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DOE OSTI2025

Learning and discovering multiple solutions using physics-informed neural networks with random initialization and deep ensemble

In this work we explore the capability of physics-informed neural networks (PINNs) to discover multiple solutions. Many real-world phenomena governed by nonlinear differential equations (DEs), such as fluid flow, exhibit multiple solutions under the same conditions, yet capturing this solution multiplicity remains a significant challenge. A key difficulty lies in providing appropriate initial conditions or guesses, as widely used time-marching schemes and Newton’s method are highly sensitive to these choices when solving complex computational problems. While machine learning models, particularly PINNs, have shown promise in solving DEs, their ability to capture multiple solutions remains underexplored. In this work, we propose a simple and practical approach using PINNs to learn and discover multiple solutions. We first demonstrate that PINNs, when combined with random initialization and deep ensemble method—originally developed for uncertainty quantification—can effectively uncover multiple solutions to nonlinear ordinary and partial DEs. Although training large ensembles of PINNs may appear computationally demanding, this can be done efficiently using vectorization techniques supported by modern deep learning frameworks, allowing many networks to be trained simultaneously. Our approach highlights the critical role of initialization in shaping solution diversity, addressing an often-overlooked aspect of machine learning for scientific computing. Furthermore, we propose utilizing PINN-generated solutions as initial conditions or initial guesses for conventional numerical solvers to enhance accuracy and efficiency in capturing multiple solutions. Extensive numerical experiments, including the Allen–Cahn equation and cavity flow, where our approach successfully identifies both stable and unstable solutions, validate the effectiveness of our method. These findings establish a general and efficient framework for addressing solution multiplicity in nonlinear DEs.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2023

Pre-equilibrium evolution of conserved charges with initial conditions in the ICCING Monte Carlo event generator

Heavy-ion collisions can be well described through relativistic viscous hydrodynamics, but questions still remain when hydrodynamics is applicable because the initial state may begin very far from equilibrium. Thus, a pre-equilibrium evolution phase is used to bridge the gap between the initial state and hydrodynamics. KøMPøST is one such pre-equilibrium model that propagates the energy-momentum tensor by decomposing it into the background and fluctuations around that background, whose evolution is captured by Green's functions. We extend this formalism to include conserved charges and calculate the corresponding nonequilibrium Green's functions in the relaxation-time approximation. The ICCING algorithm initializes conserved charges in the initial state by sampling $g$ → $q$$\overline{q}$ splitting probabilities and is, thus, perfectly positioned to implement Green's functions for charge propagation. As a result, we show that this method alters the initial-state charge geometries and is applicable in central to mid-central collisions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

MRCI - Final Technical Report

The objective of the Midwest Regional Carbon Initiative (MRCI) project was to implement a collaborative Regional Initiative to accelerate the deployment of carbon capture, utilization, and storage (CCUS) in the Midwest-Northeastern and Mid-Atlantic areas of the United States. This area encompasses a 20-state region with much of the country’s carbon dioxide (CO2) emissions. The MRCI project is co-led by Battelle Memorial Institute (Battelle) in Columbus, Ohio, and the Illinois State Geological Survey (ISGS) in Champaign, Illinois, and benefitted from the experience these organizations have acquired over the past 10-plus years leading two Department of Energy (DOE) Regional Carbon Sequestration Partnerships (RCSPs): the Midwestern Regional Carbon Sequestration Partnership (MRCSP), led by Battelle, and the Midwest Geologic Sequestration Consortium (MGSC), led by the ISGS.

CCS↗

DOE OSTI2024

Imaging the initial condition of heavy-ion collisions and nuclear structure across the nuclide chart

High-energy nuclear collisions encompass three key stages: the structure of the colliding nuclei, informed by low-energy nuclear physics, the initial condition , leading to the formation of quark–gluon plasma (QGP), and the hydrodynamic expansion and hadronization of the QGP, leading to final-state hadron distributions that are observed experimentally. Recent advances in both experimental and theoretical methods have ushered in a precision era of heavy-ion collisions, enabling an increasingly accurate understanding of these stages. However, most approaches involve simultaneously determining both QGP properties and initial conditions from a single collision system, creating complexity due to the coupled contributions of these stages to the final-state observables. To avoid this, we propose leveraging established knowledge of low-energy nuclear structures and hydrodynamic observables to independently constrain the QGP’s initial condition. By conducting comparative studies of collisions involving isobar-like nuclei—species with similar mass numbers but different ground-state geometries—we can disentangle the initial condition’s impacts from the QGP properties. This approach not only refines our understanding of the initial stages of the collisions but also turns high-energy nuclear experiments into a precision tool for imaging nuclear structures, offering insights that complement traditional low-energy approaches. Opportunities for carrying out such comparative experiments at the Large Hadron Collider and other facilities could significantly advance both high-energy and low-energy nuclear physics. Additionally, this approach has implications for the future electron-ion collider. While the possibilities are extensive, we focus on selected proposals that could benefit both the high-energy and low-energy nuclear physics communities. Originally prepared as input for the long-range plan of U.S. nuclear physics, this white paper reflects the status as of September 2022, with a brief update on developments since then.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

Feasibility of DEIM for retrieving the initial field via dimensionality reduction

When parameter estimation is solved in a high-dimensional space, the dimensionality reduction strategy becomes the primary consideration for alleviating the tremendous computational cost. Here, the discrete empirical interpolation method (DEIM) is explored to retrieve the initial condition (IC) by combining the polynomial chaos (PC) based ensemble Kalman filter (i.e. PC-EnKF), where a non-intrusive PC expansion is considered as a surrogate model in place of the forward model in the prediction step of the ensemble Kalman filter, resulting in fewer forward model integrations but with a comparable accuracy as Monte Carlo-based approaches. The DEIM acts as a hyper-reduction tool to provide the low-dimensional input for the high-dimensional initial field, which can be reconstructed using the information on the sparse interpolation grid points that is adaptively obtained through PC-EnKF data assimilation method. Thus an innovative framework to reconstruct the IC is developed. The detailed procedure at each assimilation iteration includes: the determination of the spatial interpolation points, the estimation of the initial values on the interpolation locations using the optimal observations, and the reconstruction of IC in the full space. The current study uses the reconstruction field of initial conditions of the Navier-Stokes equations as an example to illustrate the efficacy of our method. The experimental results demonstrate the proposed algorithm achieves a satisfactory reconstruction for the initial field. The proposed method helps to extend the applicable area of DEIM in solving inverse problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Variable-density buoyancy-driven turbulence with asymmetric initial density distribution

The effects of different initial density distributions on the evolution of buoyancy-driven homogeneous variable-density turbulence (HVDT) at low (0.05) and high (0.75) Atwood numbers are studied by using high-resolution direct numerical simulations. HVDT aims to mimic the acceleration-driven Rayleigh–Taylor and shock-driven Richtmyer–Meshkov instabilities and reveals new physics that arise from variable-density effects on the turbulent mixing. Here, the initial amounts of pure light and pure heavy flows are altered primarily to mimic the variable-density turbulence at the different locations of the Rayleigh–Taylor and Richtmyer–Meshkov instabilities’ mixing layers where the amounts of the mixing fluids are not equal. It is found that for the low Atwood number cases, the asymmetric initial density distribution has limited effects on both global and local flow evolution for HVDT. However, at high Atwood number, both global flow evolution and the local flow structures are strongly affected by the initial composition ratio. The flow composed of more light fluid reaches higher turbulent levels and the local statistics reach their fully-developed behavior earlier in the time evolution. In contrast, during the late time decay, where most of the flow is well-mixed, all parameters become independent of the initial composition ratio for both low and high Atwood number cases.

42 ENGINEERING↗

DOE OSTI2020

The effect of initial conditions on mixing transition of the Richtmyer–Meshkov instability

We investigate the late-time Richtmyer–Meshkov instability (RMI) growth of sinuous perturbations on an air/sulphur hexafluoride interface (Atwood number, A ~ 0.67 ) subjected to a Mach 1.2 planar shock wave at Los Alamos National Laboratory's vertical shock tube facility. Interface perturbations are established using a novel membraneless technique where cross-flowing air and SF 6 separated by an oscillating splitter plate create a perturbed density interface. The interface formed has multi-modal features and residual small perturbations, however, a dominant mode is still noticeable. The late-time perturbation growths scale with $ka_0$ initial conditions (where $k$ is the wavenumber and $a_0$ is the initial amplitude of the dominant mode) as measured at the pre-shock interface. Past nonlinear models based on potential-flow theory, heuristic/interpolation approaches, Padé approximants and numerical simulations are evaluated against present experimental results. Accounting for an explicit $ka_0$ dependence in Sadot et al.'s model, we propose an empirical rational function that captures the asymptotic behaviour of perturbation growth for a broad range of initial conditions ( 0.30 $≤ ka_0 ≤$ 0.86 ). The onset of mixing transition and its initial condition dependence are investigated with respect to the minimum state criterion ($Re =$ 1.6 $\times$ 10 5 ) for unsteady flows by Zhou. Earlier mixing transitions for higher ka0 initial conditions are noted from local and global Reynolds number estimates which are corroborated by the existence of an inertial sub-range and formation of mixing regions indicating the physical significance of the minimum state criterion in RMI flows. The transition is accompanied by the increasing teapot-like appearance of joint probability density functions of $p–q$ (invariants of the reduced velocity gradient tensor), establishing the technique as a useful tool for turbulence detection in two-dimensional diagnostics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2024

Adsorption Equilibrium, Kinetics, and Column Breakthrough Data of Acetic Acid, Butyric Acid, and Lactic Acid on IRN-78 Ion-exchange Resin at Initial pH ~3 – 7 and Temperature 25 – 55 °C

This work presents systematic aqueous-phase adsorption equilibrium, kinetics, and column breakthrough measurements with three key biointermediates that are common compounds in many bioprocesses. Adsorption equilibrium experiments were carried out with acetic acid, butyric acid, and lactic acid on a commercial ion-exchange resin, Amberlite IRN-78, at wide ranges of acid concentration (8-500 mmol/L), initial pH (similar to 3-7), and temperature (25-55 degrees C), simulating the effluent characteristics from different fermenter operations. The kinetics and column breakthrough experiments were conducted at an initial pH of 6 and a concentration of 200 mmol/L. The equilibrium study shows a higher loading at the initial pH < pK(a) and a lower loading at the initial pH > pK(a). Overall removal varies between 16 and 99% depending on the initial pH, temperature, and organic acid concentration and type. The study further indicates monolayer adsorption at the equilibrium pH > 10 and multilayer adsorption at the equilibrium pH < 6. The thermodynamic modeling of adsorption isotherm data was carried out using Langmuir and Freundlich isotherms. IRN-78 presents fast adsorption kinetics as the maximum loading was attained in <= 10 min and nearly the same breakthrough time for all three organic acids involved in this study.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2024

Adsorption Equilibrium, Kinetics, and Column Breakthrough Data for Aqueous Solutions of Binary-Acid and Ternary-Acid Mixtures of Acetic Acid, Butyric Acid, and Lactic Acid on IRN-78 Ion-Exchange Resin at Initial pH Levels of ∼3–7 and at 25–55 °C

This work is part of an effort to develop thermophysical property data and models supporting adsorptive process development for organic acid separation from a dilute aqueous solution of fermentation broth. It presents systematic experimental measurements for aqueous-phase adsorption equilibrium, kinetics, and column breakthrough for three binary-acid aqueous mixtures (acetic acid + lactic acid, butyric acid + lactic acid, and acetic acid + butyric acid) and one ternary-acid aqueous mixture (acetic acid + butyric acid + lactic acid) on Amberlite IRN-78 ion-exchange resin. The equilibrium measurements covered broad ranges of the initial acid concentration (100–400 mmol/L), initial pH (∼3–7), and temperature (25–55 °C). The equilibrium data for the binary-acid and ternary-acid aqueous mixtures indicate selective adsorption of lactic acid at initial pH levels of ∼3 and 4, while acetic acid and butyric acid are selectively adsorbed at initial pH levels of 5, 6, and 7. The subsequent kinetics and column breakthrough experiments were performed at 200 mmol/L with equimolar ratios, an initial pH of 6, and 25 °C. In conclusion, the measurements provide essential data sets for rigorous thermodynamic modeling and process simulation of adsorptive separation processes for organic acid separation from fermentation broth.

Adsorption↗

DOE OSTI2021

Discovery of widespread transcription initiation at microsatellites predictable by sequence-based deep neural network

Using the Cap Analysis of Gene Expression (CAGE) technology, the FANTOM5 consortium provided one of the most comprehensive maps of transcription start sites (TSSs) in several species. Strikingly, ~72% of them could not be assigned to a specific gene and initiate at unconventional regions, outside promoters or enhancers. Here, we probe these unassigned TSSs and show that, in all species studied, a significant fraction of CAGE peaks initiate at microsatellites, also called short tandem repeats (STRs). To confirm this transcription, we develop Cap Trap RNA-seq, a technology which combines cap trapping and long read MinION sequencing. We train sequence-based deep learning models able to predict CAGE signal at STRs with high accuracy. These models unveil the importance of STR surrounding sequences not only to distinguish STR classes, but also to predict the level of transcription initiation. Importantly, genetic variants linked to human diseases are preferentially found at STRs with high transcription initiation level, supporting the biological and clinical relevance of transcription initiation at STRs. Together, our results extend the repertoire of non-coding transcription associated with DNA tandem repeats and complexify STR polymorphism.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2021

Initiator-dependent kinetics of lyotropic liquid crystal-templated thermal polymerization

In this study, we examine the polymerization kinetics with different thermal initiators in lamellar and hexagonal lyotropic liquid crystal (LLC) structures directed by Pluronic L64. Ammonium persulfate is used to initiate the polymerization from the water phase, whereas azobisisobutyronitrile and benzoyl peroxide are employed to commence the reaction through the monomer phase. While the mesophase structure remains intact for all the initiation systems, the kinetics of polymerization and conversion vary significantly. The obtained differential scanning calorimetry (DSC) results reveal that, under same conditions, initiation from water (IFW) system results in enhanced reaction rates as well as higher monomer conversions compared to initiation from oil (IFO) one. Higher termination rate in LLC nanoconfinements induces lower reaction rates in IFO system. Moreover, our work on different LLC structures show that the effect of nanoconfinement on polymerization rate can be minimized through IFW. Chemorheology not only confirms the results obtained from DSC, but also shows that, in similar monomer conversions, the polymers obtained from IFW system exhibit an improved mechanical properties over the samples produced though IFO process.

36 MATERIALS SCIENCE↗

DOE OSTI2021

Probabilistic effects of porosity and chemical kinetics on the shock initiation of an octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine (HMX) based explosive

In this work, we investigate the effects of porosity and chemical reactivity on the shock initiation of an HMX (octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine)-based polymer bonded explosive due to thin flyer impact using two-dimensional mesoscale simulations. Microstructural samples are computationally generated, and the calculated shock response is compared to experiments. The sample dimensions are 0.1×1.05 mm 2 , and each sample accounts for grain scale heterogeneities, including randomized grain morphology and pore spacing. Simulations are performed using the Lawrence Livermore National Laboratory (LLNL) arbitrary Lagrangian–Eulerian multi-physics code ALE3D. Chemistry and equation of states are handled with LLNL's thermochemical code Cheetah. We employ a modified Arrhenius-based chemical kinetics model that scales the temperature exponent fitting parameter with flyer shock conditions. This new model is calibrated to the mean response of experimental shock initiation thresholds for three separate Kapton flyer thicknesses. A simplified shock initiation criterion is proposed, which classifies the macroscale initiation response based on cross-sectional temperature profiles of the shocked sample. This criterion is used to further investigate the probabilistic response of the sample to variations in the mean porosity and chemical kinetics rate, and we find an inverse correlation between the shock pressure and the confidence interval of the experimental threshold velocity. Furthermore, the methodology presented in this work for quantifying the probabilistic nature of shock initiation is material agnostic. These mesoscale studies may be used to inform continuum scale models and expedite the development of future, tailored energetic formulations.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Nonadiabatic dynamics with classical trajectories: The problem of an initial coherent superposition of electronic states

Advances in coherent light sources and development of pump–probe techniques in recent decades have opened the way to study electronic motion in its natural time scale. When an ultrashort laser pulse interacts with a molecular target, a coherent superposition of electronic states is created and the triggered electron dynamics is coupled to the nuclear motion. A natural and computationally efficient choice to simulate this correlated dynamics is a trajectory-based method where the quantum-mechanical electronic evolution is coupled to a classical-like nuclear dynamics. These methods must approximate the initial correlated electron–nuclear state by associating an initial electronic wavefunction to each classical trajectory in the ensemble. Different possibilities exist that reproduce the initial populations of the exact molecular wavefunction when represented in a basis. We show that different choices yield different dynamics and explore the effect of this choice in Ehrenfest, surface hopping, and exact-factorization-based coupled-trajectory schemes in a one-dimensional two-electronic-state model system that can be solved numerically exactly. Furthermore, this work aims to clarify the problems that standard trajectory-based techniques might have when a coherent superposition of electronic states is created to initialize the dynamics, to discuss what properties and observables are affected by different choices of electronic initial conditions and to point out the importance of quantum-momentum-induced electronic transitions in coupled-trajectory schemes.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Accelerating kinetic plasma simulations with machine-learning-generated initial conditions

Computational models of plasma technologies often solve for the system operating conditions by time-stepping an initial value problem to a quasi-steady solution. However, the strongly nonlinear and multi-timescale nature of plasma dynamics often necessitate millions, or even hundreds of millions, of steps to reach convergence, reducing the effectiveness of these simulations for computer-aided engineering. We consider acceleration of kinetic plasma simulations via data-driven machine-learning-generated initial conditions, which initialize the simulations close to their final quasi-steady-state, thereby reducing the number of steps to reach convergence. Three machine-learning models are developed to predict the density and ion kinetic profiles of capacitively coupled plasma discharges relevant to the microelectronics industry. The models are trained on kinetic simulations over a range of device operating frequencies and pressures. Best performance was observed when simulations were initialized with ion kinetic profiles generated by a convolutional neural network, reducing the mean number of steps to reach convergence by 17.1× when compared to initialization with a zero-dimensional global model. We also outline a workflow for continuous data-driven model improvement and simulation speedup, with the aim of generating sufficient data for full device digital twins.

Artificial neural networks↗

DOE OSTI2020

Robustness of slow contraction to cosmic initial conditions

We present numerical relativity simulations of cosmological scenarios in which the universe is smoothed and flattened by undergoing a phase of slow contraction and test their sensitivity to a wide range of initial conditions. Our numerical scheme enables the variation of all freely specifiable physical quantities that characterize the initial spatial hypersurface, such as the initial shear and spatial curvature contributions as well as the initial field and velocity distributions of the scalar that drives the cosmological evolution. In particular, we include initial conditions that are far outside the perturbative regime of the well-known attractor scaling solution. We complement our numerical results by analytically performing a complete dynamical systems analysis and show that the two approaches yield consistent results.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2020

Traces of nonequilibrium effects, initial condition, bulk dynamics, and elementary collisions in the charm observables

Heavy quarks produced in relativistic heavy-ion collisions are known to be sensitive probes of the hot and dense QCD matter they traverse. In this paper we study how their dynamics is affected by the nature of the bulk evolution of the QCD matter, the initial condition of the system, and the treatment of elementary interactions between heavy quarks and the surrounding medium. For the same initial condition and the same quark-gluon plasma (QGP) expansion scenario we discuss the consequences of the assumption of a local equilibrium by comparing the consequences for the nuclear modification factor $R_{AA}$ and the elliptic flows of charm quarks, scrutinizing the different components of the final distribution of charm quarks. For this purpose we employ the parton-hadron-string dynamics (PHSD) model, which is an off-shell microscopic transport approach, as well as the linearized-Boltzmann (LB) scheme obtained by coarse graining the PHSD bulk and assuming local equilibrium for the interactions of the charm quarks with the bulk. The $R_{AA}$ of charm quarks stemming from the later LB approach is also compared to a genuine fluid dynamics evolution initiated by the coarse grained PHSD, which allows us to further assess the consequences of reducing the full $\textit{n}$-body dynamics. We then proceed to a systematic comparison of PHSD (in its LB approximation) with MC@HQ, another transport model for heavy flavors which also relies on the LB approach. In particular, we investigate the consequences for the nuclear modification factor of charm quarks if we vary separately the initial heavy quark distribution function in matter, the expansion dynamics of the QGP, and the elementary interactions of heavy quarks of these models. We find that the results for both models vary significantly depending on the details of the calculation. However, both models achieve very similar predictions for key heavy quark observables for certain combinations of initial condition, bulk evolution, and interactions. In this work, we conclude that this ambiguity limits our ability to determine the different properties of the system based on the current set of observables.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

Gluonic hot spot initial conditions in heavy-ion collisions

The initial conditions in heavy-ion collisions are calculated in many different frameworks. The importance of nucleon position fluctuations within the nucleus and subnucleon structure has been established when modeling initial conditions for input to hydrodynamic calculations. However, there remain outstanding puzzles regarding these initial conditions, including the measurement of the near equivalence of the elliptical v 2 and triangular v 3 flow coefficients in ultracentral 0–1% Pb + Pb collisions at the CERN Large Hadron Collider. Recently a calculation termed magma incorporating gluonic hot spots via two-point correlators in the color glass condensate framework, and no nucleons, provided a simultaneous match to these flow coefficients measured by the ATLAS experiment, including in ultracentral 0–1% collisions. Our calculations reveal that the magma initial conditions do not describe the experimental data when run through full hydrodynamic sonic simulations or when the hot spots from one nucleus resolve hot spots from the other nucleus, as predicted in the color glass condensate framework. Finally, we also explore alternative initial condition calculations and discuss their implications.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2021

Constraining the initial stages of ultrarelativistic nuclear collisions

It is frequently supposed that quark-gluon plasma created in heavy-ion collisions undergoes free streaming at early times. Here, we examine this issue based on the assumption that a universal attractor dominates the dynamics already at the earliest stages, which offers a way to connect the initial state with the start of the hydrodynamic expansion in an approximate but conceptually transparent fashion. We demonstrate that the centrality dependence of the measured particle multiplicities can be used to quantitatively constrain the pressure anisotropy and find that it strongly depends on the model of the initial energy deposition. As an illustration, we compare three initial state models and show that they predict rather different early-time values of the pressure anisotropy. This suggests that while assuming free streaming prior to hydrodynamization may be compatible with some initial state models, in general, features of the prehydrodynamic flow need to be matched with the model of the initial state.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗