity, inertia, and elasticity. We present the highly compliant quadruped robot eBert, on which we identify six nonlinear normal modes (NNMs) using our new theoretical tools and validate their existence in simulation and hardware. With black-box optimization to determine step length, simulations show how each NNM naturally develops into a distinct gait, manifesting different speeds, which also largely transfers to the robotic hardware. Our experiments show that eBert can exploit its mechanics to generate task-specific movements which may serve as foundation for designing a new generation of agile and efficient robots leveraging embodied intelligence.

Operator learning is a rapidly advancing area of computational science. It is particularly well suited to problems where a partial differential equation (PDE) must be solved repeatedly under varying physical configurations. Most existing architectures represent the solution operator in a fixed basis. While this assumption is well aligned with global structures, it is less suitable for phenomena governed by local interactions in physical space. We explore an alternative perspective motivated by the observation that the continuum limit of coupled oscillator systems can describe a broad class of PDEs. Building on this idea, we introduce the Kuramoto Neural Operator (KNO), which represents the solution through the evolution of a latent field of interacting oscillators. Across a diverse collection of PDE benchmarks, KNO achieves strong predictive performance, with improvements over competing approaches. Our experimental evaluation also includes an extensive ablation study that quantifies the contribution of each architectural component incorporated into KNO. Furthermore, we show that the model's prediction error is closely linked to the collective dynamics of the latent oscillators. It varies systematically with their degree of synchronization, providing insights into the underlying mechanisms.

Forecasting non-stationary time series remains difficult due to long-range dependencies, local volatility bursts, structural shifts, and nonlinear oscillatory behaviors. Although Transformer-based forecasters are effective for modeling long-term temporal dependencies, their feed-forward blocks typically rely on smooth static activations that are insufficiently sensitive to abrupt regime changes. Motivated by quantitative Transformer designs and oscillator-based nonlinear activations, we propose QFCQT, short for Quantum-Fractal-inspired Chaotically Gated Quantformer, for robust forecasting under complex volatile dynamics. Here, "quantum-fractal-inspired" denotes a computational analogy based on soft oscillator superposition and multi-scale nonlinear responses, rather than a formal quantum-mechanical or fractal-theoretic derivation. QFCQT consists of three main components: (1) a Quantformer-style numerical encoder that directly processes multivariate inputs via linear embedding; (2) a learnable Lee-oscillator activation module that maps scalar pre-activations to dynamic oscillatory responses and summarizes them through Max-over-Time pooling; and (3) a smooth-chaotic gated fusion mechanism that adaptively balances conventional smooth activations and chaos-sensitive responses. Furthermore, instead of using a single fixed oscillator, QFCQT employs a soft superposition of eight parameterized Lee oscillator families to adaptively capture different nonlinear response patterns across regimes. Experiments on ETTh1, ETTh2, and A-share Stock Index benchmarks show that QFCQT consistently outperforms strong baselines, including Informer, LogTrans, LSTMa, HAT, and COTN.

Oscillatory signals, such as vibration, carry class-discriminative information in specific frequency bands; perturbing them in raw feature space for counterfactual analysis easily destroys their temporal structure and produces physically implausible results. In this work, we introduce IMFACT (IMF-based counterfACTuals), a model-agnostic framework for generating plausible counterfactual explanations for time series classifiers that operates in the decomposition space of Empirical Mode Decomposition. An input signal is split into Intrinsic Mode Functions (IMFs), and selected IMFs are progressively substituted with those of a Nearest Unlike Neighbour (NUN) until the classifier flips to the target class. We evaluate six IMF-selection strategies and a multi-NUN cycling extension on two UCR benchmarks (FaultDetectionA, FruitFlies). The variance-based strategy with three NUNs outperforms two prominent baseline techniques on reliability and plausibility metrics, while cycling across three NUNs yields the best proximity across both datasets.

Task B of the 1st DAFx Parameter Estimation Challenge requires estimating the frequencies, decay rates, gains, and number of modes in a dense plate-reverb impulse response. Weak and overlapping modes make sparse peak detection prone to severe undercounting. We train an ExtraTrees regressor on simulator-generated data to predict mode counts in four frequency bands. These counts define dense frequency grids, after which a differentiable all-pole resonator model refines decay and gain while keeping frequency fixed. On two separate synthetic validation sets, the system reduces a local challenge-style error by about 66% relative to the official default peak-picking baseline. The improvement is mainly associated with lower mode-count mismatch, while decay and gain remain the largest error sources. These findings support separating modal-density estimation from continuous parameter fitting.

Reservoir computing has emerged as an efficient machine learning framework for predicting time series generated by dynamical systems. In contrast to other machine and deep learning approaches, a reservoir computing trains only the output layer via linear regression, leaving the reservoir (recurrent layer) untrained. This simplification makes reservoir computers easier to train and more amenable to experimentation. However, because current reservoirs consist of networks of randomly connected nodes and require the optimization of numerous hyperparameters, a framework that precisely explains how reservoir computing operates and how it can be optimized remains missing. Here, we propose a frequency-based reservoir inspired by the brain's oscillatory dynamics and its hierarchy of timescales. The frequency-based reservoir can be interpreted as an ensemble of independent oscillatory units, each processing a portion of the input's frequency content. This allows us to understand the reservoir's internal behavior by modeling it as a single unit driven by an external input. Borrowing from the theory of a nonlinear oscillator forced by complex periodic inputs, we found that units of the frequency-based reservoir selectively amplify and store specific input frequencies, which are then used for prediction. The frequency-based reservoir performs as well as or better than equivalent random reservoirs. Furthermore, the frequency-based approach can be optimized to improve short-term prediction, a property that random reservoirs lack. Finally, we show that the frequency-based reservoir can also predict complex spatiotemporal dynamics. Our results show that reservoir computing can be designed using brain properties and theoretical insights borrowed from the physics of forced nonlinear oscillators.

Photonic reservoir computing is a promising physical machine-learning technique for predicting time-series data. The quantization of the response signal from the reservoir is required for the implementation of photonic reservoir computing, and the number of quantization bits and sampling frequency need to be optimized to achieve high performance and low energy consumption. However, few studies have been reported to investigate the effect of bit quantization and sampling frequency. In this study, we introduce a concept of approximate reservoir computing with a semiconductor laser by quantizing the amplitude of node states in the reservoir and output weights. We evaluate the performance of a chaotic time-series prediction task and energy consumption per sample. We achieve significant reduction of energy consumption by optimizing the number of quantization bits, the sampling frequency, and the injection current of the semiconductor laser, while maintaining the prediction performance.

The rapid adoption of generative AI tools has created new literacy demands for designers who must verbalize tacit knowledge through natural language prompts. Yet the micro-level cognitive processes by which designers externalize implicit intentions during iterative AI dialogue remain underexplored. This exploratory qualitative study examined five expert designers (11-20 years UI/UX experience, M = 15.4 years) using think-aloud protocols. We identified "Strategic Oscillation" -- experts' intentional return to vague language (Vibe) after progressing toward operational specifications (Code), leveraging AI's probabilistic nature for creative exploration. We observed shifts from "instruction" to "consultation" mode, deep reflection triggered by semantic collisions, and one participant's disengagement revealing conceptual alignment as a tentative boundary condition. We propose the ECRT cycle (Expectation-Collision-Reflection-Transformation) operationalizing the SECI model's "externalization" phase for human-AI collaboration. These findings yield three design patterns for metacognitive support and position AI as a "probabilistic mirror" catalyzing tacit knowledge externalization.

We suggest using the Lyapunov characteristic exponent (LCE) as a dense reward signal for the reinforcement learning problem of stabilizing the inverted pendulum with vertical motion. With LCE, the agent not only successfully found the oscillatory motion known as the Kapitza pendulum but also damped the pendulum's pivoting, leaving it in a strictly upright position.

We revisit one-zero and two-zero textures of the neutrino mass matrix under current experimental and cosmological constraints. We identify the phenomenologically viable texture structures using the latest results on neutrino oscillation parameters, the cosmological bound on the sum of neutrino masses, the kinematic bound on the effective electron-neutrino mass, and limits from neutrinoless double-beta decay. For two-zero textures, several structures are still allowed if only the CMB bound on the neutrino mass sum is imposed. Among them, the $B$-series textures show a characteristic prediction for the Dirac CP phase, with $δ_{\rm CP}$ lying around $π/2$ and $3π/2$, and are within the reach of future neutrinoless double-beta decay searches. When the stronger CMB+BAO constraint is included, however, only the $A$-series textures remain viable. Therefore, we also analyze one-zero textures by using machine learning techniques, particularly flow matching. It turns out that some of the texture structures are already excluded by current data, while the allowed ones give distinct predictions for $\sum_i m_i$, $m_{ν_e}^{\rm eff}$, $\langle m_{ee}\rangle$, and $δ_{\rm CP}$. We further discuss how the one-zero texture structures can arise from non-invertible selection rules.

Data-driven models now rival numerical weather prediction in the medium range, but extending them to sub-seasonal lead times raises challenges absent at shorter horizons. Errors accumulate over long autoregressive rollouts, systematic biases grow with lead time, and several years of data must be held out for independent verification, even though machine-learning models otherwise benefit from longer training records. To address these challenges, we adapt ECMWF's AIFS-CRPS medium-range model. AIFS-SUBS adopts a 24h autoregressive time step to reduce error accumulation, adds stratospheric levels and top-of-atmosphere thermal radiation as predictors, and reserves 2007--2011 as an independent verification window. We evaluate two config-durations: AIFS-SUBS, fine-tuned on operational analyses, and AIFS-SUBS-ERA5, trained on ERA5 alone. Across weeks 2--6, AIFS-SUBS matches the operational Integrated Forecasting System (IFS) in probabilistic skill while reducing systematic biases. For the convective (OLR) component of the Madden--Julian Oscillation (MJO), AIFS-SUBS extends skilful forecasts (correlation > 0.5) by eight days relative to the IFS, while matching or exceeding the IFS for the full multivariate RMM index. AIFS-SUBS also reproduces the observed MJO modulation of tropical cyclone activity comparably. Stratospheric skill is particularly strong with AIFS-SUBS reproducing sudden stratospheric warming (SSW) frequency and surface impact. In the AI Weather Quest, AIFS-SUBS-ERA5 attains a variable-averaged ranked probability skill score slightly ahead of the IFS at weeks 3 and 4. At inference, AIFS-SUBS uses about 200 times less energy than the IFS, opening the door to much larger real-time ensembles. AIFS-SUBS is ECMWF's first machine-learning model targeted at sub-seasonal time-scales.

Learning dissipative dynamics from discrete observations is essential for reliable long-horizon prediction and physically meaningful parameter identification. For linearly damped Hamiltonian systems, the exact flow is generally not symplectic but conformally symplectic, contracting the canonical symplectic form by a scalar factor that reflects the net dissipation. We propose Conformal Symplectic Networks with damping identification (CSympNet-ID), a discrete-time map-learning framework that learns the one-step flow map directly from snapshot pairs while enforcing exact discrete conformal symplecticity by construction, without penalty terms or projection. The architecture composes an exact symplectic neural core with explicit diagonal scaling layers whose factors are parameterized exponentially by a scalar damping-rate parameter, thereby guaranteeing positivity and interpretability of the learned dissipation factor. We establish a scaling-conjugacy factorization for conformal symplectic maps and derive a pointwise-in-step density result for CSympNet-ID. We evaluate an irregular-step damped oscillator, a damped spring-mass chain, a damped nonlinear cubic oscillator, and additional high-dimensional extensions. CSympNet-ID gives the most favorable overall results among the compared models in the reported experiments, particularly in data-scarce regimes, target contraction-law recovery, and high-dimensional tests where unstructured baselines degrade rapidly.

Oscillatory neural networks (ONNs) have emerged as a promising neuromorphic architecture, leveraging coupled dynamical systems to perform computation and represent information through phase relationships. Their interactions can be designed to support intrinsic energy-minimizing dynamics, enabling tasks such as associative memory and optimization, and positioning them as a candidate architecture for continuous learning and inference. We present a neuromorphic primitive implemented using memristive edges with inhibitory couplings as a potential design for autonomous learning, and provide circuit simulation validation that the system is capable of denoising noisy inputs on an auto-associative task. While numerical Hopfield/Ising models routinely assume signed weights, neuromorphic implementations of ONNs often fail to realize negative weights due to device and circuit constraints. A practically implementable route to inhibitory (negative) weights is particularly valuable: it expands the class of attractor structures accessible to oscillator networks beyond purely synchronous couplings, and supports phase-coded memories where anti-phase constraints are not merely transiently enforced during training but can persist autonomously after release. We provide circuit simulations and theoretical analyses demonstrating that signed effective weights are necessary for anti-phase attractors to persist autonomously.

This paper proposes a Bayesian filtering-based approach for learning the dynamics of a physical system from partial, noisy measurements. We model the system dynamics using a Lagrangian mechanics formulation. As in Lagrangian neural networks (LNNs), we parameterize the kinetic and potential energies with neural networks. The unknown external forces in the Lagrangian formulation are modeled as white Gaussian noise. The corresponding Euler--Lagrange equations then yield a continuous-time stochastic state-space model (SSM) that describes the system dynamics. The neural network parameters and system states are then jointly learned via a maximum-likelihood method using Gaussian-approximation-based Bayesian filters. The effectiveness of the proposed method is demonstrated on pendulum and Duffing oscillator examples, and its performance is compared with conventional LNNs and with approximate Bayesian filters using known system models.

Multi-agent large language model (LLM) systems often rely on verifier and critic agents to suppress hallucinations, but verification is delayed. During this delay, false claims can propagate through the agent network. We model this process as delayed consensus on a graph with grounded corrector nodes. Spectral decomposition by the grounded Laplacian yields a closed-form stability threshold for the verification dose: correction that is too strong or too delayed can turn consensus into oscillation. The most unstable regime occurs when the communication and verification delays coincide; for delay two, the threshold is the inverse golden ratio. The same framework gives a supermodular placement objective and a greedy (1-1/e)-approximation rule for assigning a limited corrector budget to influential nodes. Experiments across five open models confirm the predicted dose-delay oscillations. By contrast, grounded factual answering makes truth an absorbing boundary and eliminates the effect, suggesting that the instability is specific to signed-belief tasks while grounded verification remains stabilizing

Deep learning-based models have achieved state-of-the-art performance in Time Series Forecasting (TSF), yet their evaluation remains dominated by pointwise error metrics such as Mean Squared Error (MSE), which quantify numerical accuracy but overlook structural properties of the forecast signal, including recurrent dynamics, oscillatory behavior, and phase alignment. As a result, forecasts exhibiting over-smoothing, phase shifts, or frequency distortions may achieve favorable error scores despite substantial structural degradation. To address this limitation, we propose TopoCast, a topology-driven framework for evaluating structural fidelity in TSF. TopoCast reconstructs phase-space representations of forecast and ground-truth sequences using Takens delay embedding and applies persistent homology to characterize their intrinsic dynamics. We derive four complementary topological fidelity measures from persistence diagrams and aggregate them into a Topological Fidelity Score (TFS). We further introduce dominant cycle overlap, a novel metric that maps persistent topological features to the temporal domain to assess whether dominant oscillatory patterns occur at the correct time points. Combined with TFS, this yields the Localized Topological Fidelity Score (LTFS), a phase-aware measure that captures temporal localization errors invisible to existing evaluation metrics. Experiments on five Transformer architectures across three real-world benchmark datasets demonstrate that models with similar forecasting errors can exhibit markedly different structural fidelity profiles, revealing failure modes overlooked by conventional evaluation and highlighting the value of topology-aware forecast assessment.

Implicit Neural Representations (INRs) have been proven successful in encoding continuous signals through coordinate-based networks, yet facing a spectral dilemma: periodic activations capture fine details but act as all-pass filters that memorise noise, while spatially compact activations regularise effectively but suffer from low-frequency bias. Existing attempts to resolve this trade-off introduce computational overhead or tuning frailty. We propose to model each neuron's activation as the steady-state response of a sinusoidally-forced damped harmonic oscillator, whose amplitude naturally governs the network's spectral selectivity during training. By jointly optimising the oscillator parameters alongside the network weights, our method adapts to the target signal's spectral content without explicit regularisation. Initialised in the stopband, the network exhibits a coarse-to-fine learning curriculum that progressively expands its spectral gate, capturing low-frequency structures first and high-frequency details only when justified by the reconstruction objective. Comprehensive experiments show that our approach consistently achieves state-of-the-art or competitive results against established INRs, while requiring no task-specific tuning of any hyperparameters.

Standard continuous-depth models, such as Neural Ordinary Differential Equations (NODEs), offer significant advantages in modeling physical systems by learning continuous vector fields rather than discrete temporal steps. However, when applied to complex dynamical systems, standard NODEs frequently struggle with highly nonlinear dynamics. This paper investigates the Frequency-domain Neural ODE (FNODE), an architecture that projects continuous temporal dynamics into the frequency domain using the Fast Fourier Transform (FFT). By operating in the frequency domain, the model provides better generalization to the dynamical system. The architecture is empirically evaluated against discrete models, specifically Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs), and other continuous-depth variants, including Augmented Neural ODE (ANODE), across four distinct dynamical systems: the Lotka-Volterra model, the forced Duffing oscillator, the Van der Pol oscillator, and the Lorenz system. To rigorously assess generalization and robustness, curriculum and ensemble learning are used to evaluate the model's convergence by estimating confidence intervals across different ensemble models. The empirical results demonstrate that the FNODE architecture achieves better generalization while exhibiting remarkable convergence stability.

Extracting a flat clustering solution from a hierarchy is a common task in practical cluster analysis and can be formulated as an optimisation problem. Existing approaches focus on finding a single optimal solution. We introduce FOSC-X, a framework for extracting the top-M globally optimal flat clusterings from local, non-horizontal cuts of a hierarchical cluster tree, while optionally enforcing constraints on the number of clusters. This enables automatic identification of multiple high-quality alternative clusterings that capture different aspects of the hierarchical structure. Without constraints, the top-M problem can be solved in polynomial time using dynamic programming, exploiting the property that locally optimal partial candidates within subtrees can be combined to form globally optimal solutions while automatically determining the number of clusters. However, this can lead to solutions with numbers of clusters that are ultimately undesirable -- e.g., too large to be meaningful or practically analysed within a particular application domain. Imposing cluster-count constraints breaks the optimality property underlying the unconstrained dynamic programming approach, since locally optimal partial candidates may no longer combine into feasible globally optimal solutions. FOSC-X addresses this challenge through a dynamic programming strategy that maintains compact sets of feasible candidates using lower and upper feasibility bounds while pruning infeasible or dominated combinations. The resulting method guarantees optimal rankings of the top-M solutions with linear-time complexity in the number of cluster nodes and dataset size, both with and without cluster-count constraints. Experiments show that FOSC-X efficiently reveals alternative clustering structures overlooked by single-solution extraction methods.

Deep neural networks (DNNs) have achieved remarkable success in scientific computing, yet they often suffer from spectral bias in capturing oscillatory and multiscale behaviors. In this study, we investigate this limitation by examining the failure of shallow ReLU neural networks in fitting high-frequency functions. This observation identifies two important factors in resolving rapid oscillations: the initial slope scale and the distribution of partition points induced by the networks. Motivated by this analysis, we propose RepNN, a reparameterized neural network model with activation ReLU or tanh designed for high-frequency and multiscale problems. The key idea is to reparameterize the weights and biases in the first hidden layer, which enables effective control of the initial slope scale and provides an appropriate distribution of the initial partition points. Furthermore, treating the reparameterized weights and biases as trainable parameters allows the DNN to achieve adaptive frequency scaling during training. In addition, we derive quantitative estimates for the output and slope magnitudes of the reparameterized DNN to guide the initialization of the proposed method. Numerical experiments, including multiscale one- and four-dimensional function approximations, forward and inverse PDE problems in combination with physics-informed neural networks (PINNs), and operator learning for an earthquake problem using real data, demonstrate that RepNN improves the predicted accuracy of vanilla DNNs in capturing highly oscillatory features with slightly additional computational cost. These results indicate that RepNN provides an effective and flexible approach for overcoming spectral bias and applying DNNs to multiscale problems.

Physical reservoir computing harnesses nonlinear mechanical dynamics but, by convention, freezes the substrate and trains only a linear readout, presuming the substrate is not usefully trainable. We revisit that premise for networks of nonlinear oscillators whose mass, damping, and stiffness are learned end-to-end through a symplectic integrator. Our central result is a trilemma: memory horizon, gradient stability, and dynamical expressivity cannot be simultaneously maximized, because all three are governed by the damping. The backward gradient decays at a rate set by the damping, capping how far back credit can propagate, while forward sensitivities grow exponentially in the largest Lyapunov exponent, so usable gradients require damping above a stability floor. Since the Lyapunov exponent falls as damping rises while the memory ceiling falls as the horizon grows, stable training is confined to a band that contracts with horizon and closes at a critical point. We test every step on a twenty-oscillator network. A damping sweep finds the largest Lyapunov exponent monotone and crossing zero at a well-defined stability floor, confirming the theorem's key assumption. A compute-matched comparison of learned versus frozen substrate on delayed recall across nine horizons shows the learned substrate dominating at short horizons and the advantage closing and reversing near a horizon of eleven steps, the predicted signature of band closure; trained models settle near the stability floor, seeking the edge of chaos unprompted. The analytic ceiling overestimates the empirical crossover roughly fivefold, a gap between detectable and learnable gradient that we report rather than tune away. The contribution is a confirmed account of when training a physical substrate beats freezing it.

Inferring the forces that drive a dynamical system from partial observations is a fundamental challenge across physics, particularly when distinct underlying mechanisms produce similar observable dynamics. Here we show that the effective muscular forcing underlying avian respiratory dynamics can be reconstructed from measurements of air-sac pressure alone. Using an interpretable learning framework based on Kolmogorov-Arnold networks, we infer the governing equations of the system directly from data and uncover a nontrivial structure in the underlying forcing that is not apparent from the pressure signal, which instead suggests a relaxation-like oscillation. The reconstructed dynamics predict a two-phase activation pattern within each respiratory cycle, which we independently validate through electromyographic recordings of expiratory muscles. These results demonstrate that data-driven reconstruction of dynamical laws can reveal hidden physical structure and provide access to unobserved driving variables, establishing a general route to infer latent forces in partially observed dynamical systems.

Long-term time series forecasting benefits from inductive biases that expose recurring temporal structure. Existing periodic forecasting methods typically model recurrence through predefined periods, global spectral components, or fixed learnable templates. However, real-world temporal dynamics are rarely rigidly periodic: around a nominal cycle, oscillatory behavior often exhibits \emph{non-rigid periodicity} (NRP), where cycle magnitude, cycle alignment, and local cycle duration vary over time. Under these conditions, fixed-template periodic modeling can become fundamentally mismatched to the underlying temporal states. We propose AOSNet, a Hilbert-guided forecasting framework that reformulates periodic forecasting from fixed template matching to adaptive oscillatory-state alignment. AOSNet extracts analytic-signal descriptors from both the observed sequence and a learnable global oscillatory prior, then adaptively aligns local states through a descriptor-conditioned gate that selectively preserves reliable observations while softly correcting mismatched regions. The learned prior serves not as a rigid repeated template but as a flexible oscillatory reference interpreted through local state dynamics. Experiments on eight public benchmarks and two cloud workload traces demonstrate leading or highly competitive accuracy with a compact model size and low inference latency, supporting repeated forecasting settings such as capacity planning and autoscaling. Controlled synthetic studies that isolate cycle-magnitude and cycle-alignment variation and combine them with cycle-duration changes show that the advantage of oscillatory-state alignment increases as NRP intensifies.

We study the dynamics of gradient descent in the Edge of Stability regime, where the learning rate is large enough to induce persistent oscillations in the loss and the sharpness. We propose a continuous-time effective model that tracks the evolution of the average trajectory coupled with the time-averaged covariance of its fast oscillations. Our analysis reveals that the natural quantity to monitor in such unstable regimes is an effective free energy, which combines the original risk functional with a curvature-related "entropic" term. Our model allows us to track the envelope of the oscillations even in situations where its dynamics evolve on similar timescales as the averaged weights. Otherwise stated, we can track the spikes that occur during the training of some neural network architectures. For wide two-layer neural networks optimized under stable non-vanishing oscillations, we derive a mean-field limit that results in a novel kinetic equation describing the joint distribution of weights and their fluctuations. We show that this equation can be interpreted as a Wasserstein-2 gradient flow of a macroscopic free energy. Finally, we provide numerical evidence on matrix factorization and deep learning tasks (CIFAR-10) to demonstrate the model's accuracy in capturing the envelope of the oscillations and the predictive power of the effective free energy.

Equilibrium Propagation (EP) is a physics-based training framework that has primarily been employed in energy-based models, including continuous Hopfield networks, nonlinear resistive networks and coupled phase oscillators. However, EP's practical applications have so far remained limited to relatively small-scale problems. Predictive coding networks (PCNs), another class of energy-based models rooted in computational neuroscience, are typically trained with a specialized algorithm and have likewise not yet been demonstrated at large scale. In this work, we develop an EP-based training method for PCNs which combines the centered variant of EP with a novel equilibration scheme for PCNs. Using this approach, we train a 10-layer convolutional PCN (VGG10) on full-size ImageNet, achieving 13.23% test error rate on the top-5 classification task, close to the 12.2% backpropagation baseline. To our knowledge, this is the first demonstration of both PCNs and EP-based training at ImageNet scale. These results significantly extend the scalability of both approaches and suggest that the primary challenges in scaling EP in other physical systems may come more from the computational properties of these systems than from inherent limitations of the EP framework.

Remote photoplethysmography (rPPG) enables non-contact extraction of blood volume pulse (BVP) signals using consumer-grade cameras. Recent unsupervised rPPG methods learn BVP representations without requiring ground-truth physiological annotations, yet their optimization is often hindered by noisy and unstable gradients, resulting in slow convergence and limited cross-domain generalization. In this paper, we propose FCUS-rPPG, a fast-converging unsupervised rPPG framework with strong generalization capability. Motivated by the observation that BVP representations exhibit both multi-spectral covariation and low-dimensional manifold structure, we design a spectrally shared backbone that facilitates BVP feature disentanglement while improving optimization efficiency. To jointly enhance convergence stability and generalization performance, we further develop a unified optimization framework operating at the gradient, loss-landscape, and feature-representation levels. Specifically, a post-verification masking mechanism filters out misleading gradients according to the weak-amplitude physiological prior of BVP signals; a perturbation-based loss landscape smoothing strategy steers optimization toward more generalizable flat minima; and a noise-aware null-space regularization constrains feature updates to the orthogonal complement of the noise subspace, thereby mitigating noise-induced representation drift. Extensive experiments on five datasets demonstrate that FCUS-rPPG requires only one training epoch, whereas existing methods typically require tens to hundreds of epochs. Notably, FCUS-rPPG consistently achieves state-of-the-art (SOTA) performance in cross-dataset evaluations. This study provides an efficient and robust solution to the real-world deployment of unsupervised rPPG. The source code will be publicly available at https://github.com/JiaJieLee/FCUS-rPPG.

Oscillatory neural signals such as electroencephalography (EEG) and local field potentials (LFPs) show phase relationships that coordinate communication across brain regions. Modern recordings capture hundreds of channels across many frequency bins, yet standard phase analyses are restricted to only a few variables. The Torus Graph (TG) model, an exponential-family distribution over phases whose univariate and pairwise potentials generalize von Mises distributions, infers principled structure among oscillations but models only static, undirected dependencies and is limited to $\sim \! 100$ variables because its score matching inference scales as $\mathcal{O}(d^{6})$. We introduce a stochastic score matching procedure that reduces the per-iteration cost to $\mathcal{O}(d^{2})$, enabling inference on datasets with thousands of variables. This scalable foundation supports analyses of 1,860 frequency-phase features from multi-electrode LFPs and enables two extensions previously inaccessible to TGs or classical circular statistics: (i) a TG Hidden Markov Model capturing state-dependent phase-coupling changes (e.g., spindle-related states during sleep) and (ii) an autoregressive TG inferring directional interactions via transfer-entropy estimation. Applied to LFP recordings, these models reveal state-dependent phase-interaction patterns between wakefulness and NREM sleep. Together, they enable systematic, large-scale mapping of dynamic and directional phase relationships across brain and cognitive states.

Supervised Contrastive Learning (SupCon) has achieved strong performance by explicitly modeling pairwise relationships among samples. However, existing SupCon-based methods suffer from two key limitations: negative-sample dilution induced by the standard InfoNCE loss, and feature-space entanglement caused by the lack of explicit constraints separating category-relevant (common) and category-irrelevant (style) features. These limitations reduce feature discriminability and generalization ability. To address these issues, we propose OSCS-SupCon (Orthogonal Sigmoid-based Common and Style Supervised Contrastive Learning), a unified framework that combines a sigmoid-based pairwise contrastive objective with explicit orthogonality constraints. Specifically, we introduce a sigmoid-based contrastive loss with two learnable parameters, temperature and bias, which adaptively modulate pairwise decision boundaries and alleviate negative-sample dilution. Furthermore, we enforce orthogonality between common and style feature subspaces via a linear projection with ReLU nonlinearity, thereby reducing feature overlap and improving disentanglement of style-irrelevant representations. Extensive experiments on six benchmark datasets demonstrate that OSCS-SupCon consistently outperforms state-of-the-art supervised contrastive learning methods across multiple backbone architectures. In particular, on the fine-grained CUB200-2011 dataset with a ResNet-18 backbone, the proposed method achieves a 3.4% improvement in classification accuracy over CS-SupCon, highlighting its robustness and generalization capability. Ablation studies further confirm the effectiveness of each component.

Deep learning optimization relies heavily on the assumption of smooth loss landscapes, a condition systematically violated by modern architectures due to non-smooth components such as ReLU activations and quantization operators. In such non-smooth regimes, adaptive optimizers such as Adam suffer from gradient chattering, violent oscillations caused by conflicting signals within the Clarke subdifferential, leading to poor convergence and suboptimal generalization. To address this, we introduce Singularity-aware Adam (S-Adam), a novel optimizer that stabilizes training by dynamically modulating step sizes based on local geometric instability. Our key contribution is the Local Geometric Instability (LGI) metric, a computationally efficient estimator of the Clarke subdifferential diameter derived from the variance of randomized directional derivatives. S-Adam incorporates an adaptive damping mechanism exp(-λ$$ρ) that decelerates updates in high-instability regions while preserving fast convergence in smooth basins. We provide a rigorous convergence analysis using differential inclusions, proving that S-Adam converges almost surely to ($δ$,$ε$)-Clarke stationary points at the optimal O(1/$\sqrt(T)$) rate. Empirical evaluations on Quantization-Aware Training (QAT) and high-noise small-batch learning demonstrate that S-Adam consistently outperforms AdamW and Prox-SGD, achieving accuracy gains of up to 6 percent on CIFAR-100 and 3 percent on TinyImageNet while effectively mitigating gradient oscillations.

Existing theory of momentum assumes that gradients arrive at every parameter at a roughly constant rate, an assumption violated in practice by heavy-tailed data distributions and modern architectures. We theoretically analyze the dynamics of two tractable models of momentum under sparse updates: a least squares model with sparse inputs and a logistic regression model with a rare class. Both admit exact closed-form second-moment dynamics whose high-dimensional limits we characterize across three scaling exponents for sparsity, batch size, and momentum decay. The phase structure on both problems is governed by the ratio of two intrinsic timescales: a momentum retention timescale (how many active updates the buffer survives) and a learning timescale (how many active updates it takes to reduce the squared error). When learning is much slower than retention, the limit matches SGD; when learning is faster, the system is unstable; where the timescales coincide, we recover classical heavy-ball dynamics. The oscillatory dynamics occur at different momentum values for different token sparsity, creating a spectral conflict for global momentum across token frequencies.

Underwater robotics has advanced significantly over recent decades. however, the development of miniaturized underwater robots remains limited by low energy densities of traditional power sources. Nature offers compelling solutions-organisms like mantis shrimps and fleas utilize latch-mediated spring actuation (LaMSA) systems that achieve rapid movements through a decoupled energy storage and release mechanism. Despite extensive studies of LaMSA, replicating such rapid, asymmetric actuation within simple, compact structures remains challenging. In this work, we introduce a bioinspired, soft bistable actuator with an integrated latch mechanism that enables asymmetric energy input and release using a single motor. Coupled with fin structures, this design facilitates efficient underwater propulsion and maneuverability. Experimental results demonstrate stable periodic flapping, precise steering, and a maximum thrust of 0.528 N, impulse of 0.147 Ns, and vertical displacement of 30 mm. By modulating fin angles, the robot achieves versatile motions, including vertical ascent, diagonal forward movement, and lateral translation. This study presents a novel, energy-efficient approach for controlling motion in compact underwater robots, paving the way for advanced biomimetic designs with potential applications in exploration, environmental monitoring, and inspection.

Learning-based surrogates have become increasingly effective for wave-field prediction, and neural operators in particular have shown strong performance within observed frequency regimes. However, higher-frequency prediction under scarce target supervision remains comparatively underexplored, especially in wave problems where higher-frequency data are substantially more expensive to simulate or measure than lower-frequency data. A central difficulty is that cross-frequency transfer is inherently asymmetric: coarse amplitude structure remains relatively stable across frequencies, whereas phase-sensitive oscillatory structure deteriorates much more rapidly as frequency increases. Motivated by this asymmetry, we propose APEX, Amplitude-anchored and Phase-prior-guided Enhancement from eXtrapolated coarse predictions, a framework for target-scarce higher-frequency wave-field prediction. A lower-frequency neural operator first provides a coarse prediction in the target-frequency regime, from which we retain only the amplitude as a transferable structural anchor. A conditional flow-matching enhancer then reconstructs the target higher-frequency field under the guidance of a Green's-function-inspired phase prior. Experiments on SimpleWave, Helmholtz, and Maxwell benchmarks show that APEX consistently outperforms direct lower-to-higher extrapolation, target-adapted operator, and joint generative baselines under limited target-frequency supervision. Our results suggest that reliable higher-frequency prediction of oscillatory wave fields should not rely on direct end-to-end transfer of the full complex field, but instead on explicitly reusing transferable coarse structure while separately recovering the missing oscillatory detail.

Coordinating micro-robotic swarms in realistic, time-dependent fluid environments remains a major challenge for biomedical and environmental applications. We present a hybrid CFD-MO-MARL (Computational Fluid Dynamics-Multi Objective-Multi Agent Reinforcement Learning) framework that couples a high-fidelity incompressible Navier--Stokes solver with decentralized proximal policy optimization to learn swarm control policies in oscillatory flow. Sixteen magnetically actuated micro-robots were simulated to navigate a pulsatile arterial waveform within a 2 mm channel while jointly optimizing upstream progression, energy efficiency, and motion smoothness. Conflicting objectives are resolved using Projected Conflicting Gradient (PCGrad) surgery. Without PCGrad, energy and smoothness rewards collapse during training, demonstrating that gradient conflict resolution is essential for stable multi-objective learning. The converged policy achieves progress rewards of 6.5-7.0, energy efficiency of 0.63-0.65, and smoothness of 0.97-0.99, outperforming brute-force baselines by more than 8 reward units on the primary objective. Training reveals three emergent behaviors not encoded in the reward function: hydrodynamic throttling formations that reduce peak flow velocities, a cycle-synchronized ratchet mechanism that exploits flow reversals for upstream movement, and individualized final-approach strategies near the target boundary. These results demonstrate that physically realistic fluid--agent interactions can be integrated directly into multi-objective reinforcement learning, providing a scalable framework for micro-swarm control in biomedical navigation, environmental monitoring, and microfluidic systems.

Modern deep learning commonly relies on AdamW with prescribed learning rate schedules, but recent works challenge both components: Schedule-Free optimization removes explicit schedules via iterate averaging, and Muon improves the update geometry by orthogonalizing momentum for matrix parameters. Despite Muon's strong empirical performance, its underlying mechanism remains partially understood. We study Muon through the river-valley loss landscape, where useful training progress occurs along a flat, low-curvature bulk subspace (the river), while high-curvature dominant directions form steep valley walls that induce oscillations. We empirically show that while Muon's orthogonalization accelerates river progress by increasing the bulk component, it also amplifies dominant-direction noise, causing oscillatory trajectories. Building on this, we propose Anytime MUon with Stable gradient Evaluation (AMUSE), which integrates Muon's rapid bulk progress with the stabilizing effect of Schedule-Free averaging. AMUSE uses a time-varying interpolation coefficient that initially evaluates gradients near the fast Muon sequence for rapid adaptation, then gradually shifts toward the stable averaged sequence to suppress valley-wall oscillations. As a result, AMUSE requires no learning rate schedules and supports anytime training. Across vision tasks and large language model pretraining, AMUSE consistently improves the performance-iteration Pareto frontier over (Schedule-Free) AdamW and Muon.

Oscillations and synchronization are widely believed to play a fundamental role in representation and computation. However, existing machine learning approaches based on synchronization dynamics have largely been confined to specialized settings such as object discovery, with limited evidence of scalability to standard vision benchmarks or logic reasoning tasks. We propose the Winfree Oscillatory Neural Network (WONN), a dynamical neural architecture based on generalized Winfree dynamics. WONN evolves representations on the torus $(S^1)^d$ through structured oscillatory interactions, combining phase-based inductive biases with flexible and hierarchical interaction mechanisms instantiated as either fixed trigonometric mappings or learnable neural networks. We evaluate WONN on image recognition and complex reasoning tasks, including CIFAR, ImageNet, Maze-hard, and Sudoku. Across these domains, WONN achieves competitive or superior performance with strong parameter efficiency. In particular, WONN is, to our knowledge, the first synchronization-based oscillatory architecture to scale competitively to ImageNet-1K. Furthermore, on Maze-hard, WONN achieves 80.1% accuracy using only 1% of the parameters of prior state-of-the-art models. These results suggest that structured oscillatory dynamics provide a scalable and parameter-efficient alternative to conventional neural architectures.

Radio-frequency (RF) fingerprinting identifies wire-less transmitters using hardware-induced imperfections present in baseband I/Q signals. However, deep learning models often degrade under receiver and channel distribution shifts, particularly as transmitter populations grow. This work proposes the Hamiltonian Transformer, a physics-informed attention architecture that enforces norm preserving value dynamics within each attention head using a learned skew-symmetric generator and a Störmer-Verlet leapfrog integration step. An additional phase-increment embedding exposes oscillator dynamics at the input layer. All experiments use non-equalized raw I/Q signals from the WiSig dataset under four protocols: same-day classification, cross-receiver generalisation, cross-day generalisation, and transmitter scaling up to 150 devices. The Hamiltonian Transformer achieves 99.12% accuracy under same-day conditions and 61.64% at 150 transmitters, consistently outperforming CNN and Transformer baselines across all scale points. A controlled ablation study identifies norm-preservation in the value update as the primary inductive bias driving the scaling advantage, with the phase increment embedding providing the single largest per-component improvement. These results indicate that embedding physics-informed structural priors into attention mechanisms is an effective approach to large-scale transmitter identification on raw wireless signals.

Pulse-coupled oscillator models inspired by firefly synchronization are widely used to study decentralized time coordination in distributed systems. We analyze a discrete-time, discrete-phase firefly-inspired synchronization model and show that collective synchrony emerges only near a critical balance between the quorum threshold (fraction of pulsing neighbors required to trigger a phase update) and the pulse duration (how long agents remain detectable to others). Within this parameter region, the system exhibits bimodal performance: it either reaches near-perfect synchronization or becomes trapped in stable multi-cluster states, where symmetrically phase-offset subgroups mutually reinforce one another and prevent global synchrony. Our analysis shows that reducing connectivity or introducing noise suppresses these low-performance states by breaking such symmetric interactions, indicating that highly connected or noiseless systems are not necessarily optimal for collective synchronization.

Relative positional encodings determine which functions of query-key lag can enter the primitive attention logit. RoPE supplies a rotary phase, while ALiBi supplies an additive distance bias. Motivated by group-theoretic views of linear translation-invariant positional encodings, we study a non-semisimple case in which a complex rotary eigenvalue and a nilpotent response live in the same defective Jordan block. The resulting relative operator generates oscillatory-polynomial features such as $e^{-γd}\cos(ωd)$, $e^{-γd}\sin(ωd)$, $d e^{-γd}\cos(ωd)$, and $d e^{-γd}\sin(ωd)$, for causal lag $d=i-j\geq 0$. Thus the construction realizes a distance-modulated phase basis $d e^{iωd}$, rather than merely adding a separate distance channel to RoPE. We formulate Exact Jordan-RoPE as a non-semisimple one-parameter representation, give its real block form, and specify the contragredient query action required by non-orthogonal positional maps. We also distinguish this exact representation from stabilized variants whose bounded shear improves numerical behavior but breaks the exact group law. Kernel-level diagnostics and a Jordan-friendly synthetic language-model task show that the coupled Jordan basis is useful when the target contains distance-modulated phase interactions. On a small WikiText-103 byte language model, a scaled-exact variant improves over RoPE and direct-sum baselines within the Jordan family, while RoPE+ALiBi remains strongest overall. The evidence is structural rather than a broad performance claim.

Human cognition emerges from coordinated spiking dynamics in distributed neural circuits, where information is encoded via both firing rates and precise spike timing determined by brain rhythms. Inspired by this notion, we propose a brain-inspired learning primitive in which cognition-level neural synchrony emerges through iterative bottom-up and top-down interactions between micro-scale dynamics of spiking neurons and a macro-scale mechanism of oscillatory synchronization. Specifically, we model each parcel (e.g., a cortical region or an image pixel) in the target system as a spiking neuron embedded in a predefined connectivity scaffold. Low-level information is encoded in a spatiotemporal domain, where neurons are selectively grouped and fire spontaneously over time through self-organized dynamics. In the bottom-up route, oscillatory synchronization is formed from past spiking activity accumulated over a finite memory window. Since brain dynamics operate in a regime of partial and transient synchronization rather than global phase locking, we model oscillatory coordination using a time-delayed synchronization formulation, which enables a top-down modulation of heterogeneous neural spiking for a large-scale distributed system. Together, we devise a spiking-by-synchronization neural network (S2-Net) that uses rhythmic timing as a control mechanism for efficient information processing. Promising results have been achieved across a broad range of tasks, including neural activity decoding, energy-efficient signal processing, temporal binding and semantic reasoning.

Safety is a primary challenge in real-world reinforcement learning (RL). Formulating safety requirements as state-wise constraints has become a prominent paradigm. Handling state-wise constraints with the Lagrangian method requires a distinct multiplier for every state, necessitating neural networks to approximate them as a multiplier network. However, applying standard dual gradient ascent to multiplier networks induces severe training oscillations. This is because the inherent instability of dual ascent is exacerbated by network generalization -- local overshoots and delayed updates propagate to adjacent states, further amplifying policy fluctuations. Existing stabilization techniques are designed for scalar multipliers, which are inadequate for state-dependent multiplier networks. To address this challenge, we propose an augmented Lagrangian multiplier network (ALaM) framework for stable learning of state-wise multipliers. ALaM consists of two key components. First, a quadratic penalty is introduced into the augmented Lagrangian to compensate for delayed multiplier updates and establish the local convexity near the optimum, thereby mitigating policy oscillations. Second, the multiplier network is trained via supervised regression toward a dual target, which stabilizes training and promotes convergence. Theoretically, we show that ALaM guarantees multiplier convergence and thus recovers the optimal policy of the constrained problem. Building on this framework, we integrate soft actor-critic (SAC) with ALaM to develop the SAC-ALaM algorithm. Experiments demonstrate that SAC-ALaM outperforms state-of-the-art safe RL baselines in both safety and return, while also stabilizing training dynamics and learning well-calibrated multipliers for risk identification.

This paper provides a concise yet comprehensive review of recent advancements in millimeter-wave (mm-wave) oscillators below 100 GHz and sub-terahertz (sub-THz/THz) oscillators above 100 GHz for next-generation computing and communication systems, including 5G, 6G, and beyond. Various design approaches, including CMOS, SiGe, and III-V semiconductor technologies, are explored in terms of performance metrics such as phase noise, output power, efficiency, frequency tunability, and stability. The review highlights key challenges in achieving high-performance and reliable oscillator designs while discussing emerging techniques for performance enhancement. By evaluating recent design trends, this work aims to offer valuable insights and design guidelines that facilitate the development of robust mm-wave and sub-THz/THz oscillators for future communication, computing, and sensing applications.

Studying nonlinear dynamical systems through their state space behavior can be challenging, and one possible alternative is to analyze them via their associated Koopman operator. This turns the nonlinear problem into a linear, infinite-dimensional one. To approximate the operator in finite dimensions, extended dynamic mode decomposition (EDMD) is a commonly used algorithm. It requires a finite list of functionals and a set of snapshots from the system to compute an approximation of the operator and its corresponding spectrum. Instead of choosing the list of functionals directly, it can be implicitly defined via kernels, a method known as kernel extended dynamic mode decomposition (kEDMD). However, one still needs to define the kernel and choose its parameter values. In this paper, we aim to streamline this process by extending dictionary learning for EDMD to kernel learning in kEDMD. By simplifying kEDMD we show how to perform gradient-based optimization over the learnable kernel parameters, and demonstrate that this method leads to useful kernels for the original kEDMD. The focus of our work is a method that takes a weighted list of kernels with randomly initialized values as input and outputs a list of kernels and parameter values suitable for approximating the Koopman operator of the underlying system. We demonstrate that unimportant kernels can be removed from the list by analyzing the weights in the weighted sum. We evaluate the method across several experiments, including the Duffing oscillator and the Kuramoto-Sivashinsky PDE, showcasing the method's different strengths.

Parametrically driven oscillators provide a natural platform for neuromorphic computation, where nonlinear mode coupling and intrinsic dynamics enable both memory and high-dimensional transformation. Here, we investigate a two-mode system exhibiting 2:1 parametric resonance and demonstrate its operation as a reservoir computer across distinct dynamical regimes, including sub-threshold, parametric resonance, and frequency-comb states. By encoding input signals into the drive amplitude and sampling the resulting temporal and spectral responses, we perform one step-ahead prediction of benchmark chaotic systems, including Mackey-Glass, Rossler, and Lorenz dynamics. We find that optimal computational performance is achieved within the parametric resonance regime, where nonlinear interactions are activated while temporal coherence is preserved. In contrast, although frequency-comb states introduce increased spectral dimensionality, their performance is not consistently good across their existence band and also degrades in the chaotic comb regime due to loss of phase coherence. Mapping prediction error over parameter space reveals a direct correspondence between computational capability and the underlying bifurcation structure, with low-error regions aligned with the parametric resonance boundary. We further show that the input modulation, the detuning from the frequency matching condition, damping ratio, and input data rate systematically control the accessible dynamical regimes and thereby the computational performance. These results establish parametric resonance as a robust operating regime for oscillator-based reservoir computing and provide design principles for tuning physical systems toward optimal neuromorphic functionality.

When training neural networks with full-batch gradient descent (GD) and step size $η$, the largest eigenvalue of the Hessian -- the sharpness $S(\boldsymbolθ)$ -- rises to $2/η$ and hovers there, a phenomenon termed the Edge of Stability (EoS). \citet{damian2023selfstab} showed that this behavior is explained by a self-stabilization mechanism driven by third-order structure of the loss, and that GD implicitly follows projected gradient descent (PGD) on the constraint $S(\boldsymbolθ)\leq 2/η$. For mini-batch stochastic gradient descent (SGD), the sharpness stabilizes below $2/η$, with the gap widening as the batch size decreases; yet no theoretical explanation exists for this suppression. We introduce stochastic self-stabilization, extending the self-stabilization framework to SGD. Our key insight is that gradient noise injects variance into the oscillatory dynamics along the top Hessian eigenvector, strengthening the cubic sharpness-reducing force and shifting the equilibrium below $2/η$. Following the approach of \citet{damian2023selfstab}, we define stochastic predicted dynamics relative to a moving projected gradient descent trajectory and prove a stochastic coupling theorem that bounds the deviation of SGD from these predictions. We derive a closed-form equilibrium sharpness gap: $ΔS = ηβσ_{\boldsymbol{u}}^{2}/(4α)$, where $α$ is the progressive sharpening rate, $β$ is the self-stabilization strength, and $σ_{ \boldsymbol{u}}^{2}$ is the gradient noise variance projected onto the top eigenvector. This formula predicts that smaller batch sizes yield flatter solutions and recovers GD when the batch equals the full dataset.

We establish a mathematical correspondence between state space models, a state-of-the-art architecture for capturing long-range dependencies in data, and an exactly solvable nonlinear oscillator network. As a specific example of this general correspondence, we analyze the diagonal linear time-invariant implementation of the Structured State Space Sequence model (S4). The correspondence embeds S4D, a specific implementation of S4, into a ring network topology, in which recent inputs are encoded, as waves of activity traveling over the one-dimensional spatial layout of the network. We then derive an exact operator expression for the full forward pass of S4D, yielding an analytical characterization of its complete input-output map. This expression reveals that the nonlinear decoder in the system induces interactions between these information-carrying waves that enable classifying real-world sequences. These results generalize across modern SSM architectures, and show that they admit an exact mathematical description with a clear physical interpretation. These insights enable a new level of interpretability for these systems in terms of nonlinear oscillator networks.

Generative modeling within constrained sets is essential for scientific and engineering applications involving physical, geometric, or safety requirements (e.g., molecular generation, robotics). We present a unified framework for constrained diffusion models on generic nonconvex feasible sets $Σ$ that simultaneously enforces equality and inequality constraints throughout the diffusion process. Our framework incorporates both overdamped and underdamped dynamics for forward and backward sampling. A key algorithmic innovation is a computationally efficient landing mechanism that replaces costly and often ill-defined projections onto $Σ$, ensuring feasibility without iterative Newton solves or projection failures. By leveraging underdamped dynamics, we accelerate mixing toward the prior distribution, effectively alleviating the high simulation costs typically associated with constrained diffusion. Empirically, this approach reduces function evaluations and memory usage during both training and inference while preserving sample quality. On benchmarks featuring equality and mixed constraints, our method achieves comparable sample quality to state-of-the-art baselines while significantly reducing computational cost, providing a practical and scalable solution for diffusion on nonconvex feasible sets.

The LEOPs (Light-ERG-Oscillatory Potentials) dataset provides light-adapted (LA) electroretinogram (ERG) and Oscillatory Potentials (OPs) waveforms for typically developing Control, Autism Spectrum Disorder (ASD) and ASD + Attention Deficit Hyperactivity Disorder (ADHD) childhood and adolescent populations. The ERGs were recorded in the Right And Left eyes with skin electrodes using the handheld RETeval device at two sites in Australia and the United Kingdom. The LEOPs dataset includes 5309 single flash ERG and 4434 OPs waveforms as well as images selected from each participant showing the position of the skin electrode. The LEOPs dataset is constructed from recordings using a 9 step randomized flash series from $-0.37$ to $1.20$$Td.s$, a 2 step at 113 and 446 $Td.s$ flash strengths (2500 Control, 1730 ASD and 451 ASD + ADHD samples), as well as the $85$$Td.s$ (Light Adapted 3 $cd.s.m^{-2}$ (LA3)) equivalent International Society of Clinical Electrophysiology of Vision (ISCEV) Standard flash with 435 Control, 176 ASD and 37 ASD + ADHD waveform samples. Code for the stimulus is provided along with participant demographics, date and time of testing, and where available diagnostic scores for the ASD and ASD + ADHD groups, alongside iris color, electrode position with image files and time domain values for the ERG and summed values for the OPs. The repository contains excel file, exported JSON files on the patient level that are more suitable for machine learning tasks, images of electrode position for each recording and the protocol files for use with the RETeval.

Extrapolative prediction of complex nonlinear dynamics remains a central challenge in engineering. This study proposes a one-shot learning method to identify global frequency-response curves from a single excitation time history by learning governing equations. We introduce MEv-SINDy (Multi-frequency Evolutionary Sparse Identification of Nonlinear Dynamics) to infer the governing equations of non-autonomous and multi-frequency systems. The methodology leverages the Generalized Harmonic Balance (GHB) method to decompose complex forced responses into a set of slow-varying evolution equations. We validated the capabilities of MEv-SINDy on two critical Micro-Electro-Mechanical Systems (MEMS). These applications include a nonlinear beam resonator and a MEMS micromirror. Our results show that the model trained on a single point accurately predicts softening/hardening effects and jump phenomena across a wide range of excitation levels. This approach significantly reduces the data acquisition burden for the characterization and design of nonlinear microsystems.

Post-training quantization (PTQ) assumes that a well-converged model is a quantization-ready model. We show this assumption fails in a structured, measurable, and previously uncharacterized way. Using a calibration-free per-group INT4 probe applied to all 154 publicly available Pythia-160m training checkpoints, we identify a three-phase divergence structure: a rapid-learning phase where both FP32 perplexity and quantization robustness improve together, a meta-stable plateau lasting roughly 70,000 steps where FP32 perplexity stagnates but INT4 gap remains bounded, and an explosive divergence phase where the INT4 gap compounds from 11% to 517% while FP32 perplexity barely moves. Critically, this divergence begins not when the learning rate starts decaying, but precisely when FP32 perplexity converges a finer-grained onset predictor that implies post-convergence weight updates, rather than decay magnitude alone, are the proximate cause. We further show that INT8 quantization is entirely immune throughout all three phases, constraining the mechanism to the coarseness of the 16-level INT4 grid specifically, and rule out weight outlier accumulation as the mechanism via direct kurtosis measurement. Finally, we conduct a controlled fork experiment from the pre-divergence checkpoint comparing three learning rate schedules (cosine continuation, SGDR warm restarts, and our proposed Oscillatory Lock-In) across nine independent runs. SGDR uniformly accelerates divergence (0/9 pairwise wins against cosine), while OLI's settled cool phases reduce the INT4 gap by 2.2 percentage points on average (t = -5.46, p < 0.0001), demonstrating that schedule amplitude calibration, not oscillation alone, determines whether perturbation helps or hurts. Our code, probe implementation, and all 154-checkpoint audit results are released publicly.

Spatiotemporal neural dynamics and oscillatory synchronization are widely implicated in biological information processing and have been hypothesized to support flexible coordination such as feature binding. By contrast, most deep learning architectures represent and propagate information through activation values, neglecting the joint dynamics of rate and phase. In this work, we introduce Kuramoto oscillatory Phase Encoding (KoPE) as an additional, evolving phase state to Vision Transformers, incorporating a neuro-inspired synchronization mechanism to advance learning efficiency. We show that KoPE can improve training, parameter, and data efficiency of vision models through synchronization-enhanced structure learning. Moreover, KoPE benefits tasks requiring structured understanding, including semantic and panoptic segmentation, representation alignment with language, and few-shot abstract visual reasoning (ARC-AGI). Theoretical analysis and empirical verification further suggest that KoPE can accelerate attention concentration for learning efficiency. These results indicate that synchronization can serve as a scalable, neuro-inspired mechanism for advancing state-of-the-art neural network models. Code is avaliable at https://github.com/microsoft/Neuro-inspired_Phase_Encoding.

Parallel simulation of robotic systems requires partitioning the dynamics into coupled subsystems. Finite-iteration coupling across the partition boundary can inject spurious energy, even when each subsystem is passive. We propose an early-terminable, energy-safe coupling interface for port-Hamiltonian subsystems based on Douglas--Rachford splitting in wave (scattering) coordinates. The wave-domain formulation reduces passivity to norm inequalities and coupling to orthogonality. Within this setting, the deep correspondence between monotone operator theory and discrete passivity can be exploited to construct a Douglas--Rachford inner iteration whose Fejér monotonicity provides algorithmic dissipation. Under passivity of the subsystem integrators and an impedance-tuning condition, the proposed method guarantees discrete passivity of the augmented storage for any finite inner-iteration budget and converges to the monolithic discretization as the budget increases. Experiments on a linear--Duffing coupled-oscillator benchmark support the finite-iteration energy inequality at numerical roundoff (1e-14 in double precision), with state-error metrics decreasing over the tested inner-iteration budgets.

Electroencephalography (EEG)-based emotion recognition remains challenging in cross-subject settings due to severe inter-subject variability. Existing methods mainly learn subject-invariant features, but often under-exploit stimulus-locked group regularities shared across subjects. To address this issue, we propose the Group Resonance Network (GRN), which integrates individual EEG dynamics with offline group resonance modeling. GRN contains three components: an individual encoder for band-wise EEG features, a set of learnable group prototypes for prototype-induced resonance, and a multi-subject resonance branch that encodes PLV/coherence-based synchrony with a small reference set. A resonance-aware fusion module combines individual and group-level representations for final classification. Experiments on SEED and DEAP under both subject-dependent and leave-one-subject-out protocols show that GRN consistently outperforms competitive baselines, while additional analyses confirm the effects of prototype learning, PLV/coherence resonance, and leakage-safe reference construction.

We show that the out-of-equilibrium driving protocol of score-based generative models (SGMs) can be learned via local learning rules. The gradient with respect to the parameters of the driving protocol is computed directly from force measurements or from observed system dynamics. As a demonstration, we implement an SGM in a network of driven, nonlinear, overdamped oscillators coupled to a thermal bath. We first apply it to the problem of sampling from a mixture of two Gaussians in 2D. Finally, we train an oscillator network on the MNIST dataset to generate images of handwritten digits 0 and 1.

We address transformer attention on energy-constrained physical substrates. Softmax attention requires exponentiation and global reduction, operations with high energy cost on von Neumann hardware and no natural physical analog. We show that Kuramoto synchronization dynamics (which arise in electrical, mechanical, superconducting, and charge-density-wave oscillator arrays, among other physical systems) implement a well-defined attention operation. The resulting mechanism, \emph{fixed-query oscillator attention}, replaces softmax's arithmetic with the equilibration of a gradient flow on the sphere: queries are learned anchors fixed on the sphere, and free oscillators evolve under Kuramoto--Lohe dynamics until they settle at positions encoding attention weights via cosine similarity. Because the computation is equilibration, no global exponential normalization is needed. The fixed point is provably unique and globally attractive from almost every initial condition, a guarantee that holds across every physical realization. Empirically, at the minimal hardware configuration (oscillator dimension $d_{\mathrm{osc}} = 2$), oscillator attention matches softmax on keyword spotting, and on subject-verb agreement it trains more reliably while reaching softmax's accuracy. Softmax retains an advantage on causal language modeling, but the gap decays as a power law in $d_{\mathrm{osc}}$. The main objective of this work is not to replace softmax in software but to provide a mathematically grounded blueprint for accurate attention on physical substrates.