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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Predicting nonequilibrium Green’s function dynamics and photoemission spectra via nonlinear integral operator learning

Understanding the dynamics of nonequilibrium quantum many-body systems is an important research topic in a wide range of fields across condensed matter physics, quantum optics, and high-energy physics. However, numerical studies of large-scale nonequilibrium phenomena in realistic materials face serious challenges due to intrinsic high-dimensionality of quantum many-body problems and the absence of time-invariance. The nonequilibrium properties of many-body systems can be described by the dynamics of the correlator, or the Green's function of the system, whose time evolution is given by a high-dimensional system of integro-differential equations, known as the Kadanoff–Baym equations (KBEs). The time-convolution term in KBEs, which needs to be recalculated at each time step, makes it difficult to perform long-time numerical simulation. In this paper, we develop an operator-learning framework based on recurrent neural networks (RNNs) to address this challenge. We utilize RNNs to learn the nonlinear mapping between Green's functions and convolution integrals in KBEs. By using the learned operators as a surrogate model in the KBE solver, we obtain a general machine-learning scheme for predicting the dynamics of nonequilibrium Green's functions. Besides significant savings per each time step, the new methodology reduces the temporal computational complexity from $O(N_t^3)$ to $O(N_t)$ where N t is the number of steps taken in a simulation, thereby making it possible to study large many-body problems which are currently infeasible with conventional KBE solvers. Through various numerical examples, we demonstrate the effectiveness of the operator-learning based approach in providing accurate predictions of physical observables such as the reduced density matrix and time-resolved photoemission spectra. Moreover, our framework exhibits clear numerical convergence and can be easily parallelized, thereby facilitating many possible further developments and applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Discrete generative diffusion models without stochastic differential equations: A tensor network approach

Diffusion models (DMs) are a class of generative machine learning methods that sample a target distribution by transforming samples of a trivial (often Gaussian) distribution using a learned stochastic differential equation. In standard DMs, this is done by learning a “score function” that reverses the effect of adding diffusive noise to the distribution of interest. Here we consider the generalisation of DMs to lattice systems with discrete degrees of freedom, and where noise is added via Markov chain jump dynamics. We show how to use tensor networks (TNs) to efficiently define and sample such “discrete diffusion models” (DDMs) without explicitly having to solve a stochastic differential equation. We show the following: (i) by parametrising the data and evolution operators as TNs, the denoising dynamics can be represented exactly; (ii) the auto-regressive nature of TNs allows to generate samples efficiently and without bias; (iii) for sampling Boltzmann-like distributions, TNs allow to construct an efficient learning scheme that integrates well with Monte Carlo. We illustrate this approach to study the equilibrium of two models with non-trivial thermodynamics, the d = 1 constrained Fredkin chain and the d = 2 Ising model. Published by the American Physical Society 2025

Causer, Luke (ORCID:0000000194243473)↗
DOE OSTI2026

Time at Temperature Experiments on Neutron Irradiated Zircaloy-2 using Conventional & Flash DSC

Dryout events in Boiling Water Reactors (BWRs) are currently treated by NRC regulations as automatic disqualification for continued fuel rod operation, even though this criterion does not account for the rate or duration of power increases, associated changes in material behavior, or the possibility of rewetting. Operational history from Anticipated Operational Occurrences (AOOs) shows that short, transient power excursions often demand only modest heat removal, and industry experience suggests that fuel can briefly enter dryout yet return to safe, stable operation. The lack of detailed understanding of the material response during such events motivates the present series of experiments. This study uses unique and innovative methods to investigate microstructural evolution in irradiated Zircaloy-2 exposed to high temperatures in inert environments. Differential Scanning Calorimetry (DSC) and FlashDSC are combined to build a comprehensive experimental framework capable of identifying the a–ß phase transformation in zirconium and examining defect annealing under steep thermal gradients. FlashDSC enables rapid heating and cooling of Focused Ion Beam (FIB)–prepared large-area lift-outs (LALOs) of irradiated Zircaloy-2 at rates of 1,000 K/s to peak temperatures of 600°C, 750°C, and 900°C. The goal is to determine whether these conditions produce measurable microstructural changes that could influence cladding performance in typical BWR environments. Microstructural characterization includes quantifying dislocation density and assessing secondary phase particle (SPP) size and distribution using Transmission Electron Microscopy (TEM). Ongoing analysis, such as diffraction pattern indexing and 4D STEM processing, will further refine these observations.

36 - MATERIALS SCIENCE↗
DOE OSTI2024

The root cause of disruptive NTMs and paths to stable operation in DIII-D ITER baseline scenario plasmas

Analyses of the DIII-D ITER Baseline Scenario database support that the disruptive m,n=2,1 magnetic islands are pressure gradient driven, non-linear instabilities seeded in a sequence of stochastic transient magnetic perturbations, and that the current profile relaxation does not affect the m,n=2,1 island onset rate. At low torque, these Neoclassical Tearing Modes are most commonly seeded by non-linear 3-wave coupling when the differential rotation between the q=1 & q=2 rational surfaces approaches zero. Lack of statistically significant difference between the current profiles of stable and unstable states, as well as lack of correlation between the tearing mode onset rate and the current profile relaxation both reject causality between the current profile evolution and the 2,1 magnetic island onsets in these plasmas. These support that preserving the differential rotation between the q=1 and q=2 rational surfaces is key to long pulse stable operation in the plasma scenario planned for ITER, while optimization of the current profile within the explored parameter space may lead to much weaker improvements than sustaining the differential rotation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Monitoring the long-term performance of organic redox flow battery by a distribution of relaxation time analysis

Organic redox flow batteries hold great promise as an energy storage technology, but their intricate chemistry makes them vulnerable to various degradation mechanisms. Monitoring this degradation is essential for identifying the limiting processes within the cells. Electrochemical impedance spectroscopy (EIS) offers a straightforward, in-situ method for measuring the total resistance of an operating cell. However, to pinpoint the limiting processes during long-term cycling, EIS data must be complemented by other techniques. Distribution of relaxation time (DRT) analysis is particularly effective for differentiating resistance components. Here, in this study, we perform a comprehensive analysis of resistance evolution and the separation of anode and cathode contributions during long-term cycling of a full cell employing 7,8-dihydroxyphenazine-2-sulfonic acid (DHPS) as the anolyte. Separate analyses of the DHPS anolyte and ferri-/ferrocyanide catholyte were conducted using a symmetric cell setup. The relaxation times derived from symmetric cells facilitate the identification of peaks in the DRT profiles from the full cell. Importantly, the DRT profiles indicate a correlation between the evolution of charge transfer resistance and the chemical degradation of DHPS. The methodologies and results outlined in this study offer significant insights for developing diagnostic tools applicable to other types of redox flow batteries.

Distribution of relaxation time↗
DOE OSTI2024

Analytic solutions of the DGLAP evolution and theoretical uncertainties

The energy dependence for the singlet sector of Parton Distributions Functions (PDFs) is described by an entangled pair of ordinary linear differential equations. Although there are no exact analytic solutions, it is possible to provide approximated results depending on the assumptions and the methodology adopted. These results differ in their sub-leading, neglected terms and ultimately they are associated with different treatments of the theoretical uncertainties. In this work, a novel analytic approach in Mellin space is presented and a new methodology for obtaining closed and exponentiated analytic solutions is devised. Different results for the DGLAP evolution at Next-Leading-Order are compared, discussing advantages and disadvantages for each solution. The generalizations to higher orders are addressed.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Recurrent convolutional neural networks for modeling nonadiabatic dynamics of quantum-classical systems

Recurrent neural networks (RNNs) have recently been extensively applied to model the time evolution in fluid dynamics, weather predictions, and even chaotic systems due to their ability to capture temporal dependencies and sequential patterns in data. Here we present an RNN model based on convolutional neural networks for modeling the nonlinear nonadiabatic dynamics of hybrid quantum-classical systems. The dynamical evolution of the hybrid systems is governed by equations of motion for classical degrees of freedom and von Neumann equation for electrons. The Physics-Aware Recurrent Convolution (PARC) neural network structure incorporates a differentiator-integrator architecture that inductively models the spatiotemporal dynamics of generic physical systems. Here, we apply our RNN approach to learn the space-time evolution of a one-dimensional semiclassical Holstein model after an interaction quench. For shallow quenches (small changes in electron-lattice coupling), the deterministic dynamics can be accurately captured using a single-CNN-based recurrent network. In contrast, deep quenches induce chaotic evolution, making long-term trajectory prediction significantly more challenging. Nonetheless, we demonstrate that the PARC-CNN architecture can effectively learn the statistical climate of the Holstein model under deep-quench conditions.

Holstein model↗
DOE OSTI2025

An isotopic labeling investigation into the influence of the nitro group on LLM-105 thermal decomposition

Here, this work presents the first application of isotopically labeled LLM-105 (2,6-diamino-3,5-dinitropyrazine-1-oxide) to investigate thermal decomposition pathways. Specially synthesized LLM-105 isotopologues were utilized to isolate the influence of labeled 15 NO 2 nitro groups on the formation of lightgas products. Simultaneous differential scanning calorimetry, thermo-gravimetric, and mass spectrometry measurements were employed to track the evolution of product gases, enabling the direct comparison of isotopically shifted species with unlabeled LLM-105. Key findings show that C 2 N 2 production is mainly dependent on nitrogen sources from either the amine groups or the pyrazine ring (i.e., not the nitro groups). The formation of NO, N 2 , and N 2 O all involves the nitro groups to some extent. NO (nitric oxide) was found to be the predominant gas species directly formed from the nitro group of LLM-105. In contrast, mixed nitrogen isotopologues of N 2 and N 2 O (i.e., 14 N 15 N and 15 NNO) formed more readily in comparison to their pure counterparts (i.e., 15 N 2 and 15 N 2 O). This indicates the amine and/or pyrazine groups of LLM-105, in addition to the nitro group, are involved in the decomposition pathways forming N 2 and N 2 O. In addition, our investigation led to the discovery of two previously unreported decomposition products (CHO and HNCO), which were confirmed through hydrogen labelling utilizing deuterium isotopes. These results provide detailed speciation trends of gaseous products during LLM-105 decomposition, offering new insights into reaction pathways. Experimental data reported here will support the development of a detailed chemical kinetics model for LLM-105, essential for the safe handling of high explosives.

Chemistry - Chemical explosives↗
DOE OSTI2026

Causality-respecting adaptive refinement for PINNs: enabling precise interface evolution in phase field modeling

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving physical systems described by partial differential equations (PDEs). However, their accuracy in dynamical systems, particularly those involving sharp moving boundaries with complex initial morphologies, remains a challenge. Here, this study introduces an approach combining residual-based adaptive refinement (RBAR) with causality-informed training to enhance the performance of PINNs in solving spatio-temporal PDEs. Our method employs a three-step iterative process: initial causality-based training, RBAR-guided domain refinement, and subsequent causality training on the refined mesh. Applied to the Allen-Cahn equation, a widely-used model in phase field simulations, our approach demonstrates significant improvements in solution accuracy and computational efficiency over traditional PINNs. Notably, we observe an ‘overshoot and relocate’ phenomenon in dynamic cases with complex morphologies, showcasing the method’s adaptive error correction capabilities. This synergistic interaction between RBAR and causality training enables accurate capture of interface evolution, even in challenging scenarios where traditional PINNs fail. Our framework not only resolves the limitations of uniform refinement strategies but also provides a generalizable methodology for solving a broad range of spatio-temporal PDEs. The enhanced performance of the RBAR–causality combined framework demonstrates its strong potential for advancing PINN-based modeling of physical systems characterized by complex, evolving interfaces.

Allen-Cahn equations↗
DOE OSTI2025

Understanding the Cathode Electrochemistry of Humidified Solid‐State Lithium‐Oxygen Batteries

Lithium-oxygen batteries (LOBs) possess a high theoretical energy density, making them potential candidates for next-generation energy storage. However, challenges such as reactive oxygen species-induced component degradation hinder their practical use. Inorganic solid-state electrolytes offer an alternative to degradation-prone aprotic electrolytes, while also protecting lithium anodes from potential atmospheric reactants. Here, this study explores the cathode electrochemistry of solid-state LOBs using humidified oxygen, which forms an aqueous catholyte during initial cycling, thereby improving cathode-electrolyte contact. To quantitatively analyze the cathode electrochemistry, a ‘Humidity-Incorporated’ Differential Electrochemical Gas Monitoring System (HiDEMS) is developed to control humidity and monitor gas consumption and evolution in real time. When studying a Li-O 2 cell that employs a NASICON-type Li 1.3 Al 0.3 Ti 1.7 (PO 4 ) 3 (LATP) solid electrolyte and a porous carbon cathode, a shift in discharge products from Li 2 O 2 to LiOH is observed over repeated cycles. While Li 2 O 2 evolves O 2 during charging, LiOH oxidation leads to minimal O 2 release and increased CO 2 production, originating from oxidation of carbon electrodes. Further, dissolution of Al and P from LATP is observed, likely driven by the formation of the alkaline catholyte. The findings highlight the need for carbon-free cathode materials and more stable solid-state conductors to minimize side reactions and improve rechargeability in humidified solid-state Li-O 2 batteries.

LATP degradation↗
DOE OSTI2024

Formation of the {gamma}ʹʹʹ-Ni2(Cr, Mo, W) phase during a two-step aging heat treatment in HAYNES® 244® Alloy

Precipitation hardening is the dominant method of achieving high strength in most Ni-based superalloys. The formation of nanoscale precipitates during thermal exposure is often studied to determine the optimal methods of attaining high strength. The commercial Ni-based superalloy, HAYNES® 244® alloy, is strengthened through a novel -Ni2(Cr, Mo, W) intermetallic phase that forms during a two-step aging cycle. The precipitation kinetics of this intermetallic phase are sluggish for single-step aging in comparison to the γʹ phase in precipitation-strengthened Ni-based alloys, but a two-step aging treatment has shown to reliably harden the alloy and improve high-temperature properties compared to a single-step aging heat treatment. To investigate the formation and coarsening of this phase, heat-treated samples of the 244 alloy were analyzed with high-energy in situ and ex situ X-ray techniques such as small angle X-ray scattering and wide angle X-ray scattering as well as Vickers micro-hardness, electron microscopy, and atom probe tomography. The relationship between hardness, aging parameters, and microstructure evolution is discussed. The enthalpy of formation and precipitate solvus temperature were determined with high-temperature differential scanning calorimetry and dilatometry analysis.

Ni-based Superalloys↗
DOE OSTI2025

Voltage Probability Density Function Shaping Control Strategy Considering Grid Operational Uncertainties

It is well-known that power systems operation always affected by various uncertainties which make the bus voltage a random process that can be characterized by its probability density function (PDF) at any time instant. In this context, this paper presents a novel PDF-based voltage control framework for power systems. By modeling voltage as a stochastic process, we formulate a stochastic differential equationthat captures grid uncertainties. The associated Fokker-Planck-Kolmogorov equation is derived to describe the evolution of the voltage PDF, which enables the formulation of a PDF-shaping control strategy. To simplify the PDF control formulation, a B-spline neural network is introduced for real-time estimation and regulation of the voltage distribution. The proposed PDF control law updates voltage references for energy storage systems and synchronous generators using real-time PDF measurements and feedback signals. The proposed method is validated on a modified Kundur’s two-area system. Simulation results demonstrate that the controller can significantly improve the voltage stability under stochastic conditions, highlighting its effectiveness in modern inverter-rich grids.

Gui, Yonghao [ORNL] (ORCID:0000000250435534)↗
DOE OSTI2025

A small core in Vesta inferred from Dawn’s observations

Vesta’s large-scale interior structure had previously been constrained primarily using the gravity and shape data from the Dawn mission. However, these data alone still allow a wide range of possibilities for the differentiation state of the body. The moment of inertia is arguably the most diagnostic parameter related to the radial density distribution of a planetary body, making it crucial for assessing the body’s state of internal differentiation. Determining the moment of inertia requires additional measurements of the amplitudes of small rotational motions, such as precession and nutation. Here we report an updated estimate of the moment of inertia of Vesta inferred from Dawn’s Doppler tracking via the Deep Space Network and onboard imaging data. The recovered value for Vesta’s normalized polar moment of inertia is $\overline{C}$/MR 2 = 0.4208 ± 0.0047 (where M is the mass of Vesta and R is the reference radius), which is only 6.6% lower than the homogeneous value of 0.4505. This value, combined with the gravity field and global shape, suggests that Vesta’s interior has limited density stratification beneath its howardite–eucrite–diogenite-dominated crust. We propose two possible origin scenarios that are consistent with the observed constraints. In the first scenario, Vesta’s interior did not undergo full differentiation due to late accretion. In the second scenario, Vesta originated as an impact remnant of a larger differentiated body re-accreted with non-chondritic bulk composition produced from a catastrophic impact. Vesta did not experience complete differentiation in either scenario, suggesting that its current state reflects a complex interplay between its accretion timing, thermal evolution, redistribution of 26 Al bearing melt and/or impact processes.

CNEOS 2014-01-08 bolide↗
DOE OSTI2024

Effect of Zn Addition on Phase Evolution in AlCrFeCoNiZn High–Entropy Alloy

The addition of Zn to AlCrFeCoNi high-entropy alloy (HEA) poses intriguing questions as to how it would affect phase evolution. Herein, the phase evolution in AlCrFeCoNiZn is studied using a combination of experimental techniques (X-ray diffraction, scanning electron microscopy, energy-dispersive spectroscopy, and differential scanning calorimetry) and computational (density-functional theory [DFT], calculation of phase diagrams, and machine-learning) methods. Mechanically alloyed and spark-plasma-sintered AlCrFeCoNiZn assumes a metastable single-phase, body-centered-cubic (BCC) structure that undergoes diffusion-controlled phase separation upon subsequent heat treatment to form separate (Al, Cr)-rich, (Fe, Co)-rich, and (Zn, Ni)-rich phases. The formation of (Al, Cr)-rich phase, not reported previously in AlCrFeCoNi-based HEAs, is attributed to strong clustering tendency of Cr–Zn and Cr–Ni pairs, combined with the strong ordering of Zn–Ni pair, driving out Cr that in turn combines with Al to form a (Al, Cr)-rich phase. In the DFT results, the formation of thermodynamically stable L1 2 phase is shown wherein Cr–Fe–Zn [Al–Ni-Co] preferably occupy1a (000) [3c (0 ½ ½)] positions. Furthermore, the sluggish diffusional transformation to L1 2 phase from BCC precursors is attributed to the small stacking-fault energy of AlCrFeCoNiZn. The equilibrated HEA exhibits a high microhardness of 8.24 GPa with an elastic modulus of 184 GPa.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Nonintrusive projection-based reduced order modeling using stable learned differential operators

Nonintrusive projection-based reduced order models (ROMs) are essential for dynamics prediction in multi-query applications where underlying governing equations are known but the access to the source of the underlying full order model (FOM) is unavailable; that is, FOM is a glass-box. This article proposes a learn-then-project approach for nonintrusive model reduction. In the first step of this approach, high-dimensional stable sparse learned differential operators (S-LDOs) are determined using the generated data. In the second step, the ordinary differential equations, comprising these S-LDOs, are used with suitable dimensionality reduction and low-dimensional subspace projection methods to provide equations for the evolution of reduced states. This approach allows easy integration into the existing intrusive ROM framework to enable nonintrusive model reduction while allowing the use of Petrov–Galerkin projections. The applicability of the proposed approach is demonstrated for Galerkin and LSPG projection-based ROMs through four numerical experiments: 1-D scalar advection, 1-D Burgers, 2-D scalar advection and 1-D scalar advection–diffusion–reaction equations. In conclusion, the results indicate that the proposed nonintrusive ROM strategy provides accurate and stable dynamics prediction.

42 ENGINEERING↗
DOE OSTI2025

Conceptual Designs for Irradiation Creep Testing of SiC in HFIR

Understanding irradiation creep of nuclear fuel cladding is important to properly size the initial fuel-cladding gap and understand when pellet-cladding contact is expected to occur due to a combination of fuel swelling and cladding creep-down. Irradiation creep also plays a role in relaxing stresses that develop in-pile. Silicon carbide fiber–reinforced silicon carbide matrix (SiC/SiC) composites are the leading long-term accident-tolerant fuel cladding concept for light-water reactors (LWRs). Although some limited data are available regarding irradiation creep of the individual constituents (fibers, matrix), data regarding irradiation creep of SiC/SiC composites are currently insufficient. Additional data regarding irradiation creep compliance and the rupture lifetime (combination of creep and slow crack growth) are needed to understand material limitations. This work describes the design and development of two irradiation vehicles that are being pursued for testing SiC/SiC concepts in the High Flux Isotope Reactor (HFIR). The first is a passive experiment, referred to as the PRECISE experiment, that leverages the constant coolant pressure of HFIR to compress a metallic bellows and provide a well-characterized load to drive creep in a SiC/SiC dog bone specimen. The total creep strain would be quantified post-irradiation by measuring dimensional changes of the specimen length as well as local dimensional changes within the gauge region. Non-stressed specimens would also be irradiated under the same conditions to provide an indication of dimensional changes due to radiation-induced swelling in the absence of creep. A second, more complex experiment, referred to as the INSITE experiment, is being designed in parallel that would use pneumatics to pressurize a metal bellows and linear variable differential transformers (LVDTs) to measure the specimen displacement in situ during irradiation. Such an experiment would provide significantly more data regarding the evolution of the creep compliance as a function of dose and applied stress within a single experiment but would require significantly more development time and cost to execute. The primary concern with the INSITE experiment is the accuracy, reliability, and expected lifetime of the LVDTs during irradiation at elevated temperatures. Efforts are being made to adjust the experiment design and operating procedure to limit LVDT temperatures and mitigate or otherwise compensate for uncertainties due to factors such as temperature fluctuations, creep in the surrounding structural materials, and drift of the LVDTs. This work describes the experiment designs, thermal and structural analysis that were performed to ensure that the desired temperature and stress conditions can be achieved, some initial sensitivity analyses to predict the evolution of the radiation-induced specimen displacements, and potential sources of uncertainty in the measurements. Out-of-pile testing is being performed in parallel to confirm that the test trains achieve the expected stress states in the specimens and do not result in prohibitive stress concentrators (e.g., in the grip regions) that might risk pre-mature failure. The PRECISE experiments are proceeding toward fabrication and assembly with HFIR insertion planned during fiscal year 2026. The INSITE experiment is progressing toward out-of-pile demonstrations, which will provide more conclusive evidence regarding the feasibility of executing these tests in HFIR or whether alternative displacement monitoring techniques may need to be considered.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2026

A Tensor Network-Based Quantum Algorithm for the Nonlinear 1D Burgers' Equation

In this work, we implement a tensor network-based quantum algorithm to solve unsteady, nonlinear partial differential equations (PDEs). The challenge lies in how to effectively represent, encode, process, and evolve the nonlinear system of PDEs on quantum computers. We will discuss the new techniques using the compressible 1-dimensional (1D) Burgers' equation as an example, because it represents the fundamental nonlinear feature and yet removes certain complexity in physics, allowing us to focus on the design of quantum algorithms. Previous attempts to solve nonlinear PDEs in quantum computation have often involved storing multiple copies of solutions or employing linearizations. Neither is practical due to exponential scaling with evolution time or insufficient solution accuracy. Our framework is based on matrix product states (MPSs) and matrix product operators (MPOs). For example, the velocity field is represented by MPS, whereas the linear and nonlinear spatial differential terms of the velocity field are processed by MPOs. Our primary focus herein is to verify and validate the various tensor network components of the algorithm using solutions obtained by the classical algorithms on high performance computing (HPC) architectures. We use a classical time marching method to demonstrate the functionality of the tensor network operations to model the PDE and their robustness with the time evolution of the system. Our classical simulation results demonstrate the utility of tensor network-based operations in modeling nonlinear PDEs and highlight the necessity as well as potential advantages of using quantum simulations for these techniques.

Gopalakrishnan Meena, Murali [ORNL] (ORCID:0000000↗
DOE OSTI2025

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↗