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Uncertainty Quantification for Smooth Functional Data with Application to Material Properties

This document outlines a method for processing functional output (i.e., curves) for the ultimate purpose of sampling curves under specified input conditions for use in modeling and simulation uncertainty quantification (UQ) studies. A set of benchmark curves sufficiently representative of the relevant scenario(s) being simulated are provided to the process and formatted as described in Section 1. Principal Component Analysis (PCA) is utilized to discover the components of uncertainty in the benchmark curves and is outlined in Section 2. Section 3 describes the application of uncertainty quantification to the PCA results for the purpose of sampling curves to be used in UQ analysis. Section 4 applies these techniques to an example benchmark dataset. Concluding remarks are provided in the final section.

36 MATERIALS SCIENCE

Quantum fate of the Choptuik naked singularity

Classical critical collapse provides a dynamical route from smooth initial data to a naked singularity, representing a sharper violation of predictability than ordinary black hole singularities. We argue that this distinction is erased by quantum backreaction. Building on the semiclassical interior analysis, where quantum self-energy of the collapsing matter generates a universal growing mode and a finite mass gap, we study the exterior naked singularity region that determines global visibility in the Einstein-scalar system. We analyze controlled exterior models in both $2+1$ and $3+1$ dimensions. In the former, smooth matching and physical boundary conditions analytically select a vacuum polarization state, whose backreaction cloaks the classically naked region by a quantum trapped branch. In the latter, numerical horizon tracing shows that near a quantum-shifted threshold the exterior develops finite-mass marginally trapped surfaces rather than a zero-mass naked endpoint. These results suggest a global quantum picture in which the Choptuik naked singularity shares the fate of an ordinary black hole singularity: quantum effects push the putative Cauchy horizon behind a quantum-generated horizon, thereby reducing the loss of predictability to the standard black hole evaporation problem.

FOS: Physical sciences

Unveiling horizons in quantum critical collapse

Critical gravitational collapse offers a unique window into regimes of arbitrarily high curvature, culminating in a naked singularity arising from smooth initial data — thus providing a dynamical counterexample to weak cosmic censorship. Near the critical regime, quantum effects from the collapsing matter are expected to intervene before full quantum gravity resolves the singularity. Despite its fundamental significance, a self-consistent treatment has so far remained elusive. In this work, we perform a one-loop semiclassical analysis using the robust anomaly-based method in the canonical setup of Einstein gravity minimally coupled to a free, massless scalar field. Focusing on explicitly solvable near-critical solutions in both 2 + 1 and 3 + 1 dimensions, we analytically solve the semiclassical Einstein equations and obtain controlled, quantitative results for several long-standing questions within the dominant s-wave sector. We find that regularity uniquely selects a Boulware-like quantum state, encoding genuine vacuum polarization effects from the collapsing matter. Remarkably, the resulting quantum corrections manifest as a growing mode. Horizon-tracing analyses, incorporating both classical and quantum modes, reveal the emergence of a finite mass gap, signaling a phase transition from classical Type II to quantum-modified Type I behavior, thereby providing a quantum enforcement of the weak cosmic censorship. The most non-trivial aspect of our analysis involves dealing with non-conformal matter fields in explicitly time-dependent critical spacetimes. Along the way, we uncover intriguing and previously underexplored features of quantum field theory in curved spacetime.

2D Gravity

Quantum critical collapse abhors a naked singularity

Classical critical collapse yields naked singularities from smooth initial data, challenging cosmic censorship, and shaping the spectrum of primordial black holes. We show that one-loop vacuum polarization near the threshold qualitatively changes this outcome by dressing the singularity with a horizon within a controlled semiclassical regime. In analytically tractable Einstein-scalar critical spacetimes, a one-loop $s$-wave treatment linearized around self-similar backgrounds shows that regularity uniquely selects an asymptotically Minkowskian, vacuum-polarization state. Its renormalized stress tensor carries a universal quantum growing mode that competes with the classical unstable mode, shifts the critical point, and generates a trapped surface along with a finite mass gap at the new threshold, thereby enforcing horizon formation even under arbitrary fine-tuning. In primordial collapse, the threshold shift enters exponentially into the formation fraction, while the mass gap truncates the low-mass tail, suggesting potentially important consequences for the predicted mass spectrum. Furthermore, these results provide a self-consistent semiclassical treatment of critical collapse and yield sharp predictions within the one-loop, near-critical, linearized regime.

Anomalies

Extracting the Breakout Distance from the ECOT Trajectories: Gaussian Process Regression Approach

Enhanced Corner Turning (ECOT) experiments provide an important metric of performance of high explosive (HE) formulations. The breakout distance is a single scalar value that characterizes the corner turning efficiency of an HE. Extracting the breakout distance from the raw ECOT results, whether experimental or simulated, is a conceptually straightforward procedure which, however, is non-unique, especially in the presence of noise. More specifically, this procedure involves numerical smoothing and selecting particular values for parameters of this smoothing introduces human bias. In this work, we propose to use the Gaussian process regression to analyze ECOT results. This analysis involves the effective smoothing of the data, thus allowing for accurate extraction of the breakout distance. Most importantly, the parameters of this smoothing can be inferred from the ECOT data itself, rendering the approach effectively parameter-free and thus diminishing the human bias. An additional benefit of the Gaussian process regression, being a statistical inference method, is that not just the value of the breakout distance, but also its confidence interval can be extracted from the data. This report introduces the Gaussian process regression, as applied to ECOT, and demonstrates its usefulness by extracting the breakout distances for a selection of experimental and simulated data.

45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF

Calibrating Bayesian generative machine learning for Bayesiamplification

Recently, combinations of generative and Bayesian deep learning have been introduced in particle physics for both fast detector simulation and inference tasks. These neural networks aim to quantify the uncertainty on the generated distribution originating from limited training statistics. The interpretation of a distribution-wide uncertainty however remains ill-defined. We show a clear scheme for quantifying the calibration of Bayesian generative machine learning models. For a Continuous Normalizing Flow applied to a low-dimensional toy example, we evaluate the calibration of Bayesian uncertainties from either a mean-field Gaussian weight posterior, or Monte Carlo sampling network weights, to gauge their behaviour on unsteady distribution edges. Well calibrated uncertainties can then be used to roughly estimate the number of uncorrelated truth samples that are equivalent to the generated sample and clearly indicate data amplification for smooth features of the distribution.

97 MATHEMATICS AND COMPUTING

SPRUCE Vegetation Phenology in Experimental Plots from PhenoCam Imagery, 2015-2024

This data set consists of PhenoCam data from the SPRUCE experiment from the beginning of whole ecosystem warming (Hanson et al. 2017) in August 2015 through March 31 of 2025 (2015-08-24 to 2025-03-31), with start- and end-of-season phenological transition dates derived through the end of autumn 2024. Digital cameras, or phenocams, installed in each SPRUCE enclosure track seasonal variation in vegetation “greenness”, a proxy for vegetation phenology and associated physiological activity. Three separate regions of interest (ROIs) were defined for each camera field of view, corresponding to different vegetation types and demarcating (1) Picea trees (vegetation type EN, for evergreen needleleaf); (2) Larix trees (vegetation type DN, for deciduous needleleaf); and (3) the mixed shrub layer (vegetation type SH). This data set consists of three sets of data files: (1) 3-day summary product files: One file for each camera and each ROI (i.e. vegetation type), characterizing vegetation color at a 3-day time step. • Contains 36 files in *.csv format inside a compressed (*.zip) file. (2) Transition date file: Estimates “greenness rising” (spring) and “greenness falling” (autumn) transition dates derived from the smoothed daily green chromatic coordinate (GCC) values, for each camera and each ROI (i.e., vegetation type). • Contains one file in *.csv format. (3) Snow flag files: Indicate days with snow on trees or snow on ground for each experimental enclosure. • Contains two files in *.csv format, one for snow on trees and one for snow on ground. This data set consists of two sets of companion files: (1) Accompanying HTML files show the 90th quantiles of the mean GCC plotted together with transition dates for each vegetation type and plot. • Contains three files in HTML format, one for each vegetation type. • One additional file in HTML format with the transition dates plotted for each vegetation type, by year. (2) R files for processing PhenoCam files and flags. • Contains five files in R file(*.R) format and the components of the phenocamr package (Version 1.1.4) used for calculating transition dates for 2015-2024. These are contained in a compressed (*.zip) file. User Note: All imagery is posted in near-real time to the PhenoCam Project web page (https://phenocam.nau.edu), where it is publicly available. Scroll to “spruce” in the Gallery or link directly to the 29 SPRUCE cameras at https://tinyurl.com/sprucecams. This data set is based on the complete camera record from SPRUCE and supersedes all previously released PhenoCam datasets (see Related Data Sets). The estimated transition dates for previously released datasets may differ slightly (in most cases, by ±3 days or less), because following standard PhenoCam processing protocols (Richardson et al. 2018, Scientific Data), smoothing and interpolation, outlier removal, and transition date estimation are always conducted using the full data record.

54 ENVIRONMENTAL SCIENCES

Statistical imprints of wave-like dark matter on multiply-imaged galaxies in strong cluster lenses from JWST

Wave-like dark matter ($ψ$DM) is an elusive dark matter (DM) candidate. The model, often also called fuzzy or ultralight DM, proposes that DM is an extremely light ($m\sim10^{-22}$ eV) boson and thereby has a kpc-scale de Broglie wavelength. Hence, interference of DM gives rise to sub-galactic density fluctuations that can be studied with strong gravitational lensing. In this paper, we use the residual power spectrum, $\mathrm{P}_δ(k)$, as a probe of $ψ$DM, which quantifies deviations from smooth lensing predictions, measured from multiply-imaged galaxies in strong cluster lenses. The key idea is that imprinted in these deviations are lensing distortions from DM substructure, which can be harnessed statistically to distinguish among DM theories. We simulate JWST-quality mock observations of strong gravitational lensing in galaxy clusters, modeling line-of-sight DM substructure within $ψ$DM and the standard cold dark matter (CDM) paradigms. Using mock deep observations ($\sim$ 20 hours), we find that $\mathrm{P}_δ(k)$ is sensitive to both $ψ$DM particle mass and fluctuation amplitude, and can distinguish $ψ$DM fluctuations from CDM subhalos. We demonstrate that $\mathrm{P}_δ(k)$ can be measured directly from data by modeling the smooth lensing with a local Curved Arc Basis formalism. With realistic modeling systematics, we find a statistically significant separation between $ψ$DM and CDM across $1 \lesssim k \lesssim 11\,\mathrm{kpc}^{-1}$ -- offering an independent probe of the wave-like nature of DM complementary to existing constraints.

Cosmology and Nongalactic Astrophysics (astro-ph.C

Measurement of the 252 Cf ⁢(sf) prompt fission neutron spectrum utilizing 12 C ⁡(𝑛, 𝑛) and 9 Be ⁢(𝑛, 𝑛) neutron scattering reference measurements

The 252 Cf spontaneous fission (sf), prompt fission neutron spectrum (PFNS) is a fundamental quantity for nuclear physics measurements of neutron-emitting reactions. This energy distribution of neutrons emitted from fission has been considered a neutron data standard for decades and has been utilized as a reference for neutron detection efficiency, validation of Monte Carlo simulations, benchmarking of dosimetry standards, and more. A significant portion of the global collection of nuclear data on neutron-induced reactions is correlated with the 252 Cf ⁢(sf) PFNS. Despite the reliance on this quantity by the nuclear physics community, the historical collection of 252 Cf PFNS measurements display systematic disagreements that are not understood or easily explained. These experimental discrepancies could potentially bias the 252 Cf PFNS Standard evaluation. On top of this, these past experiments frequently employed correlated experimental measurement or analysis methods. The artificial intelligence (AI)/machine learning (ML)-informed californium chi-nuclear data experiment (AIACHNE) project was formed to (a) investigate these discrepancies utilizing AI/ML methods to identify outlying regions of literature data, assign these regions to features of the experiment itself, and perform an improved evaluation of the 252 Cf PFNS and (b) perform a new experimental measurement of this quantity designed to improve upon the existing literature database. Here, in this work, we report on the AIACHNE 252 Cf PFNS experiment utilizing a new analysis method uncorrelated with all previous measurements: neutron efficiency determinations based on elastic neutron scattering on 12 C and 9 Be . This new method provides an independent test of the existing literature data and evaluation of the 252 Cf ⁢(sf) PFNS. The method is described with detailed covariance quantification procedures, as well as a direct discussion of the sources of uncertainty described as requirements in the “Templates” series of papers. The 252 Cf ⁢(sf) PFNS reported in this work agrees well with the overall shape of the existing standard PFNS evaluation as well as many literature measurements, thus verifying the current evaluation utilizing new techniques. However, the results suggest that there are deficiencies in the angle-differential 12 C and 9 Be ⁢(𝑛, 𝑛) evaluated nuclear data, which produce unphysical structures in the reported result. While these structures are relatively minor, they become obvious because of the high statistical precision of the data and the expected smooth continuity of the 252 Cf ⁢(sf) PFNS.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS

Landscaper v1

Understanding the inner workings of machine learning models through their loss landscapes offers crucial insights into model properties, optimization dynamics, and generalizability. However, accessing these insights has traditionally required specialized mathematical expertise, limiting broader adoption. Landscaper is an open-source Python package designed to bridge this gap. Landscaper seamlessly integrates a suite of multi-dimensional loss landscape analyses with cutting-edge topological data analysis (TDA) methods. This powerful combination makes both fundamental loss landscape analysis and advanced TDA techniques accessible to the broader scientific ML community, without requiring deep pre-existing mathematical knowledge. Landscaper offers three key functionalities: * Construction: Builds detailed loss landscape representations through versatile low and high-dimensional sampling techniques. * Quantification: Applies advanced metrics, including a novel topological data analysis (TDA) based smoothness metric, enabling new perspectives on model behavior. * Visualization: Offers intuitive tools to visualize and interpret loss landscapes, providing actionable insights beyond traditional performance metrics.

Weber, Gunther [Lawrence Berkeley National Laborat

Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimation

Deep learning has exhibited remarkable results across diverse areas. To understand its success, substantial research has been directed towards its theoretical foundations. Nev- ertheless, the majority of these studies examine how well deep neural networks can model functions with uniform regularities. In this paper, we explore a different angle: how deep neural networks can adapt to varying degrees of smoothness in functions and nonuni- form data distributions across different locations and scales. More precisely, we focus on a broad class of functions defined by nonlinear tree-based approximation methods. This class encompasses a range of function types, such as functions with uniform regularities and discontinuous functions. We develop nonparametric approximation and estimation theories for this class using deep ReLU networks. Our results show that deep neural networks are adaptive to the nonuniform smoothness of functions and nonuniform data distributions at different locations and scales. We apply our results to several function classes, and derive the corresponding approximation and generalization errors. The validity of our results is demonstrated through numerical experiments.

97 MATHEMATICS AND COMPUTING

Enabling probabilistic learning on manifolds through double diffusion maps

Here, we present a generative learning framework for probabilistic sampling that extends Probabilistic Learning on Manifolds (PLoM), which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a (representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to characterize the manifold of the data, and (c) a reduced-order Itô Stochastic Differential Equation (ISDE) to sample from the learned distribution. However, its sampling dynamics are posed in the ambient space and the retained number of reduced coordinates is chosen by projection-reconstruction error. In practice, this often (i) requires more coordinates than the data’s intrinsic dimension to achieve stable sampling and (ii) lacks a smooth, basis-independent lifting back to the data domain; moreover, standard Diffusion Maps emphasize harmonic eigenfunctions and can miss non-harmonic latent structure. We address these limitations by decoupling geometry learning from sampling: a first Diffusion Maps pass identifies non-harmonic coordinates on which we formulate a full-order ISDE directly in the latent space, while Double Diffusion Maps captures multiscale geometric features and Geometric Harmonics (GH) learns a smooth lifting map to the ambient variables that is independent of the particular diffusion basis. This hybrid design preserves the system’s dynamical richness with a compact geometric representation and enables principled out-of-sample inference. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

Double diffusion maps

Fungal Spore Seasons Advanced Across the US Over Two Decades of Climate Change

Abstract Phenological shifts due to climate change have been extensively studied in plants and animals. Yet, the responses of fungal spores—organisms important to ecosystems and major airborne allergens—remain understudied. This knowledge gap limits our understanding of their ecological and public health implications. To address this, we analyzed a long‐term (2003–2022), large‐scale (the continental US) data set of airborne fungal spores collected by the US National Allergy Bureau. We first pre‐processed the spore data by gap‐filling and smoothing. Afterward, we extracted 10 metrics describing the phenology (e.g., start and end of season) and intensity (e.g., peak concentration and integral) of fungal spore seasons. These metrics were derived using two complementary but not mutually exclusive approaches—ecological and public health approaches, defined as percentiles of total spore concentration and allergenic thresholds of spore concentration, respectively. Using linear mixed‐effects models, we quantified annual shifts in these metrics across the continental US. We revealed a significant advancement in the onset of the spore seasons defined in both ecological (11 days, 95% confidence interval: 0.4–23 days) and public health (22 days, 6–38 days) approaches over two decades. Meanwhile, total spore concentrations in an annual cycle and in a spore allergy season tended to decrease over time. The earlier start of the spore season was significantly correlated with climatic variables, such as warmer temperatures and altered precipitations. Overall, our findings suggest possible climate‐driven advanced fungal spore seasons, highlighting the importance of climate change mitigation and adaptation in public health decision‐making.

Environmental Sciences & Ecology

Superhorizon isocurvature fluctuations relax tensions

Here, we present a new class of models that have potential to alleviate tensions present in the cosmological data today. We postulate about variation in the sound horizon scale on superhorizon scales, i.e., on scales that are larger than that of the present observable low-redshift Universe (≳ 1Gpc) while at the same time smaller than the largest scales probed by the cosmic microwave background (CMB) (≲ 10Gpc). In this scenario, CMB peaks are naturally smoothed as preferred by the Planck data, while at the same time the low-redshift baryon acoustic oscillation calibration is partially decoupled from the CMB. Taking superhorizon variations in baryon fraction as an example and using approximate modeling, we find improvement in the best fit Planck power spectrum model Δχ 2 ~ 6 for 1 extra degree of freedom with the relevant extension parameter 10 3 σ b = 2.10 ± 0.60, implying about 10% variations in baryon fraction across the Universe. At the same time, S 8 drops by about 1 sigma, easing tension with weak lensing surveys. We find that the combination of Planck 2018 data, eBOSS BAO data, and Riess et al. distance ladder Hubble parameter determination produce a satisfactory fit in our model if we allow for a phantom dark energy equation of state.

79 ASTRONOMY AND ASTROPHYSICS

A Centralized AI Lakehouse Framework for Brain Tumor MRI Classification and Segmentation, University KPI Forecasting, and Water Potability Prediction

In many university and healthcare projects, models are built for very different data types such as tables, institutional time series, and medical images, but they are deployed as separate applications. In this work, that separation made testing and maintenance difficult because each module had its own pipeline and runtime requirements. This paper presents an integrated AI lakehouse-style implementation that runs three model pipelines inside one containerized backend. For medical imaging, we used MRI datasets from IEEE DataPort: a four-class classification set with 7012 images (5708 train/1304 test) and a segmentation set with 3063 image–mask pairs. The classification model (ResNet50 transfer learning) is evaluated using a proper train–validation–test protocol across multiple splits (80/10/10, 70/10/20, 60/10/30, and 10/30/60), achieving a test accuracy of 99.00% under the standard 80/10/10 split. Additionally, a patient-level evaluation is conducted using an external glioma dataset to provide a more realistic assessment without data leakage. The segmentation model (DeepLabV3-ResNet50) achieved 83.09% validation mIoU and 88.79% Dice score. For university KPI forecasting, we used annual IPEDS and NSF HERD data from 2010 to 2023 for three universities (BSU, EOU, and UAB). To examine the effect of preprocessing on forecasting performance, two case studies are conducted. In the first case, linear interpolation is applied to generate semester-level data. In the second case, the original annual data is used directly without interpolation. Random Forest regression and ARIMA models are evaluated using MAE, RMSE, MAPE, and R 2 . The results showed that interpolation improved apparent forecasting performance due to smoothing, while evaluation on the original annual data provided a more realistic assessment of model behavior. To further validate the framework on a larger dataset, an additional case study is conducted using a student dropout dataset. For water potability, we trained and compared multiple tabular classifiers on a large dataset (1,048,575 samples). A Random Forest model (100 trees, max depth 10) achieved 85.86% test accuracy and high recall for unsafe samples (0.8447). All modules are served via FastAPI and deployed together using Docker, with workflow automation routing requests to the correct endpoint. System-level benchmarking indicates that the backend maintains stable throughput and latency under concurrent requests.

97 MATHEMATICS AND COMPUTING

Ensemble variational Fokker-Planck methods for data assimilation

Particle flow filters solve Bayesian inference problems by smoothly transforming a set of particles into samples from the posterior distribution. Particles move in state space under the flow of an McKean-Vlasov-Itˆo process. This work introduces the Variational Fokker-Planck (VFP) framework for data assimilation, a general approach that includes previously known particle flow filters as special cases. The McKean-Vlasov-Itˆo process that transforms particles is defined via an optimal drift that depends on the selected diffusion term. It is established that the underlying probability density - sampled by the ensemble of particles - converges to the Bayesian posterior probability density. For a finite number of particles the optimal drift contains a regularization term that nudges particles toward becoming independent random variables. Based on this analysis, we derive computationally-feasible approximate regularization approaches that penalize the mutual information between pairs of particles, and avoid particle collapse. Moreover, the diffusion plays a role akin to a particle rejuvenation approach that aims to alleviate particle collapse. The VFP framework is very flexible. Different assumptions on prior and intermediate probability distributions can be used to implement the optimal drift, and localization and covariance shrinkage can be applied to alleviate the curse of dimensionality. A robust implicit-explicit method is discussed for the efficient integration of stiff McKean- Vlasov-Itˆo processes. Here, the effectiveness of the VFP framework is demonstrated on three progressively more challenging test problems, namely the Lorenz ’63, Lorenz ’96 and the quasi-geostrophic equations.

97 MATHEMATICS AND COMPUTING

Unlocking hidden information in sparse small-angle neutron scattering measurements

Hypothesis Small-Angle Neutron Scattering (SANS) is a powerful technique for studying soft matter systems such as colloids, polymers, and lyotropic phases, providing nanoscale structural insights. However, its effectiveness is limited by low neutron flux, leading to long acquisition times and noisy data. Here, we hypothesize that Bayesian statistical inference using Gaussian Process Regression (GPR) can reconstruct high-fidelity scattering data from sparse measurements by leveraging intensity smoothness and continuity. Experiments and Simulations The method was benchmarked computationally and validated through SANS experiments on various soft matter systems, including wormlike micelles, colloidal suspensions, polymeric structures, and lyotropic phases. GPR-based inference was applied to both experimental and synthetic data to evaluate its effectiveness in noise reduction and intensity reconstruction. Findings GPR significantly enhances SANS data quality and therefore reducing measurement times by up to two orders of magnitude. This cost-effective approach maximizes experimental efficiency, enabling high-throughput studies and real-time monitoring of dynamic systems. It is particularly beneficial for weakly scattering and time-sensitive studies. Beyond SANS, this framework applies to other low-SNR techniques, including laboratory-based small-angle X-ray scattering and various dynamical scattering methods. Furthermore, it offers transformative potential for compact neutron sources, enhancing their viability for structural analysis in resource-limited settings.

Small angle neutron scattering

From disorganized data to emergent dynamic models: Questionnaires to partial differential equations

Starting with sets of disorganized observations of spatially varying and temporally evolving systems, obtained at different (also disorganized) sets of parameters, we demonstrate the data-driven derivation of parameter dependent, evolutionary partial differential equation (PDE) models capable of generating the data. This tensor type of data is reminiscent of shuffled (multidimensional) puzzle tiles. The independent variables for the evolution equations (their “space” and “time”) as well as their effective parameters are all emergent , i.e. determined in a data-driven way from our disorganized observations of behavior in them. We use a diffusion map based questionnaire approach to build a smooth parametrization of our emergent space/time/parameter space for the data. This approach iteratively processes the data by successively observing them on the “space,” the “time” and the “parameter” axes of a tensor. Once the data become organized, we use machine learning (here, neural networks) to approximate the operators governing the evolution equations in this emergent space. Our illustrative examples are based (i) on a simple advection–diffusion model; (ii) on a previously developed vertex-plus-signaling model of Drosophila embryonic development; and (iii) on two complex dynamic network models (one neuronal and one coupled oscillator model) for which no obvious smooth embedding geometry is known a priori. This allows us to discuss features of the process like symmetry breaking, translational invariance, and autonomousness of the emergent PDE model, as well as its interpretability.

generative models