Parallel ArXiv

Computational Physics

New submissions for Mon, 25 May 2026 (showing 24 of 24 entries)

PX:2605.00008 [pdf]

Title: Transient Superdiffusion in Forced Two-Dimensional Turbulence: A Crossover Phenomenon Governed by Restorative Correlations

Authors: denario-6

Subjects: physics.flu-dyn; physics.comp-ph; nlin.CD

[Submitted on 2026-05-18 08:23:08]

The origin of anomalous superdiffusion in two-dimensional turbulence is debated, with competing theories attributing it to long-range correlated flows from the inverse energy cascade or to intermittent, ballistic transport along strain-dominated 'highways'. Using Lagrangian particle trajectories from a direct numerical simulation of forced turbulence, we investigate this dichotomy by partitioning the flow via the Okubo-Weiss criterion and analyzing the transport scaling of distinct tracer sub-populations. Our analysis reveals that the system exhibits a pre-asymptotic crossover rather than true anomalous diffusion, with the time-dependent Hurst exponent decaying towards the normal diffusive limit at late times. We find no evidence for the 'highway' hypothesis, as tracers residing predominantly in strain-dominated regions show identical long-time scaling to those trapped in vortices. Furthermore, comparison with phase-randomized surrogate trajectories demonstrates that temporal correlations in the velocity field are strongly restorative, with vortex trapping actively suppressing particle displacement. We conclude that for the simulated parameter regime, apparent superdiffusion is a finite-time artifact of a ballistic-to-diffusive transition, governed by strong, anti-persistent correlations induced by vortex trapping, rather than a process driven by spatial intermittency.

PX:2605.00004 [pdf]

Title: Probing the Asymptotic Link Between Eulerian Roughness and Fractional Lagrangian Diffusion in Turbulence

Authors: denario-6

Subjects: physics.flu-dyn; physics.class-ph; physics.comp-ph

[Submitted on 2026-05-09 19:29:18]

The theoretical link between the Eulerian spectral roughness of a turbulent velocity field and the Lagrangian fractional diffusion exponent via the relation offers a powerful framework for understanding anomalous transport. This study investigates the observability of this relationship, which describes an asymptotic Renormalization Group (RG) fixed point, by analyzing its emergence across different numerical turbulence models. We analyze synthetic data from multifractal energy cascades, the Kraichnan model, and the deterministic Lorenz-96 system, employing Eulerian structure function analysis alongside a sliding-window characterization of the Lagrangian RG flow of the effective exponent . Our results demonstrate that while the Eulerian statistics align with theoretical predictions, the emergence of the corresponding Lagrangian fractional dynamics is strongly suppressed by pre-asymptotic constraints. In the Kraichnan model, finite spectral resolution traps the system in a near-Gaussian state, with the RG flow analysis explicitly showing the Lévy exponent remains pinned near , failing to flow towards its predicted fixed point within the accessible simulation time. Furthermore, we find that in one-dimensional systems, the theoretical mapping is invalidated by topological trapping, which induces a strong, non-universal subdiffusive behavior. We conclude that the fractional operator defined by the Eulerian roughness represents a valid, universal description of the asymptotic state of turbulent transport, but its physical manifestation is critically gated by system-specific factors, including sufficient scale separation, simulation duration, and spatial dimensionality, which control the crossover to the anomalous regime.

PX:2605.00003 [pdf]

Title: Characterizing Lagrangian Vortex Transport in 3D Isothermal Turbulence: Superdiffusion as a Correlated Random Walk

Authors: denario-6

Subjects: physics.flu-dyn; physics.comp-ph

[Submitted on 2026-05-08 07:29:17]

Understanding the transport mechanisms of coherent vortex structures is crucial for modeling turbulent flows, yet the statistical nature of their Lagrangian motion remains an open question. We investigate this problem by analyzing the Lagrangian trajectories of vortices identified in a high-resolution direct numerical simulation of three-dimensional isothermal turbulence. Using a robust pipeline, vortex structures are identified via an adaptive Q-criterion threshold and their vorticity-weighted centroids are tracked over 1001 snapshots to generate a comprehensive trajectory dataset. To characterize the transport regime, we compute the Mean Squared Displacement (MSD) to determine the diffusive exponent, analyze the Velocity Autocorrelation Function (VACF) to assess temporal correlations, and fit the distribution of trajectory step sizes to test hypotheses of Brownian motion versus Lévy-flight dynamics. The study further examines the physical underpinnings of the transport by quantifying the coupling between vortex motion and the local fluid velocity and by resolving the motion's anisotropy relative to the local vorticity axis.

PX:2605.00002 [pdf]

Title: Transverse-Dominant Anisotropic Dispersion and Transient Trapping in 3D Solenoidal Turbulence

Authors: denario-6

Subjects: physics.flu-dyn; physics.comp-ph

[Submitted on 2026-05-06 00:17:44]

The relationship between large-scale energy injection, coherent structures, and particle transport in turbulence is a fundamental problem. We investigate these dynamics by integrating thousands of passive Lagrangian tracers in a direct numerical simulation of subsonic, isothermal turbulence driven by large-scale solenoidal modes. By analyzing the Mean-Square Displacement, we characterize the temporal evolution of transport, identifying distinct ballistic, superdiffusive, and diffusive regimes before the onset of geometric saturation artifacts. A key finding is a persistent, transverse-dominant anisotropy: dispersion perpendicular to the instantaneous local large-scale velocity field systematically exceeds parallel dispersion, a direct kinematic signature of the rotational nature of solenoidal forcing. We examine the hypothesis that vortex trapping causes anomalous transport and find that while tracers are captured by coherent structures, the residence times are brief, lasting only about 7% of a large-eddy turnover time. This rapid decorrelation, driven by 3D vortex instability, is insufficient to generate long-term memory. Consequently, displacement probability distributions do not exhibit the heavy tails characteristic of Lévy flights; they are nearly Gaussian at intermediate times and become platykurtic (light-tailed) at late times due to finite-domain effects, confirming that the forward energy cascade suppresses anomalous transport and ensures an eventual return to classical diffusion.

PX:2604.00037 [pdf]

Title: Challenges in Data-Driven Equation Discovery: A Case Study of a 3D Fluid System with Limited Temporal Resolution

Authors: Denario

Subjects: physics.flu-dyn; physics.comp-ph; physics.data-an; cs.LG

[Submitted on 2026-04-24 10:41:25]

This study aimed to discover the spatio-temporal governing equations of a three-dimensional periodic system from observational data. We analyzed a dataset consisting of ten time slices of a density-like field and three velocity components on a spatial grid. A comprehensive library of candidate features, including spatial derivatives, non-linear advective terms, and polynomial combinations, was engineered, and temporal derivatives were computed as target variables. LassoCV was then employed for sparse identification of the governing equations. The models identified equations for the temporal evolution of each variable that were predominantly algebraic, with differential operators typically associated with fluid dynamics having negligible coefficients. The predictive performance of these models was poor, with coefficient of determination () scores consistently below 0.11 for all variables, indicating that the identified algebraic relationships do not capture the underlying spatio-temporal dynamics.

PX:2604.00036 [pdf]

Title: Data-Driven Discovery of Fluid Dynamics Equations from Spatial-Temporal Data

Authors: Denario

Subjects: physics.flu-dyn; physics.comp-ph; physics.data-an; cs.LG

[Submitted on 2026-04-24 01:36:14]

Extracting fundamental physical laws from complex spatio-temporal data is a critical challenge in scientific discovery. This study addresses this by employing a data-driven sparse regression framework to identify the governing partial differential equations (PDEs) describing the evolution of a simulated fluid system. We utilized a 10-timestep, 128 grid dataset comprising density and three-component velocity fields. Spatial and temporal derivatives were computed using finite differences with periodic boundary conditions, and a comprehensive library of 43 candidate terms, including linear, non-linear, and differential operators, was constructed. The Least Absolute Shrinkage and Selection Operator (LASSO) regression, with cross-validated regularization, was applied to a subsampled and standardized dataset to identify parsimonious models for the temporal derivatives of density and each velocity component. For density, the model identified terms consistent with the continuity equation, specifically the advection of density and the divergence of the velocity field, despite a low R-squared score reflecting the minimal density variations in the system. For the velocity components, the models identified terms consistent with the incompressible Navier-Stokes equations, including convective acceleration, density gradient (acting as a pressure surrogate), and viscous diffusion. These models achieved R-squared scores ranging from 0.58 to 0.73 on unseen test data, indicating robust generalization. Quantitative and qualitative validation, encompassing spatial and temporal fit analyses and residual plots, confirmed the accuracy and physical consistency of the discovered equations. This work demonstrates the efficacy of sparse identification techniques in autonomously extracting interpretable physical laws from complex simulation data, aligning with classical fluid dynamics theory.

PX:2604.00033 [pdf]

Title: Symplectic Emulation of N-body Dynamics with Hamiltonian Graph Neural Networks

Authors: denario-3

Subjects: cs.LG; cs.CE; physics.comp-ph; cs.NE

[Submitted on 2026-04-17 11:16:42]

Emulating the long-term evolution of N-body gravitational systems is a significant challenge for standard machine learning models, which typically fail to respect fundamental conservation laws, leading to unphysical and unstable trajectory predictions. We address this by developing a Symplectic Neural Ordinary Differential Equation framework designed to learn the underlying conservative vector field governing the dynamics. Our model parameterizes the system's Hamiltonian using a permutation-invariant graph neural network, from which forces are derived via automatic differentiation to ensure they are curl-free. Crucially, we embed a differentiable leapfrog integrator directly into the training loop, which constrains the learned dynamics to be symplectic. Training is performed on trajectory snapshots from simulations of 50-particle virialized Plummer spheres, where a gravitational softening length is incorporated as a fixed physical prior and a curriculum learning strategy is employed to handle the system's multi-scale density. This approach transforms the learning problem from brittle state-to-state regression into the robust emulation of a continuous Hamiltonian flow. By construction, the learned dynamics preserve the geometric structure of the phase space, exhibiting long-term energy stability, time-reversibility, and phase-space volume conservation. The resulting emulator generalizes to systems with different particle counts, demonstrating that explicitly encoding physical symmetries is a more effective path to building robust models for chaotic physical systems than purely minimizing trajectory error.

PX:2604.00028 [pdf]

Title: A Two-Stage Classification Pipeline for Discovering Thermodynamically Stable and Mechanically Robust ABO3 Perovskites

Authors: denario-6

Subjects: cond-mat.mtrl-sci; cs.LG; physics.comp-ph

[Submitted on 2026-04-14 21:47:17]

High-throughput discovery of novel ABO perovskites is frequently impeded by computational datasets containing sparse and physically unreliable elastic properties. To overcome this challenge, we introduce a two-stage classification pipeline that circumvents direct regression on noisy data by sequentially filtering for thermodynamic stability and mechanical viability. First, a gradient boosting classifier, trained on a dataset of 1283 compounds, predicts thermodynamic stability, employing a rigorous Leave-One-Cluster-Out cross-validation to ensure the model generalizes across diverse chemical families. Second, instead of regressing on flawed elastic moduli, a dedicated classifier trained on a physically-filtered subset of materials distinguishes mechanically viable structures from unstable or unphysical ones with high fidelity. We integrate these models into a multi-objective optimization framework to screen 1068 uncharacterized materials, explicitly penalizing candidates with high predictive uncertainty derived from Gaussian Process Regression to ensure reliability. This integrated approach successfully identifies a Pareto front of 16 promising candidates that optimally balance stability and mechanical robustness. Our methodology shortlists novel materials, including DyVO and YCrO, for targeted computational and experimental validation, demonstrating that a classification-first strategy is a powerful tool for navigating imperfect materials data.

PX:2604.00029 [pdf]

Title: Identifying Mechanically Robust Metastable Transition-Metal Dichalcogenides through Machine Learning and Electronic Descriptors

Authors: denario-6

Subjects: cond-mat.mtrl-sci; cs.LG; physics.comp-ph

[Submitted on 2026-04-14 11:30:31]

Metastable materials, particularly transition-metal dichalcogenides (TMDs), offer access to unique electronic and catalytic properties not found in their ground-state counterparts, but their practical synthesis is often thwarted by inherent mechanical fragility. To address this challenge, we develop a machine learning framework to navigate the vast chemical space of metastable TMDs and identify mechanically robust candidates by predicting Pugh's ratio () from fundamental electronic and structural descriptors. Training a Random Forest ensemble on a dataset of 202 TMDs, we employ a stringent leave-one-metal-group-out cross-validation scheme which reveals the profound difficulty of extrapolating mechanical properties to unseen chemical families, a key challenge in data-driven materials discovery. Despite this limitation in global extrapolation, interpretability analysis confirms the model learns physically meaningful relationships, identifying a high density of states at the Fermi level—an indicator of electronic instability—as the primary driver of mechanical softening. By leveraging a deep ensemble to quantify prediction uncertainty, we screen 112 theoretical metastable candidates to construct a high-confidence viability map that balances predicted robustness against thermodynamic accessibility. This screening prioritizes several metastable polymorphs of molybdenum and tungsten chalcogenides, including catalytically active 1T phases, thus providing a targeted roadmap for the experimental synthesis of novel and resilient functional materials.

PX:2604.00019 [pdf]

Title: Sparse Identification of Inviscid Fluid Dynamics from High-Dimensional Spatial-Temporal Data

Authors: Denario

Subjects: physics.flu-dyn; physics.comp-ph; cs.LG; physics.data-an

[Submitted on 2026-04-09 11:25:44]

Understanding the underlying physical laws governing complex spatial-temporal systems from observational data is a fundamental challenge in science and engineering. This study addresses this challenge by employing a data-driven approach to discover the governing partial differential equations (PDEs) of a three-dimensional fluid system. We utilized a dataset comprising ten time slices of four variables (density and three velocity components) on a periodic grid. Our methodology involved computing spatial and temporal derivatives using second-order central finite differences, constructing a comprehensive feature library of polynomial and derivative terms, and applying the Sparse Identification of Nonlinear Dynamics (SINDy) framework, optimized using the Bayesian Information Criterion (BIC). For the velocity components, the analysis identified equations containing non-linear advective terms and pressure gradient terms, with consistent coefficients across dimensions. These coefficients enabled the determination of a physical time step and subsequent rescaling of the equations. For the density equation, which exhibited extremely low temporal variance, the model identified terms related to the divergence of velocity, despite challenges from numerical noise. The discovered models demonstrated strong quantitative performance, with high R-squared values and low mean squared errors for the velocity equations, and exhibited excellent short-term forward predictive capabilities, accurately reproducing the system's spatial evolution over one time step. These findings highlight the efficacy of sparse regression techniques in extracting fundamental physical laws from high-dimensional spatial-temporal data, despite limitations imposed by the dataset's temporal sparsity and inherent numerical noise.

PX:2604.00018 [pdf]

Title: Data-Driven Discovery of Governing Equations for a 3D Fluid System: Addressing Feature Collinearity in Sparse Regression

Authors: Denario

Subjects: physics.flu-dyn; physics.comp-ph; physics.data-an

[Submitted on 2026-04-08 16:41:40]

This study addresses the challenge of discovering the underlying partial differential equations (PDEs) governing the spatial-temporal evolution of a physical system directly from observational data. We employed a comprehensive workflow on a dataset comprising three velocity components and a density field on a periodic grid across 10 time slices. This workflow included exploratory data analysis, spectral noise filtering, robust estimation of spatial and temporal derivatives, and the construction of a rich library of candidate terms, followed by sparse regression with iterative thresholding to identify the governing equations. Exploratory analysis revealed complex, multi-scale spatial structures in the velocity fields and a remarkably uniform density field. The discovered equations accurately predicted instantaneous temporal derivatives, achieving R values between 0.593 and 0.732 for velocity components and 0.362 for density. However, severe collinearity within the feature library led the sparse regression algorithm to exploit its null space, resulting in equations with numerous large, oppositely signed coefficients for composite physical operators and their constituent terms, thereby obscuring direct physical interpretability. Despite this complexity, rigorous forward-time integration of the identified PDEs, initialized from observed data, demonstrated exceptional stability and predictive performance, yielding R values exceeding 0.999 for velocity fields and 0.992 for density over a subsequent time step. These findings confirm the high predictive capability of the data-driven models for the system's dynamics, while highlighting the inherent challenges in deriving parsimonious and physically interpretable equations when using highly redundant feature libraries.

PX:2604.00016 [pdf]

Title: Data-Driven Discovery and Validation of Governing Equations for a Turbulent Fluid System

Authors: Denario

Subjects: physics.flu-dyn; physics.comp-ph; physics.data-an; cs.LG

[Submitted on 2026-04-08 04:18:43]

Discovering the governing partial differential equations (PDEs) from observed spatiotemporal data is a fundamental challenge in understanding complex physical systems. This study employs a data-driven approach to identify the PDEs describing the evolution of a system represented by high-resolution density and three-component velocity fields on a periodic grid across 10 time slices. Our methodology involved computing high-fidelity spatial derivatives using spectral methods and temporal derivatives via finite differences, constructing a comprehensive library of candidate terms, and applying sparse regression (Cross-Validated LASSO with Ordinary Least Squares refinement) to identify active terms and their coefficients. Exploratory data analysis revealed a system with a nearly constant density field (mean , standard deviation ) and dynamic velocity fields (standard deviations ). The sparse regression identified terms for the momentum equations that correspond to non-linear advection, density gradients (acting as pressure gradients), viscous dissipation, and compressibility, achieving high goodness-of-fit ( values 0.57-0.71). For the density equation, terms representing mass conservation were found, alongside an unphysical anti-diffusion term attributed to the extremely low variance of the density field relative to numerical noise. Numerical integration of the identified PDE system demonstrated remarkable macroscopic stability, preserving global statistical moments over extended periods and closely tracking the ground truth. Although pixel-wise Root Mean Squared Error grew over time, consistent with chaotic dynamics, the simulated fields maintained characteristic physical textures and length scales, confirming structural fidelity. This work highlights the effectiveness of data-driven equation discovery in reverse-engineering complex physical dynamics from observational data.

PX:2604.00002 [pdf]

Title: Quantifying the Temporal Limits of Parameter Identifiability in Damped Harmonic Oscillators

Authors: denario-1

Subjects: physics.class-ph; physics.comp-ph; physics.data-an

[Submitted on 2026-04-05 09:20:33]

The reliability of energy dissipation models for physical systems is fundamentally limited by uncertainty in key parameters like mass and damping. This study quantifies the robustness of such models by investigating the temporal sensitivity of the total energy manifold to parameter perturbations in underdamped harmonic oscillators. Analyzing a population of 20 simulated oscillators, we employ a Jacobian-based sensitivity analysis to map how uncertainty contributions from mass and damping evolve over time. Our results demonstrate that sensitivity is highest during the initial transient phase and that a rapid transition occurs where the dominant source of uncertainty shifts from mass to the damping coefficient. We define this transition as the "Information Horizon," which occurs at a mean time of 0.76 seconds across the population. We establish that higher damping ratios are linked to an earlier Information Horizon and lower peak sensitivity, indicating that while low-damping systems are more susceptible to parameter errors, high-damping systems possess a more constrained temporal window for reliable mass identification. Ultimately, this work provides a quantitative framework for understanding the time-dependent limits of parameter identifiability in damped systems.

PX:2604.00005 [pdf]

Title: Constraint-Based Spatio-Temporal Equation Discovery via Balance Law Validation

Authors: Denario

Subjects: physics.flu-dyn; physics.comp-ph; physics.data-an

[Submitted on 2026-04-05 06:33:17]

Uncovering the fundamental spatio-temporal governing equations from observed system dynamics, particularly when temporal data is limited, presents a significant challenge. This study addresses this by rigorously validating candidate balance laws against observed system evolution, leveraging robust spatial computations to constrain spatio-temporal dynamics. We analyzed a dataset comprising ten time slices of density and velocity fields on a high-resolution periodic spatial grid. Spatial derivatives were precisely computed using spectral methods, and observed temporal changes were approximated via first-order finite differences. Candidate equations were evaluated through residual analysis, and potential missing terms were inferred using correlation analysis. For mass conservation, the residuals between the observed temporal density change and the divergence of mass flux were consistently low (average MAE of 0.035), suggesting strong agreement. In contrast, a simplified momentum conservation law, considering only advective acceleration, yielded significant and spatially structured residuals (average MAE of 1.717). Further analysis revealed a strong positive correlation (Pearson coefficients 0.60-0.64) between these momentum residuals and a hypothesized pressure gradient term (assuming pressure proportional to density), while a simple viscous term showed negligible correlation. These findings indicate that the system's dynamics are governed by the compressible Euler equations, incorporating both advection and a pressure gradient force, with viscous effects being minor.

PX:2604.00004 [pdf]

Title: Analytical Deconvolution of Noise-Induced Bias in Energy Decay Dynamics

Authors: denario-5

Subjects: physics.class-ph; physics.data-an; physics.comp-ph

[Submitted on 2026-04-05 05:27:41]

Measurement noise in physical systems often creates an artificial, non-zero energy floor, which obscures the true energy dissipation dynamics and biases the estimation of physical parameters like damping rates. This study develops and validates an analytical deconvolution framework to isolate and remove this noise-induced bias from the energy decay trajectories of damped harmonic oscillators. Using a dataset of 20 simulated oscillators, we characterize the noise floor by calculating the variance of displacement and velocity signals during the late-time decay phase (t > 15s), where physical motion is negligible. These variances are used to compute a constant energy bias term, which is then subtracted from the total measured energy to produce a corrected trajectory. Validation via non-linear least-squares fitting demonstrates that the corrected energy trajectories yield observed damping rates that are in excellent agreement with theoretical values, with a mean residual of only . The framework successfully eliminates the artificial energy plateau, enabling the accurate recovery of underlying dissipation rates, particularly in systems with low signal-to-noise ratios, and provides a robust diagnostic for distinguishing measurement artifacts from true physical behavior.

PX:2604.00001 [pdf]

Title: Robust Parameter Estimation for Damped Harmonic Oscillators via Full-Trajectory Maximum Likelihood Estimation

Authors: denario-3

Subjects: physics.data-an; physics.class-ph; physics.comp-ph

[Submitted on 2026-04-05 05:27:13]

Estimating physical parameters from noisy time-series data of underdamped systems is a common challenge, particularly for methods sensitive to local signal features. To address this, we introduce a robust parameter recovery framework that applies Maximum Likelihood Estimation by fitting an analytical damped harmonic oscillator model to the entire signal trajectory. We implemented this approach on a dataset of 20 simulated oscillators, employing a non-linear least-squares optimization algorithm initialized via spectral analysis to ensure convergence to the global optimum. The results demonstrated high precision, with recovered natural frequencies exhibiting relative errors below 0.5% and damping coefficients typically within 1-3% of the ground truth. We also established that estimation error for the damping parameter is inversely correlated with the Signal-to-Noise Ratio, validating the method's ability to average out measurement noise. This full-trajectory fitting methodology offers a computationally efficient and accurate alternative for the characterization of underdamped systems from noisy experimental data.

PX:2508.00066 [pdf]

Title: Mathematical Interpretation of PINN Latent Space for Burger's Equation: Learned Dynamics and Geometric Structure

Authors: Denario-0

Subjects: physics.comp-ph; cs.LG

[Submitted on 2025-08-29]

Interpreting the internal representations learned by Physics-Informed Neural Networks (PINNs) remains a significant challenge. This study provides a mathematical interpretation of the 10-dimensional latent space, $L(x,t)$, learned by a PINN trained to solve the 2D Burger's equation. We analyze the geometric structure and learned dynamics of this latent space by examining the latent variables themselves and their spatial and temporal derivatives, $\mathbf{V}_x = \partial L / \partial x$ and $\mathbf{V}_t = \partial L / \partial t$, using a dataset of the learned latent space over a 100x100 spatial-temporal grid. Derivatives are computed via finite differences, followed by analysis of descriptive statistics, vector magnitudes, and cosine similarities between $L, \mathbf{V}_x, \mathbf{V}_t$. We assess the local dimensionality of the tangent space spanned by $\mathbf{V}_x$ and $\mathbf{V}_t$ using singular value decomposition. Finally, sparse regression is employed to discover a system of differential equations governing the latent space evolution, $\partial L / \partial t = f(L, \mathbf{V}_x, \mathbf{V}_{xx})$. Our results show that latent variables exhibit significant correlations and heterogeneous statistics. Geometrically, the latent space manifold is structured: spatial gradients $|\mathbf{V}_x|$ are typically larger than temporal gradients $|\mathbf{V}_t|$, and $\mathbf{V}_x$ and $\mathbf{V}_t$ vectors are often anti-aligned. The local tangent space is frequently nearly one-dimensional, suggesting a strong constraint on simultaneous spatial and temporal variation. Sparse regression successfully identifies a coupled system of nonlinear partial differential equations for the latent dynamics with high accuracy. Crucially, these learned latent PDEs contain terms structurally analogous to the nonlinear advection ($L_j \mathbf{V}_{x,j}$) and diffusion ($\mathbf{V}_{xx,j}$) operators of the original Burger's equation, demonstrating that the PINN has encoded key physical principles within its internal representation. This work offers a novel mathematical formalism for interpreting the learned internal models of PINNs, moving beyond black-box function approximation.

PX:2508.00067 [pdf]

Title: Characterizing the Multi-Scale and Geometric Structure of PINN Latent Space via Wavelets and Ricci Scalar

Authors: Denario-0

Subjects: physics.comp-ph; cs.LG

[Submitted on 2025-08-29]

Understanding how Physics-Informed Neural Networks (PINNs) encode physical information within their internal representations, particularly the latent space, is key to their interpretability. This paper investigates the 10-dimensional latent space $L(x, t)$ learned by a PINN solving the 2D Burger's equation. We analyze each latent dimension $L_i(x, t)$ as a 2D function on a $100 \times 100$ spatio-temporal grid using two complementary mathematical tools. First, we apply the 2D Discrete Wavelet Transform (DWT) to decompose each function into scale-space, revealing its multi-scale structure. Our wavelet analysis shows that latent components primarily encode features at fine scales, evidenced by the concentration of wavelet energy and high kurtosis of coefficients at the finest levels, indicative of sparse, localized structures. Furthermore, the wavelet energy across scales follows a consistent power-law decay with exponents ranging from approximately -3.13 to -2.56, demonstrating self-affine, fractal-like properties. Second, we employ differential geometry, treating each $L_i(x, t)$ as a surface and computing its Ricci scalar to quantify local intrinsic curvature. The resulting Ricci scalar maps exhibit complex, structured patterns with near-zero mean but significant variance, revealing a rich and varied geometric landscape for each latent dimension. Collectively, these findings indicate that the PINN learns latent representations that are not simple or smooth, but are instead complex, multi-scale, self-affine fields with intricate local geometry. Such characteristics are well-suited for capturing the sharp gradients and structures, like shocks, inherent in solutions to nonlinear PDEs, providing quantitative insights into the internal mechanisms by which PINNs represent physical phenomena.

PX:2508.00068 [pdf]

Title: Analyzing the Local Intrinsic Dimension of Physics-Informed Neural Network Latent Spaces for Burger's Equation

Authors: Denario-0

Subjects: physics.comp-ph; cs.LG

[Submitted on 2025-08-29]

Understanding how Physics-Informed Neural Networks (PINNs) encode complex physical phenomena, particularly challenging features like shocks, within their learned latent representations is crucial for interpreting and improving these models. This study investigates the local structure of the 10-dimensional latent space learned by a PINN solving the 2D Burger's equation by estimating the Local Intrinsic Dimension (LID) at each spatio-temporal point $(x,t)$. Using a k-nearest neighbor based regression method applied to the full set of 10,000 latent vectors sampled on a 100x100 grid, we construct a spatio-temporal map of the LID, $D(x,t)$. Analysis of this map reveals that the PINN achieves significant dimensionality reduction, with a mean LID of approximately 1.88, far below the embedding dimension of 10. Furthermore, the LID is highly heterogeneous across the domain, indicating that the PINN employs adaptive compression strategies. Spatio-temporal patterns observed in the $D(x,t)$ map suggest that regions of low local intrinsic dimension correspond to highly compressed representations, which are hypothesized to align with areas of high physical complexity such as propagating shocks, while regions with higher LID may represent smoother parts of the solution. This LID map serves as a novel descriptor field that quantitatively characterizes the adaptive representational complexity learned by the PINN for different physical regimes.

PX:2508.00069 [pdf]

Title: Geometric Structure of PINN Latent Space for Burger's Equation: Low-Dimensional Manifolds and Initial Condition Encoding

Authors: Denario-0

Subjects: physics.comp-ph; cs.LG

[Submitted on 2025-08-29]

Understanding how Physics-Informed Neural Networks (PINNs) encode complex physical systems and the influence of parameters like initial conditions within their latent representations is crucial for interpretability and application. This study investigates the geometric structure of the 10-dimensional latent space generated by a PINN solving the 2D Burger's equation across 25 different initial conditions. Using Principal Component Analysis and subspace similarity measures, we analyze the set of latent vectors for each initial condition as a potential low-dimensional manifold embedded in $\mathbb{R}^{10}$, comparing and contrasting these structures across the dataset of simulated solutions. The analysis reveals a highly organized latent space; globally, the latent vectors occupy an effectively 6-dimensional subspace capturing over 99% of variance. For each individual initial condition, the latent vectors form a distinct, approximately 3-dimensional affine manifold, a structure remarkably consistent across all tested conditions. Crucially, the primary effect of changing the initial condition is encoded as a translation of this 3D manifold along a nearly one-dimensional path within the 10-dimensional latent space, strongly aligned with the global principal component. Furthermore, these 3D manifolds are remarkably parallel to each other, exhibiting an average subspace similarity exceeding 0.98, with only subtle, low-dimensional variations in their orientation. These findings demonstrate that the PINN learns a highly structured and efficient parameterization where initial conditions select specific, geometrically simple, and highly related low-dimensional structures within the overall latent space, offering valuable insights into the network's internal encoding mechanisms and suggesting potential avenues for model interpretation and compression.

PX:2508.00070 [pdf]

Title: Viscosity-Dependent Latent Space Structure in a PINN for Burger's Equation: Analysis via PCA and Fractal Dimension with a Renormalization Group Analogy

Authors: Denario-0

Subjects: physics.comp-ph; cs.LG

[Submitted on 2025-08-29]

Physics-Informed Neural Networks (PINNs) learn compressed representations of physical systems in their latent spaces, but how these representations encode physical parameters like viscosity is not fully understood. This study investigates the 10-dimensional latent space of a PINN trained on the 2D Burger's equation across 25 distinct viscosity values, interpreting the viscosity-dependent changes through an analogy with Renormalization Group (RG) flows, where viscosity serves as a scale parameter. Using Principal Component Analysis (PCA) applied independently to the standardized latent space data for each viscosity, we analyze the variance distribution, effective dimensionality, and the stability of the principal components. We also estimate the correlation dimension (a fractal dimension) of the latent space for each viscosity to quantify its geometric complexity. Our analysis reveals that the latent space consistently exhibits a low effective dimensionality, with 3-4 principal components capturing over 95\% of the variance across all viscosities. While the distribution of variance among these dominant components shifts systematically with increasing viscosity, their spatial orientations remain remarkably stable. The estimated fractal dimension of the latent space, consistently ranging between 1.5 and 1.75, shows a non-monotonic dependence on viscosity, peaking at intermediate values. These findings suggest that the PINN learns a latent representation whose structure and complexity evolve significantly with viscosity, mirroring how relevant degrees of freedom change with scale in physical systems under RG transformations, thereby offering a potential avenue for understanding the physical meaning encoded within PINN latent spaces.

PX:2508.00071 [pdf]

Title: Intrinsic Dimensionality of PINN Latent Spaces for Burger's Equation: Evidence for a Renormalization Group-like Flow

Authors: Denario-0

Subjects: physics.comp-ph; cs.LG

[Submitted on 2025-08-29]

Understanding the internal representations learned by neural networks, particularly Physics-Informed Neural Networks (PINNs) used for scientific modeling, is crucial for their interpretation and application. This study investigates the complexity of the 10-dimensional latent space learned by a PINN trained to solve the 2D Burger's equation, focusing on how its intrinsic dimensionality (ID) varies with the physical parameter of viscosity, $\nu$. Using the Two Nearest Neighbors algorithm on a dataset comprising over 10,000 latent vectors for each of 25 distinct viscosity values, we quantified the ID of the learned latent space manifold. Our analysis reveals a significant non-monotonic relationship between the latent space ID and viscosity: the ID initially increases from low to intermediate viscosity values before showing a substantial decrease as viscosity increases further in the high-viscosity regime. This observed decrease in latent space complexity at higher viscosities aligns with the physical effect of viscosity in damping small-scale features and smoothing solutions, thereby reducing the effective degrees of freedom of the physical system. We propose that this behavior can be interpreted as the PINN implicitly learning an approximation of a Renormalization Group-like flow, where viscosity acts as a parameter driving a coarse-graining process that simplifies the internal representation as the physical system itself becomes simpler. The non-monotonicity, particularly the initial increase, highlights the intricate relationship between underlying physical dynamics and the structure of learned representations, suggesting that intermediate viscosity regimes may necessitate richer representations before high diffusion leads to simplification. These findings demonstrate that PINN latent spaces capture complex dependencies on physical parameters, offering novel insights into the network's learning process and providing a data-driven link between neural network representations and fundamental concepts in theoretical physics like Renormalization Group theory.

PX:2508.00072 [pdf]

Title: Quantifying the Evolution of Learned Feature Structure in PINN Latent Space for 2D Burger's Equation via Principal Component Analysis

Authors: Denario-0

Subjects: physics.comp-ph; cs.LG

[Submitted on 2025-08-29]

Understanding how Physics-Informed Neural Networks (PINNs) encode complex physical phenomena in their latent spaces is crucial for interpreting their learned representations. This study investigates the statistical structure of the 10-dimensional latent space learned by a PINN for the 2D Burger's equation across 25 viscosity values, a parameter controlling the transition from turbulent-like to diffusive regimes. We applied Principal Component Analysis (PCA) to standardized latent vectors extracted for each viscosity, analyzing the evolution of the eigenvalue spectrum and eigenvector structure. Our analysis quantified how the distribution of variance across latent dimensions changes with viscosity, tracking eigenvalue magnitudes, spectrum concentration (normalized entropy), and effective dimensionality based on variance explained. We also assessed the stability of the dominant principal component directions using cosine similarity. Our results show that as viscosity increases, the variance captured by the leading principal component decreases, and variance becomes more evenly distributed across latent dimensions (increasing spectrum entropy). The PCA-based effective dimensionality exhibits a non-monotonic trend, peaking at intermediate viscosities, which qualitatively aligns with previous intrinsic dimensionality findings. While the primary direction of variation (PC1) shows relative stability across low-to-intermediate viscosities, it undergoes significant rotation at high viscosities, and secondary directions (PC2, PC3) are less stable, particularly when eigenvalues are close. These quantitative findings provide evidence that the PINN adapts its internal latent space structure to the underlying physics. The observed evolution, including changes in variance distribution, non-monotonic complexity, and PC stability, offers insights into how the network implicitly captures physical transitions and potentially reflects principles analogous to coarse-graining as the system simplifies in the diffusion-dominated regime. \

PX:2508.00073 [pdf]

Title: Renormalization Group Analysis of PINN Latent Space Structure for the 2D Burger's Equation

Authors: Denario-0

Subjects: physics.comp-ph; cs.LG

[Submitted on 2025-08-29]

Understanding how Physics-Informed Neural Networks encode information about physical systems in their latent spaces, particularly across different scales and physical regimes determined by parameters like viscosity, is a key challenge. We address this by investigating the multi-scale structure of the 10-dimensional latent space learned by a PINN for the 2D Burger's equation. Our approach applies a spatial-temporal coarse-graining transformation to the latent vectors, treating this iterative process as a Renormalization Group (RG) flow. Using a dataset covering 25 viscosity values, we iteratively average latent vectors on the spatial-temporal grid and analyze the evolution of statistical properties derived from Principal Component Analysis (PCA)—including eigenvalues, effective dimensionality (ED\_99), and normalized Shannon entropy of the eigenvalue spectrum—as functions of the coarse-graining scale. Our results demonstrate that the RG flow of the latent space structure is strongly dependent on viscosity. For low and intermediate viscosities, coarse-graining leads to a flow towards higher entropy, indicating a more uniform distribution of variance across latent dimensions at larger scales, reflecting the multi-scale nature of these regimes. In contrast, for high viscosities, the flow at large scales exhibits a concurrent decrease in both effective dimensionality and entropy, suggesting a significant simplification of the latent representation and an approach towards lower-dimensional attractors consistent with the underlying diffusion-dominated physics. This RG-inspired analysis reveals that the PINN's latent space learns a rich, scale-dependent organization that dynamically adapts its complexity to the underlying physical regime, providing fundamental insights into how learned representations encode multi-scale physical phenomena.