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arXiv2026

A Framework Integrating the Dynamic Stiffness Matrix with Physics-Informed Neural Networks for Solving Eigenvalue Problems and Analysing Dynamic Response

This paper introduces a framework that integrates the dynamic stiffness matrix (DSM) with physics-informed neural networks (PINN). The DSM-PINN embeds physical constraints within the model and demonstrates robustness, particularly when addressing limited datasets across diverse investigations. In this approach, deep neural network outputs approximate the displacement fields of element nodes. Unlike the finite element method (FEM), the element shape functions are homogeneous solutions to the governing partial differential equation, forming the basis of the exact dynamic stiffness matrix, thereby avoiding high-order derivative terms. This matrix also serves as a frequency-domain spectral element, resulting in a strong-form PINN. The loss function is produced by connecting neural networks with dynamic stiffness matrices. We focus on utilising PINNs to resolve eigenvalue problems by employing the Wittrick-Williams algorithm, which overcomes the challenge of neural networks failing to converge to higher-order eigenvalues. Additionally, the frequency domain-PINN method is used to analyse structural dynamic responses under moving and impulsive loads, addressing the limitation of neural networks in handling complex numbers. Theoretical convergence stability of the suggested approach is also analysed even DSM is an indefinite matrix after implementing the boundary condition. The numerical results validate the practicality and efficacy of the recommended approach.

math.NA↗

arXiv2026

Rock, Paper, Scissors, ... Dynamite - A Model of Disruption from New Technologies

We seek to understand the effect of adding disruptive highly-capable new technologies to competitions by assessing the addition of Dynamite to Rock-Paper-Scissors. We find that providing a versatile Dynamite move to only one player provides limited value (win probability increases from 50% to 55.5%) and is played rarely. That value decreases further if the game is expanded beyond just the original three moves. We also observe several mechanisms by which prior moves can become strategically unplayable, or obsolete. We hope that this model illustrates some non-intuitive aspects of developing new versatile technologies. We also hope that it illustrates some pitfalls for developers and integrators to avoid in order to create value rather than merely capability.

physics.soc-ph↗

arXiv2026

A Spectral Identifiability Threshold for Dissipative Rate Recovery from Truncated Liouvillian Spectra

Open quantum systems lose energy and phase coherence through different dissipative processes, but these processes can produce overlapping dynamical signatures. The Liouvillian spectrum summarizes how such a system relaxes, yet it is not obvious how much of that spectrum is needed to distinguish the underlying dissipation rates. We study this question for amplitude damping and dephasing in a six-qubit Lindblad model whose spectrum can be derived analytically. We retain only the slowest non-steady spectral modes and ask how many are required before each dissipative rate becomes recoverable. We show that population modes contain no dephasing information, which creates a lower bound of D = 2^n retained modes for uniform dephasing identifiability in the relevant rate regime. The measured recovery threshold reaches this bound at n = 4,5,6, while n = 3 remains above it. At n = 6, least squares achieves a mean joint absolute error of order 10^-9, compared with 4.355 x 10^-4 for four tabular learning methods. Robustness tests show that this advantage weakens when the spectra are perturbed and when a transverse field breaks the commuting structure. These results show that the amount and structure of retained spectral information can determine whether dissipative parameters are recoverable, independently of the estimator used. The present conclusions apply to noise-free simulator spectra rather than measurement-derived spectra.

cs.LG↗

arXiv2026

Examining the robustness of Physics-Informed Neural Networks to noise for Inverse Problems

Approximating solutions to partial differential equations (PDEs) is fundamental for the modeling of dynamical systems in science and engineering. Physics-informed neural networks (PINNs) are a recent machine learning-based approach, for which many properties and limitations remain unknown. PINNs are widely accepted as less computationally efficient and accurate than traditional methods for solving PDEs, such as the finite element method. However, PINNs are commonly claimed to show promise in solving inverse problems and handling noisy or incomplete data. We compare the performance of PINNs in solving inverse problems with that of a traditional approach using the finite element method combined with a numerical optimizer. The models are tested on viscosity identification in 1D Burgers' equation and in 2D/3D Taylor-Green Vortex, in all cases with additive Gaussian noise applied to training and validation data. We find that while PINNs may require less human effort and specialized knowledge, they are outperformed by the traditional approach. For example, for 2D Taylor-Green Vortex with $σ$=1 noise, the baseline has a mean prediction RMSE of 0.0013 compared to 0.01 for the best PINN variation. However, PINNs scale better than the baseline with the computational complexity of the problem. We identify failures during training to be addressed if the PINN performance on noisy inverse problems is to become more competitive.

physics.comp-ph↗

arXiv2026

SDF-Aware Weighting: Adaptive Eikonal Regularisation for Three-Dimensional Level-Set Physics-Informed Neural Networks

Adaptive loss-balancing schemes for physics-informed neural networks rest on a premise that every residual should be driven to zero. For level-set advection with an eikonal regulariser that premise fails: the eikonal term penalises deviation of $\lVert\nablaϕ\rVert$ from unity, a property transport preserves only under rigid motion; where the exact solution departs from a signed-distance function the eikonal residual of the correct answer is nonzero, and driving it to zero moves the network away from that answer. We show that standard gradient-norm balancing fails in exactly this way, its weight remaining near its initial value throughout training on benchmarks where the property is violated, and we introduce SDF-Aware Weighting (SAW), which combines a residual-quantile gate with a gradient-norm ratio so that points exhibiting legitimate departure are excluded before the surviving term is scaled. Across four three-dimensional benchmarks SAW selects an eikonal weight within an order of magnitude of the value located by an eighteen-run manual sweep, spanning four decades from $10^{-1}$ to $10^{-5}$ with a single fixed configuration. On the slotted sphere, where the initial field is non-differentiable at reentrant edges, SAW attains a lower error than any weight in that sweep. Two smooth rigid benchmarks serve as controls: SAW is worse there, as expected when its premise does not hold. An ablation with the gate disabled shows the slot is nearly entirely filled while the relative $L_2$ error reads $1.06\%$, indistinguishable from a field that never represented the slot. We give a feature-restricted measure that separates the two cases.

physics.flu-dyn↗

arXiv2022

Hypergraph reconstruction from noisy pairwise observations

The network reconstruction task aims to estimate a complex system's structure from various data sources such as time series, snapshots, or interaction counts. Recent work has examined this problem in networks whose relationships involve precisely two entities-the pairwise case. Here we investigate the general problem of reconstructing a network in which higher-order interactions are also present. We study a minimal example of this problem, focusing on the case of hypergraphs with interactions between pairs and triplets of vertices, measured imperfectly and indirectly. We derive a Metropolis-Hastings-within-Gibbs algorithm for this model and use the algorithms to highlight the unique challenges that come with estimating higher-order models. We show that this approach tends to reconstruct empirical and synthetic networks more accurately than an equivalent graph model without higher-order interactions.

cs.SI↗

arXiv2026

Solving the Incompressible Navier-Stokes Equations on Oriented Curved Surfaces Discretized by Point Clouds

We present a meshfree numerical solver for the incompressible Navier-Stokes equations on oriented curved surfaces that are represented by surface point clouds. On curved surfaces, numerical challenges pertaining to stiffness and pressure-velocity coupling are exacerbated. Moreover, vector calculus on curved surfaces differs from its Euclidean counterpart. The presented method operates on surface point clouds in an Eulerian frame of reference without requiring a computational grid or mesh. It achieves consistent approximation in space and time with high order of accuracy; we demonstrate up to order six. The incompressibility constraint is locally imposed as a weak artificial compressibility approximation, avoiding global matrix inversion. We show that the method provides consistent and convergent approximations of surface vector fields and differential operators. We study the relationship between error, spatial resolution, and artificial Mach number and characterize the frequency spectrum of the artificial oscillations. We provide numerical solutions of the incompressible Navier-Stokes equations on symmetric surfaces, such as the sphere and torus, and on parametric and non-parametric asymmetric surfaces. Since the proposed method works directly on unstructured surface point clouds, it provides a promising approach for simulations on image-derived geometries, such as in biological morphogenesis from microscopy videos.

math.NA↗

arXiv2026

Arctic Dispersion Interruption Phenomenon and Sound Source Depth Estimation

The sound speed profile in the deep Arctic Ocean causes the surface layer to form a normal mode waveguide. When the depth of the sound source or receiver is near a node of the eigenfunction, the source cannot excite the mode, or the receiver cannot detect it, resulting in the modal amplitude at the receiver being approximately zero. This manifests as a dispersion interruption in the dispersion structure, which can be observed through time-frequency analysis of the received acoustic signal. Based on the interruption frequency identified from the received signal, combined with the relationship between node depth and modal frequency calculated from the ocean sound speed profile, the source depth can be estimated if the receiver depth is known. The phenomenon of dispersion interruption and the method of estimating source depth have been validated through simulations and experiments.

physics.app-ph↗

arXiv2026

Operator Learning for Predicting Bulk Wave Parameters of Spectral Wave Models

The impact of wave-induced forcing on the mean water level and nearshore currents is typically modeled through excess momentum fluxes, also known as radiation stresses, and their spatial gradients. Accurate storm surge prediction requires coupled circulation and wave models, but the high computational cost of numerical wave models limits their temporal resolution. In this work, we explore a proof-of-concept application of Deep Operator Networks (DeepONets) as a surrogate for the Simulating WAves Nearshore (SWAN) numerical wave model. Unlike grid-dependent surrogate models, DeepONets learn the underlying continuous operator, and thus, can provide highly efficient prediction while enabling discretization-invariant inference. The proposed surrogate model is evaluated using two distinct 1-D and 2-D steady-state numerical examples with variable boundary wave conditions and wind fields. When applied to a realistic numerical example of steady-state wave simulation in Duck, NC, the DeepONet surrogate improves computational efficiency by four orders of magnitude. Furthermore, the model demonstrates consistently high accuracy in predicting the significant wave height and the x- and y- components of the radiation stress gradient, by achieving relative L_2 errors bounded by 1.91%, 10.98%, and 6.88%, respectively, across all unseen test scenarios.

physics.comp-ph↗

arXiv2026

Wigner-Eckart Factorization of the Polyatomic Boltzmann Collision Operator

We extend the Wigner-Eckart factorization of the spectral Boltzmann collision operator to polyatomic gases with continuous internal energy. Because internal energies are invariant under spatial rotations, the SO(3) reduction survives the Borgnakke-Larsen energy exchange, and the twelve-dimensional collision integral collapses onto a nine-dimensional kinematic core. The core splits into a sparse geometric tensor, evaluated exactly, and a dense physical tensor, integrated by singularity-resolving Gauss rules with an auxiliary Laplace representation of the fractional energy couplings. The quadrature attains near machine precision at the fractional exponents of real gases. The collision invariants are embedded exactly, preserving the translational-internal energy exchange. The factorization compresses the operator by three to nearly four orders of magnitude and accelerates its evaluation 40-fold over dense formulations. The method is validated against the exact monatomic limit, Landau-Teller relaxation, and an analytic frozen-channel Prandtl number, and it matches a published calibration of the same kernel for N2, CO, and H2.

math.NA↗

arXiv2026

Quantum matrix arithmetics with Hamiltonian evolution

The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. We develop a suite of methods to perform matrix arithmetics -- with the result encoded in the off-diagonal blocks of a Hamiltonian -- using Hamiltonian evolutions of input operators. We show how to maintain this $\textit{Hamiltonian block encoding}$, so that matrix operations can be composed one after another, and the entire quantum computation takes $\leq 2$ ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian block encoded operators, analogous to the well known Hadamard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain of interest, while the constructed polynomial is upper bounded by the target function over the entire unit interval. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the doubly factorized tensor hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with $1$ ancilla qubit.

quant-ph↗

arXiv2026

Attention as Conditioning: What Classical Learning Theory Predicts About Linear Transformers

Attention is widely understood as an associative memory, but that description alone does not predict how the memory will behave. Predictive theories do exist, but in the literature on animal learning. We show that the state updates of the major linear-attention families are term-for-term identical with named models from a century of animal learning theory: linear attention implements Hebbian contiguity, DeltaNet implements Rescorla--Wagner error correction, and decay variants such as RetNet implement contiguity with a stimulus trace. This dictionary turns conditioning phenomena into testable statements about the in-context behavior of linear transformers, while distinguishing algebraic consequences from empirical measurements. Algebraically, it yields an exact closed form for Kamin blocking, verified in simulation to $<10^{-7}$ across five learning rates. Empirically, it predicts a dissociation that survives training on generic in-context association: error-correcting attention exhibits cue competition, whereas contiguity-based attention does not. A single state also has two capacity regimes, with measured scaling exponents of 1.22 for faithful retrieval and 1.89 for identification, consistent with linear and near-quadratic predictions. Across the full head grid, retrieval error is governed primarily by total state size rather than its partition across heads, indicating that heads provide capacity rather than redundant copies. We also prove no spontaneous recovery for the analyzed single-state recurrences under cue-orthogonal retention trials; with a never-presented-cue control and probes within the trained positional range, we likewise find no recovery in trained models. Finally, we introduce PH-attention, a Pearce--Hall-inspired rule with an explicit feature-indexed associability state that yields cue-dependent learning rates and is absent from the token-computed gates we compare.

cs.LG↗

arXiv2026

Analysis, thermodynamics, and a numerical solver for a pressure-temperature equilibrium closure of the four-equation model

We analyze an often used closure model for multi-material hydrodynamics where pressure-temperature equilibrium (PTE) is assumed for every state; emphasis is placed on tabular equations of state. This multi-material model is often referred to as the four-equation model. The identification of the admissible set is presented and is proven to be convex, setting the foundation for development of invariant-domain preserving methods for this model. A novel numerical method is presented for solving the highly nonlinear system for the equilibrated pressure and temperature with an arbitrary number of materials. This new method is compared with some traditional iterative solvers through a collection of different tests. Additionally, we provide a detailed analysis of the thermodynamics of the mixture model for general equations of state and prove existence and uniqueness of the pressure-temperature equilibrium solution under some thermodynamic assumptions.

math.NA↗

arXiv2026

ButterMamba: Butterworth-Enhanced Spatial-Temporal Mamba for Efficient Traffic Flow Prediction

Accurate traffic flow prediction is fundamental to intelligent transportation systems, playing a pivotal role in urban mobility optimization and smart city development. While Graph Neural Networks (GNNs) integrated with time series forecasting have emerged as promising solutions, two critical limitations persist: (1) the quadratic complexity of attention-based architectures hinders real-time deployment in large-scale networks, and (2) high-frequency noise in sensor data significantly degrades prediction reliability. These challenges are particularly acute in metropolitan scenarios where both computational efficiency and noise robustness are paramount. To address these limitations, we introduce \textbf{ButterMamba}, a novel and efficient framework based on State Space Models (SSMs). ButterMamba consists of two key components: (1) a Butterworth Spectral Filtering module that preprocesses the data by removing high-frequency noise, allowing the model to focus on significant underlying trends, and (2) a Spatial-Temporal State Mixer that uses a parallel Mamba architecture to efficiently capture both long-range temporal dependencies and complex spatial correlations across the road network. By decoupling noise filtering from spatial-temporal modeling, ButterMamba achieves superior predictive accuracy with linear computational complexity. Extensive experiments on three public datasets demonstrate that ButterMamba not only outperforms existing state-of-the-art models in terms of prediction accuracy but also considerably reduces training time and memory usage.

physics.soc-ph↗

arXiv2026

Dilemmas and trade-offs in the diffusion of conventions

Outside ideal settings, conventions are shaped by heterogeneous competing processes that can challenge the emergence of norms. In order to acknowledge this complexity, this paper develops a generalized account of conventions and identifies three trade-offs involved in their diffusion: (I) the trade-off between the imperatives of social, sequential, and contextual consistency that individuals balance when choosing between conventions; (II) the competition between local (bottom-up) and global (top-down) coordination, depending on whether individuals coordinate their behavior via interactions throughout a social network or external factors transcending the network; and (III) the balance between decision optimality (e.g., collective satisfaction) and decision costs when collectives with conflicting preferences choose a convention. A broadly applicable statistical physics framework for exploring these trade-offs is developed and applied to a sign convention in physics. The method can infer the structure of the underlying coordination game, the networks of social interactions involved, and the processes through which conflicts are resolved. This shows that the purpose of conventions may exceed coordination, and that individual preferences towards conventions are concurrently shaped by cultural factors and multiple social networks. Finally, this work emphasizes the role of leadership in the resolution of conflicts.

physics.soc-ph↗

arXiv2026

Criticality and universality in network dismantling

Identifying the smallest set of elements whose removal dismantle a complex network, known as the network dismantling problem, is a fundamental task with many practical applications. Whereas network dismantling has been extensively studied over the past decade, most work has focused on developing efficient algorithms for large but finite networks. By contrast, the physics of the network dismantling process, namely how the network structural connectivity is affected by the removal of nodes or edges, remains largely unexplored in the thermodynamic limit. Here, we shed light on this understudied aspect of network dismantling by introducing an adaptive biased percolation process able to optimally dismantle a network. Through a systematic analysis of synthetic network models, we find that the proposed percolation process displays a universal phase transition, characterized by the abrupt and simultaneous disappearance of both the giant connected component and the largest 2-core, across networks with markedly different degree distributions. Simulations on real networks further support this universality, indicating that the physics of network dismantling is insensitive to a broad range of topological properties. Together, these results suggest that a topology-agnostic theory could be developed to explain the critical behavior of network dismantling.

physics.soc-ph↗

arXiv2026

Toward Interaction Dynamics: A Predictive Framework for Safe Physical Human Robot Interaction

Physical human-robot interaction requires yielding transiently to contact yet recovering the commanded reference under sustained load. Finite-stiffness impedance control retains a static deflection there, while predictive alternatives typically optimize a nonlinear robot or impedance model online. Operational-space cancellation instead exposes a translational error double integrator with a fixed transition matrix and a configuration-scheduled input map, making interaction a predictive quantity rather than a property re-derived per configuration. We build on it a compact offset-free interaction-error MPC for torque-controlled manipulators: a force-domain random-walk state estimates persistent interaction and model error, and a 30-variable convex QP maps the correction through the current task inertia while constraining the applied joint torque. Conditional results establish impedance equivalence of the unconstrained passive feedback, offset-free regulation at feasible frozen configurations, and quadratic stabilizability of the scheduled backbone. In a 1kHz MuJoCo simulation of a 7-DOF Franka FR3, the estimator cuts steady-state error under a repeated 15N step from 2.77mm to 0.042mm when added to the otherwise identical 100Hz MPC. A stiffness-and-damping-calibrated impedance baseline attains 2.59mm but briefly saturates and needs 3.3x the peak positive joint power. Adding ideal measured-force cancellation to that baseline gives 1.39mm, so constant-load rejection is not unique to MPC; the sensorless controller still reaches 0.042mm in the moving task, a 65x reduction without force sensing and without the baseline's saturation or power cost. Demonstrated in simulation under a shared actuator budget, the contribution is an efficient operational-space realization complementing rather than replacing broader interaction-control architectures.

cs.RO↗

arXiv2026

Greedy recursion parameter selection for one-way spatial integration of hyperbolic equations

Solutions to hyperbolic systems comprise waves propagating at finite speeds. When wave propagation is predominantly unidirectional, one-way wave equations can be used to evolve only the right-going solution by removing support for left-going waves. The One-Way Navier-Stokes (OWNS) approach, which was originally developed for systems of first-order hyperbolic equations, constructs one-way approximations to the linearized Navier-Stokes equations using a recursive filter to remove left-going waves. The computational cost scales with the number of recursion parameters, which must be carefully chosen to ensure accuracy and stability of the resulting one-way equation. Previous work has chosen parameters based on heuristic estimates of key eigenvalues, which requires trial-and-error tuning while also yielding slow error convergence. We propose a greedy algorithm for automatic parameter selection, which we show yields faster convergence and a net decrease in computational cost for linear and nonlinear disturbance evolution in boundary-layer flows. We review the OWNS projection (OWNS-P) and recursive (OWNS-R) methods, comparing their convergence properties, and show through our numerical analysis and experiments that OWNS-P yields superior convergence and stability properties. Although we demonstrate the method for Navier-Stokes equations, we perform our analyses on systems of linear first-order hyperbolic equations and emphasize that the greedy algorithm is applicable to such systems.

math.NA↗