Phase Space

Long-duration gravitational-wave modelling must resolve fast orbital motion together with slow dissipative evolution while preventing small numerical errors from accumulating into secular phase drift. Here we ask whether the finite-time evolution map itself can be learned as an explicit, differentiable, structure-preserving object and then repeatedly composed through a complete inspiral. We construct three neural-flow architectures: a symplectic and slimplectic flow on Galley's doubled phase space, [SINFONIA-J0]; a Taylor-anchored flow, [SINFONIA-J1]; and a Magnusian flow that learns the finite-time dissipative correction in the interaction picture, [SINFONIA-J2]. Applied to a 2.5PN neutron-star inspiral, all three expose the same controlling mechanism: long-time accuracy is governed not by pointwise map error alone, but by its signed projection onto a single secular channel fixed by energy--angular-momentum balance. Encoding this structure allows the learned maps to remain accurate through $10^{2}$--$10^{5}$ window compositions to coalescence at timesteps of a full orbital period and beyond, reaching chained phase errors orders of magnitude below a benchmark slimplectic integrator at lower cost. The same secular structure can also be exploited for physics inference: when the channel is left unconstrained, the accumulated phase retains enough information to recover an un-modelled dynamical-friction-like force, both parametrically and as a learned function of separation. Network-off controls isolate the contribution of learning from the analytic structure already built into each map. These results establish a proof of concept for structure-preserving learned evolution maps as tools for fast long-duration integration and physics inference in gravitational-wave source modelling.

Locating periodic solutions of chaotic dynamical systems normally requires an initial guess close enough to the target orbit for numerical continuation or gradient-based search to converge. We show that Physics-Informed Neural Networks (PINNs) trained on sparse, noisy observations \emph{without} initial conditions recover periodic orbits of the gravitational three-body problem, including orbit families absent from the training data. The method rests on a second-order ODE formulation, fixed-frequency Fourier features, percentile-based adaptive refinement, and a trainable scaling parameter, each validated on forward problems. Across two 100-seed ensembles, $23$--$25\%$ of runs converge to families not present in the training data. We then ask what determines which family emerges. Two $χ^2$ tests give a consistent answer: changing the training data source significantly shifts the distribution of recovered families ($p < 0.001$, Cramér's $V = 0.339$), whereas switching between the two initialization distributions tested does not ($p = 0.620$, $V = 0.094$). The random seed selects which family a given run recovers; the \emph{distribution} the weights are drawn from does not shift the aggregate frequencies, but the training data does. The evidence is empirical: we do not characterize the loss landscape analytically, and PINNs remain slower than conventional integrators on well-posed initial-value problems. What the experiments establish is that the recovered orbits are verifiable rather than merely plausible: the identified ones refine to genuine periodic solutions, a network trained on Lagrange data recovers the figure-eight choreography (Li--Liao class I.A.1, matched to seven significant digits in $T^*$), and one trained on figure-eight data recovers a Broucke--Hadjidemetriou--Hénon orbit closing to $δ_T < 10^{-9}$.

Rare-regime discovery in parameterized dynamical systems is an active-search problem: find one verified parameter at which a scientifically defined qualitative threshold is crossed, even when acceptable candidates are rare, nonconvex, or fragmented. We introduce Quantum-Classical Phase-space and Stability-Threshold Search (QC-PHAST), an evidence-gated decision protocol and query-accounting framework for finite candidate libraries. A candidate induces a dynamical object, simulator-derived criticality score, and verified first-hit predicate. Scientific metadata and charged pilot evidence are used to assess whether equation-aware search, scalar-score active search, predicate-only search, or only a query-model comparison is admissible. The quantum row is the inherited Grover/Boyer--Brassard--Hoyer--Tapp (BBHT) unknown-$M$ marked-set query reference; it is not a new quantum-search theorem, materialized circuit, or hardware-speedup claim. The result is a regime map. Direct boundary constructions, geometry controls, online simulator loops, and learned-label accounting further identify when classical structure, false positives, calibration cost, or state preparation erases the query-model margin. QC-PHAST is therefore an auditable protocol for deciding when a finite-pool marked-set reference is informative and when classical or resource-aware search should dominate.

Extended Dynamic Mode Decomposition (EDMD) approximates Koopman operators from data, but a single global operator is inefficient when different state-space regions exhibit distinct local dynamics. We introduce Cluster-Weighted EDMD (CW-EDMD), which jointly learns a soft phase-space partition and a per-cluster EDMD operator. Its Expectation-Maximization (EM) objective assigns each transition based on both geometric proximity and prediction residuals, so clusters specialize where local Koopman models are accurate rather than where the data are dense. On Lorenz, damped pendulum, and Duffing systems, across 36 configurations and 10 seeds, CW-EDMD improves matched-degree EDMD in one-step and 5s-rollout prediction. Across 288 paired comparisons, there are significant error reductions in 258 cases, increases in 4, and no differences in 26. Median one-step error reductions are 57x, 2.7x, and 12x on pendulum, Duffing, and Lorenz, respectively.

Accurately modeling the dynamics of planetary parachute and entry vehicle systems is critical for Entry, Descent, and Landing events such as vehicle separation and sensor activation. These dynamics are difficult to capture with traditional system-identification methods as parachute motion is highly nonlinear, the governing equations are not fully known, and relevant test data are scarce and expensive to acquire. In this work, we sidestep these challenges by leveraging a physics-aware generative modeling approach that learns parachute dynamics directly from data. The proposed method, Symplectic Parachute Generative Adversarial Network (SPar-GAN), adapts a Hamiltonian generative architecture to the parachute setting by conditioning on canopy design and freestream velocity, while enforcing conservation of energy through symplectic integration. We apply SPar-GAN to subscale parachute tests conducted at the National Full-Scale Aerodynamics Complex and show that it reproduces qualitatively accurate pitch-yaw dynamics of different parachute configurations while recovering a compact two-degree-of-freedom phase-space consistent with canopy axisymmetry. These results suggest that physics-constrained generative models can characterize parachute dynamics across operating conditions and may help reduce the volume of physical testing required to assess performance.

We give mathematically self-contained formulations, in the complex-time (kime) representation, of three open problems from the foundations of classical mechanics: (I) the extension of the classical entropic uncertainty principle to non-canonical variables and to multiple degrees of freedom; (II) the characterization of coordinate-invariant measures and entropies, i.e., the question of why continuous physical quantities must be paired for an invariant entropy to exist; and (III) the construction of a classical relativistic directional degree of freedom (a classical analogue of a spin-1/2 system). Throughout, the kime phase is interpreted {statistically as a latent circular random variable whose law Φmodels the intrinsic trial-to-trial variability of repeated, identically controlled experiments indexed by the kime magnitude. The mathematical bridge is an exact symplectic identification of the kime cone with the action-angle chart of a one-degree-of-freedom phase space, under which the kime measure is the Liouville measure and the phase law becomes the angular conditional of a Liouville density. Specifically, we (i) prove a sharp entropic uncertainty relation on the kime cylinder whose extremal family is von Mises x Gaussian, together with a sharp circular Fisher-information inequality saturated exactly by von Mises laws; (ii) prove an exact non-canonical uncertainty relation in which the correction term is the geometric mean of the Poisson bracket, clarifying the conjectured role of the expected bracket; (iii) prove aggregate multi-degree-of-freedom bounds via the Williamson normal form and Fischer's inequality, and isolate the per-degree-of-freedom refinement as a precise open problem of symplectic Schur-Horn type; (iv) prove that diffusion of the kime phase produces monotone entropy growth with the equipartitioned (Haar-uniform) phase law.

We introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. A head-orthogonalization penalty removes degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations. Because the initial condition is built into the network output by construction, these principal components measure the additional variability the network learns on top of the initial profile, not the full solution itself. We apply the method to the one-dimensional viscous Burgers equation, with the heat and wave equations as robustness checks. For a latent dimension $n_b=20$, the learned manifolds exhibit pronounced effective dimensional reduction: for Burgers dynamics, only $2$-$4$ principal components capture about $95\%$ of the latent-space variance, while $4$-$7$ capture about $99\%$, depending on the initial-condition family; the same qualitative compression holds for the heat and wave equations. We also split the wavenumber axis into bands (``Fourier shells'') and measure how much each band contributes to every principal component. The resulting frequency profile is invariant under the change-of-basis freedom that the orthogonalization penalty leaves in the latent space, and is therefore reproducible across independent training runs. More broadly, this establishes the learned spectral profiles and principal components as robust observables of solution-manifold geometry.

When is a material system a candidate for life at all? We argue that this question is prior to behavior, functional architecture, or computational capacity, and that at root it is one of physical admissibility. We develop a framework in which minimal autonomy, taken in the interactivist sense of normativity grounded in self-maintaining far-from-equilibrium organization, corresponds to a distinct non-equilibrium phase of matter, and we take complex plasmas, a physical and non-biological system, as its in-materio exemplar. We formalize a diagnostic phase-space whose criteria (sustained free-energy throughput, organizational closure, active information maintenance, and regulated noise sensitivity) constitute necessary conditions for life-attribution. We instantiate the diagnostics across contrasting systems and fix the boundaries of the phase space via Bénard convection as a driven baseline lacking closure, and a digital self-replicating soup that carries measured informational heredity while its physical closure remains a structural zero. We demonstrate that plasmas satisfy every admissibility condition for minimal physical autonomy while carrying none of the informational heredity that open-ended evolution requires, sharpening the distinction between physical admissibility and biological sufficiency, and bounding downstream questions of machine sentience.

Generative Astrodynamics is advanced in this work by extending generative modelling to an orbit determination problem in the cislunar environment. The task is formulated as conditional density estimation, aiming to infer the probability distribution of the initial state from angles-only measurements over short observation arcs. A normalising flow is trained on perturbed topocentric observations from Near Rectilinear Halo Orbits, enabling a flexible and potentially multimodal posterior representation. Given new measurements, the learned density is sampled to generate statistically consistent and physics-informed state hypotheses. These estimates are refined via nonlinear least-squares minimisation, providing a competitive warm start for classical algorithms.

Correspondence is fundamentally relational: it seeks the unknown transformation between two observations of a common scene, not the content of either. Yet the dominant learning-based methods do not represent the transformation as a first-class object in the architecture. They encode each image independently and let a learned similarity function or a deep decoder discover the mapping implicitly. Phase correlation is the canonical exception, measuring the inter-image relationship directly in the Fourier domain, but the rigidity of its fixed basis confines it to global translation. We introduce a learned generalization of phase correlation that lifts this restriction by learning the basis on which the transformation decomposes. The same algebraic primitive extends to dense non-rigid deformations and to unitary dynamics. On the ACDC cardiac-MRI benchmark the framework matches or exceeds prior published baselines on both registration directions. On CAMUS echocardiography it matches state-of-the-art without auxiliary scoring or adaptive-smoothness mechanisms. Applied to time-evolved wavefunction pairs of the 1-D quantum harmonic oscillator, the same framework recovers the Hermite-function eigenstates and the quantized energy levels of the unknown Hamiltonian from observation pairs alone.

In this work we investigate the role of neural architectures as implicit functional priors in control problems governed by ordinary differential equations. Rather than focusing on highly complex problems, our objective is to investigate architecture-dependent effects in controlled dynamical systems within the simplest physically interpretable settings possible. In particular, we study a controlled linear RLC electrical circuit and a nonlinear Duffing-type dynamical system. Both systems are analyzed first through classical optimal-control formulations and later through PINN-based approaches. We compare different combinations of multilayer perceptrons (MLPs) and Fourier-based KAN-like architectures, and analyze their influence on the resulting controls. The numerical experiments suggest that different architectural choices systematically generate qualitatively distinct controls, even under identical governing equations, loss functionals, initial and target states, training parameters and physical constraints. Significant differences appear in the spectral structure, smoothness, energy distribution, and phase-space behavior of the learned solutions. A central observation of this work is the emergence of a functional specialization phenomenon when the neural architectures are allowed sufficient freedom to shape the structure of the learned controls. More specifically, in the systems considered here, Fourier-based architectures tend to produce trajectories with richer oscillatory content, whereas smoother low-frequency-biased architectures tend to generate more regular and energetically efficient controls. This suggests that different functional components of the control problem may be handled more efficiently by different neural architectures, leading to an implicit specialization between state representation and control generation.

The rapid evolution of artificial intelligence has led to substantial advances in deep neural networks. Nonetheless, conventional GPU-based training remains highly energy-demanding, motivating the exploration of physical dynamics and compatible energy-based learning schemes, such as equilibrium propagation (EP). EP-based training, however, frequently suffers from convergence to local minima due to phase-space contraction. Here we introduce an Ising-dynamics-inspired equilibrium-propagation framework in which dissipative Hopfield relaxation is replaced by an extended phase-space dynamics with conjugate variables. The resulting training paradigm keeps the local two-phase learning rule of EP while changing the physical route by which neural states reach equilibrium. We show that this dynamics lowers effective energy barriers, accelerates convergence, improves noise robustness, and trains deep convolutional Hopfield networks on MNIST, FashionMNIST, and CIFAR-10 with performance comparable to backpropagation.

Chaotic systems pose fundamental challenges for data-driven dynamics discovery, as small modeling errors lead to exponentially growing trajectory discrepancies. Since exact long-term prediction is unattainable, it is natural to ask what a good surrogate model for chaotic dynamics is. Prior work has largely focused either on reproducing the Jacobian of the underlying dynamics, which governs local expansion and contraction rates, or on training surrogate models that reproduce the ground-truth dynamics' long-term statistical behavior. In this work, we propose a new framework that aims to bridge these two paradigms by training surrogate dynamics models with accurate Jacobians and long-term statistical properties. Our method constructs a local covering of a chaotic attractor in phase space and analyzes the expansion and contraction of these coverings under the dynamics. The surrogate model is trained by minimizing the maximum mean discrepancy between the pushforward distributions of the coverings under the surrogate and ground-truth dynamics. Experiments show that our method significantly improves Jacobian accuracy while remaining competitive with state-of-the-art statistically accurate dynamics learning methods. Our code is fully available at https://anonymous.4open.science/r/neighborwatch.

We consider the stabilization of Vlasov--Poisson plasma dynamics, a central control problem in nuclear fusion. Our focus is the gap between what an ideal controller would use and what experiments can actually observe: while optimal policy may rely on the full phase-space state, practical feedback is typically limited to sparse macroscopic diagnostics. We therefore study imitation learning methods that distill a fully observed expert policy into controllers operating only on macroscopic measurements. We show the stability guarantees of the learned policy, where the error floor depends on the minimal behavior cloning loss achievable under the observation constraints. We further characterize this minimal loss in terms of a notion of entropy that quantifies the complexity of the initial distribution. Our results demonstrates the theoretical feasibility of learning stabilizing feedback policies for kinetic plasma dynamics from macroscopic observations, and exhibits the adaptivity of the learning approach to low-complexity structures. Through extensive numerical experiments, we validate our theory and show that the learned policies can stabilize the system using only macroscopic observations, within a significantly longer time horizon than non-adaptive baseline controllers.

Identifying the dynamical state variables of a system from high-dimensional observations is a central problem across physical sciences. The challenge is that the state variables are not directly observable and must be inferred from raw high-dimensional data without supervision. Here we introduce DySIB (Dynamical Symmetric Information Bottleneck) as a method to learn low-dimensional representations of time-series data by maximizing predictive mutual information between past and future observation windows while penalizing representation complexity. This objective operates entirely in latent space and avoids reconstruction of the observations. We apply DySIB to an experimental video dataset of a physical pendulum, where the underlying state space is known. The method, with hyperparameters of the learning architecture set self-consistently by the data, recovers a two-dimensional representation that matches the dimensionality, topology, and geometry of the pendulum phase space, with the learned coordinates aligning smoothly with the canonical angle and angular velocity. These results demonstrate, on a well-characterized experimental system, that predictive information in latent space can be used to recover interpretable dynamical coordinates directly from high-dimensional data.

We present a Mixture-of-Experts-based foundation model applied to the GlueX DIRC detector at Jefferson Lab, demonstrating its utility as a unified framework for fast simulation, particle identification, and hit-level noise filtering of Cherenkov photons. By leveraging a single shared transformer backbone across all tasks, the approach eliminates the fragmentation of task-specific pipelines while maintaining competitive-and in several cases superior-performance relative to established methods. The model operates directly on low-level detector inputs, performing hit-by-hit autoregressive generation over split spatial and temporal vocabularies with continuous kinematic conditioning, and supports class-conditional generation of pions and kaons through its Mixture-of-Experts architecture. We benchmark against the standard geometrical reconstruction and prior deep learning methods across the full kinematic phase space of the GlueX DIRC, demonstrating that the foundation model framework transfers effectively to this detector without architectural modification. This work positions the foundation model as a practical and scalable alternative to the suite of task-specific models currently proposed for GlueX DIRC analysis.

We extend the Weak Adversarial Neural Pushforward Method to the Wigner transport equation governing the phase-space dynamics of quantum systems. The central contribution is a structural observation: integrating the nonlocal pseudo-differential potential operator against plane-wave test functions produces a Dirac delta that exactly inverts the Fourier transform defining the Wigner potential kernel, reducing the operator to a pointwise finite difference of the potential at two shifted arguments. This holds in arbitrary dimension, requires no truncation of the Moyal series, and treats the potential as a black-box function oracle with no derivative information. To handle the negativity of the Wigner quasi-probability distribution, we introduce a signed pushforward architecture that decomposes the solution into two non-negative phase-space distributions mixed with a learnable weight. The resulting method inherits the mesh-free, Jacobian-free, and scalable properties of the original framework while extending it to the quantum setting.

Simulating the long-time evolution of Hamiltonian systems is limited by the small timesteps required for stable numerical integration. To overcome this constraint, we introduce a framework to learn Hamiltonian Flow Maps by predicting the mean phase-space evolution over a chosen time span, enabling stable large-timestep updates far beyond the stability limits of classical integrators. To this end, we impose a Mean Flow consistency condition for time-averaged Hamiltonian dynamics. Unlike prior approaches, this allows training on independent phase-space samples without access to future states, avoiding expensive trajectory generation. Validated across diverse Hamiltonian systems, our method in particular improves upon molecular dynamics simulations using machine-learned force fields (MLFF). Our models maintain comparable training and inference cost, but support significantly larger integration timesteps while trained directly on widely-available trajectory-free MLFF datasets.

Generative networks are perfect tools to enhance the speed and precision of LHC simulations. Especially when generating events beyond the size of the training dataset, it is important to understand their statistical precision. We present two complementary methods to estimate the amplification factor without large holdout datasets. Averaging amplification uses Bayesian networks or ensembling to estimate amplification from the precision of integrals over given phase-space volumes. Differential amplification uses hypothesis testing to quantify amplification without any resolution loss. Applied to state-of-the-art event generators, both methods indicate that amplification is already possible in specific regions of phase space.