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DOE OSTI2023

Resonant instabilities mediated by drag and electrostatic interactions in laboratory and astrophysical dusty plasmas

Dusty plasmas are known to support a diverse range of instabilities, including both generalizations of standard plasma instabilities and ones caused by effects specific to dusty systems. It has been recently demonstrated that a novel broad class of streaming instabilities, termed resonant drag instabilities (RDIs), can be attributed to a particular resonance phenomenon, manifested by defective eigenvalues of the linearized dust/fluid system. In this work, it is demonstrated that this resonance phenomenon is not unique to RDIs and can be used as a framework to understand a wider range of instabilities, termed resonant instabilities. Particular attention is given to the filamentary ionization instability seen in laboratory dusty plasmas and to the two-stream instability. It is shown that, due to the commonalities in underlying physics between the dust-ion-acoustic two-stream instability and the acoustic RDI, these instabilities should be relevant in strongly overlapping regimes in astrophysical dusty plasmas. Further, it is proposed that a similar overlap in the experimental accessibility of these modes (and of the filamentary instability) allows for the possibility of experimental investigation in the laboratory of complex and astrophysically relevant instability dynamics.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1989

Time analysis of polydisperse sprays in complex turbulent environments

The two-phase flow field of a small swirl-stabilized burner has been investigated, using a phase Doppler particle analyzer to obtain the velocity, drop size, and time of arrival components. Water/kerosene comparisons were made for the case of nonreacting flow. It is found that the spray flow field is strongly influenced by the aerodynamic flow field and by the reaction and fluid properties. It is noted that vortex shedding produces clusters of drops.

Rudoff, R. C.↗

DOE OSTI2021

Analysis and prediction of reaction kinetics using the degree of rate control

“Degree of rate control” (DRC) analysis provides a quantitative approach for analysing the kinetics of multi-step reaction mechanisms that has been widely applied to both heterogeneous and homogeneous catalysis research, as well as electrocatalysis. The DRC of any given transition state or intermediate is defined as a partial derivative such that it approximately equals the fractional increase in net rate to the product of interest per differential decrease in its standard-state free energy for that species (÷RT), holding constant the standard-state free energies of all other transition states and intermediates. Even very complex mechanisms usually have only a few species with non-zero DRCs and are thus the species whose interactions with the catalyst most strongly affect the net rate. These key DRC values thus offer a simple and intuitive route to optimize catalyst materials, especially with the assistance of computational methods like density functional theory (DFT). These high-DRC species are also the species whose energetics must be most accurately measured or calculated to achieve an accurate kinetic model for any reaction mechanism. In simple cases with a single “rate-determining step”, the DRC for its transition state (TS) is + 1. Catalyst-bound intermediates, on the other hand, often have negative DRCs equal to a small integer times their fractional occupancy of catalyst sites. The apparent activation energy equals a weighted average of the standard-state enthalpies (relative to reactants) of all the species (intermediates, transition states and products) in the reaction mechanism, each weighted by its DRC (+RT). It has been shown that the apparent transfer coefficient in electrocatalysis, an inverted form of the Tafel slope, is a weighted average of the number of electrons transferred to generate each intermediate or product species in the mechanism, weighted again by the DRC. Quantitative analysis of kinetic isotope effects (KIEs, or the ratio of net rates for different reactant isotopes) in complex mechanisms has shown that the logarithm of the KIE equals the weighted average over all species in the mechanism of the difference between the two isotopes in their standard-state free energies (÷RT), again weighted by the DRC. The reaction orders with respect to fluid-phase concentrations of reactants, products and intermediates have also been proven to be directly related to DRCs. Thus, there are numerous experimental observables which equate to short linear combinations of DRCs, so that combinations of experimental measurements might provide access to DRC values. Since its invention for transition states in 1994 and its generalization to include intermediates in 2009, the DRC has thus far mainly been calculated for microkinetic models based either entirely on DFT or on DFT with the key energies (i.e., those for high-DRC species) being fine-tuned to match experiments. The relationships summarized in this work provide new opportunities for using experiments earlier in the development and optimization of microkinetic models that require input from computational catalysis.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1986

AGARD Fluid Dynamics Panel Symposium on Applications of Computational Fluid Dynamics in Aeronautics: Proceedings

The Fluid Dynamics Panel of AGARD arranged a Symposium on Applications of Computational Fluid Dynamics in Aeronautics, on 7 to 10 April 1986 in Aix-en-Provence, France. The purpose of the Symposium was to provide an assessment of the status of CFD in aerodynamic design and analysis, with an emphasis on emerging applications of advanced computational techniques to complex configurations. Sessions were devoted specifically to grid generation, methods for inviscid flows, calculations of viscous-inviscid interactions, and methods for solving the Navier-Stokes equations. The 31 papers presented at the meeting are published in AGARD Conference Proceedings CP-412 and are listed in the Appendix of this report. A brief synopsis of each paper and some general conclusions and recommendations are given.

Mccroskey, W. J.↗

NASA NTRS1986

Multispecies CARS measurements in turbulent combustion

A coherent anti-Stokes Raman scattering (CARS) instrument was upgraded to more accurately measure the number density of oxygen molecules. This instrument is designed to simultaneously measure nitrogen number density, oxygen number density, and the ro-vibrational temperature of nitrogen. The CARS is a nonintrusive diagnostic technique which utilizes a pulsed Nd:YAG laser, broadband dye lasers, a complex optical system, and an intensified photodiode array detector to sample hostile combustion environments. Measurements were made in a flame produced by a coaxial subsonic diffusion hydrogen-air burner. These data were used as a data base for comparison with the results of two separate computational fluid dynamics calculations. Complications which had arisen in previous studies with regard to interpretation of the oxygen data were eliminated.

Antcliff, Richard R.↗

NASA NTRS1992

Processing lunar soils for oxygen and other materials

Two types of lunar materials are excellent candidates for lunar oxygen production: ilmenite and silicates such as anorthite. Both are lunar surface minable, occurring in soils, breccias, and basalts. Because silicates are considerably more abundant than ilmenite, they may be preferred as source materials. Depending on the processing method chosen for oxygen production and the feedstock material, various useful metals and bulk materials can be produced as byproducts. Available processing techniques include hydrogen reduction of ilmenite and electrochemical and chemical reductions of silicates. Processes in these categories are generally in preliminary development stages and need significant research and development support to carry them to practical deployment, particularly as a lunar-based operation. The goal of beginning lunar processing operations by 2010 requires that planning and research and development emphasize the simplest processing schemes. However, more complex schemes that now appear to present difficult technical challenges may offer more valuable metal byproducts later. While they require more time and effort to perfect, the more complex or difficult schemes may provide important processing and product improvements with which to extend and elaborate the initial lunar processing facilities. A balanced R&D program should take this into account. The following topics are discussed: (1) ilmenite--semi-continuous process; (2) ilmenite--continuous fluid-bed reduction; (3) utilization of spent ilmenite to produce bulk materials; (4) silicates--electrochemical reduction; and (5) silicates--chemical reduction.

Knudsen, Christian W.↗

NASA NTRS2010

NASA Tech Briefs, March 2010

Topics covered include: Software Tool Integrating Data Flow Diagrams and Petri Nets; Adaptive Nulling for Interferometric Detection of Planets; Reducing the Volume of NASA Earth-Science Data; Reception of Multiple Telemetry Signals via One Dish Antenna; Space-Qualified Traveling-Wave Tube; Smart Power Supply for Battery-Powered Systems; Parallel Processing of Broad-Band PPM Signals; Inexpensive Implementation of Many Strain Gauges; Constant-Differential-Pressure Two-Fluid Accumulator; Inflatable Tubular Structures Rigidized with Foams; Power Generator with Thermo-Differential Modules; Mechanical Extraction of Power From Ocean Currents and Tides; Nitrous Oxide/Paraffin Hybrid Rocket Engines; Optimized Li-Ion Electrolytes Containing Fluorinated Ester Co-Solvents; Probabilistic Multi-Factor Interaction Model for Complex Material Behavior; Foldable Instrumented Bits for Ultrasonic/Sonic Penetrators; Compact Rare Earth Emitter Hollow Cathode; High-Precision Shape Control of In-Space Deployable Large Membrane/Thin-Shell Reflectors; Rapid Active Sampling Package; Miniature Lightweight Ion Pump; Cryogenic Transport of High-Pressure-System Recharge Gas; Water-Vapor Raman Lidar System Reaches Higher Altitude; Compact Ku-Band T/R Module for High-Resolution Radar Imaging of Cold Land Processes; Wide-Field-of-View, High-Resolution, Stereoscopic Imager; Electrical Capacitance Volume Tomography with High-Contrast Dielectrics; Wavefront Control and Image Restoration with Less Computing; Polarization Imaging Apparatus; Stereoscopic Machine-Vision System Using Projected Circles; Metal Vapor Arcing Risk Assessment Tool; Performance Bounds on Two Concatenated, Interleaved Codes; Parameterizing Coefficients of a POD-Based Dynamical System; Confidence-Based Feature Acquisition; Algorithm for Lossless Compression of Calibrated Hyperspectral Imagery; Universal Decoder for PPM of any Order; Algorithm for Stabilizing a POD-Based Dynamical System; Mission Reliability Estimation for Repairable Robot Teams; Processing AIRS Scientific Data Through Level 3; Web-Based Requesting and Scheduling Use of Facilities; AutoGen Version 5.0; Time-Tag Generation Script; PPM Receiver Implemented in Software; Tropospheric Emission Spectrometer Product File Readers; Reporting Differences Between Spacecraft Sequence Files; Coordinating "Execute" Data for ISS and Space Shuttle; Database for Safety-Oriented Tracking of Chemicals; Apparatus for Cold, Pressurized Biogeochemical Experiments; Growing B Lymphocytes in a Three-Dimensional Culture System; Tissue-like 3D Assemblies of Human Broncho-Epithelial Cells; Isolation of Resistance-Bearing Microorganisms; Oscillating Cell Culture Bioreactor; and Liquid Cooling/Warming Garment.

Source record↗

NASA NTRS2012

Multi-Mission Radioisotope Thermoelectric Generator Heat Exchangers for the Mars Science Laboratory Rover

The addition of the Multi-Mission Radioisotope Thermoelectric Generator (MMRTG) to the Mars Science Laboratory (MSL) Rover requires an advanced thermal control system that is able to both recover and reject the waste heat from the MMRTG as needed in order to maintain the onboard electronics at benign temperatures despite the extreme and widely varying environmental conditions experienced both on the way to Mars and on the Martian surface. Based on the previously successful Mars landed mission thermal control schemes, a mechanically pumped fluid loop (MPFL) architecture was selected as the most robust and efficient means for meeting the MSL thermal requirements. The MSL heat recovery and rejection system (HRS) is comprised of two Freon (CFC-11) MPFLs that interact closely with one another to provide comprehensive thermal management throughout all mission phases. The first loop, called the Rover HRS (RHRS), consists of a set of pumps, thermal control valves, and heat exchangers (HXs) that enables the transport of heat from the MMRTG to the rover electronics during cold conditions or from the electronics straight to the environment for immediate heat rejection during warm conditions. The second loop, called the Cruise HRS (CHRS), is thermally coupled to the RHRS during the cruise to Mars, and provides a means for dissipating the waste heat more directly from the MMRTG as well as from both the cruise stage and rover avionics by promoting circulation to the cruise stage radiators. A multifunctional structure was developed that is capable of both collecting waste heat from the MMRTG and rejecting the waste heat to the surrounding environment. It consists of a pair of honeycomb core sandwich panels with HRS tubes bonded to both sides. Two similar HX assemblies were designed to surround the MMRTG on the aft end of the rover. Heat acquisition is accomplished on the interior (MMRTG facing) surface of each HX while heat rejection is accomplished on the exterior surface of each HX. Since these two surfaces need to be at very different temperatures in order for the fluid loops to perform efficiently, they need to be thermally isolated from one another. The HXs were therefore designed for high in-plane thermal conductivity and extremely low through-thickness thermal conductivity by using aluminum facesheets and aerogel as insulation inside a composite honeycomb core. Complex assemblies of hand-welded and uniquely bent aluminum tubes are bonded onto each side of the HX panels, and are specifically designed to be easily mated and demated to the rest of the RHRS in order to ease the integration effort.

Mastropietro, A. J.↗

NASA NTRSDate not supplied

CFD Simulation of Ignition Overpressure and Plume-Water Interaction in the SLS Scale Model Acoustics Test

The solid rocket boosters (SRBs) on the Space Launch System (SLS) generate powerful ignition overpressure (IOP) waves that can potentially damage the vehicle and surrounding ground structures. Based on historical Shuttle experience, the IOP will be mitigated by utilizing an IOP/Sound Suppression (SS) water system which sprays water in and around the SRB and RS-25 engine plumes. However, the complex multiphase physics of the resulting plume-water interaction can lead to undesirable effects, such as high velocity water spray onto the engine nozzles and SRB nozzle side loads. Prior to launch, insight into this environment can be achieved through scale model testing and predictive computational fluid dynamics (CFD) modeling. The Scale Model Acoustic Test (SMAT) was conducted at Marshall Space Flight Center to provide the data necessary to develop the SLS liftoff acoustic environments by using a 5% scale SLS vehicle, fully integrated with a scaled IOP/SS water system, mobile launcher (ML), and tower. NASA engineers have put forth significant effort in validating the Loci/CHEM-Real Fluids multiphase CFD solver on a wet hold down test from the SMAT series. In this paper, the IOP acoustics and plume-water interactions predicted by the CFD model are compared and validated against the experiment, enabling relatively inexpensive, rapid turnaround for multiphase launch environment analyses to support future flights and design evolutions of SLS and other launch vehicles.

Computational Fluid Dynamics↗

DOE OSTI2021

Development and Demonstration of a Reduced-Order PWR Power Dispatch Simulator

This report describes the development, modeling, and results of a reduced-order thermal power dispatch pressurized water reactor (RO-TPD-PWR) power plant simulator that incorporates coupled electrical and thermal power dispatch to an industrial process. The simulator is built on models that have been developed in previous work, including a Rancor Microworld model previously developed by INL and the University of Idaho [Ref] and reactor core and secondary system models that have been published previously [Ref]. The simulator also includes a scalable model of a high temperature electrolysis (HTE) system that has been developed at INL. A particular benefit of the Rancor Microworld model is that it is compatible with a human-system interface (HMI) that has already been developed by the Human System Simulation Laboratory (HSSL) at INL that can emulate the control room of a PWR for realistic human-in-the-loop studies with mock nuclear power plant operating procedures. The results of these human-in-the-loop tests will be compared with similar tests using a commercial full-scope generic PWR (GPWR) simulator that has been modified to include thermal power dispatch, referred to as the TPD-GPWR [ , ]. The simplicity of the RO-TPD-PWR helps to better understand and verify the results of the TPD-GPWR, which employs >1,000 variables and has much greater complexity. A second key purpose of the RO-TPD-PWR simulator is that its relative simplicity allows it scale between lab-size equipment (~100 kW) and full-scale nuclear power plants (~1 GW) by adjusting only a few parameters in the models, such as fluid masses and heat exchanger areas. The RO-TPD-PWR simulator will be used to guide the controls of lab tests, so that equipment in those tests, such as the Thermal Energy Distribution System (TEDS) and pilot-scale high temperature electrolysis (HTE) units at scales of 50-250 kW, may operate in a manner that is relevant to full-scale PWR operations.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

DOE OSTI2025

A Multiscale Approach to Simulate Non‐Isothermal Multiphase Flow in Deformable Porous Materials

Coupled thermal, hydraulic, and mechanical processes in porous materials play important roles in several energy and environmental technologies. The Darcy-Brinkman-Biot (DBB) framework has proven effective in modeling multiphase fluid flow in deformable porous solids across both pore and Darcy scales, including in systems where fractures coexist with a porous matrix. In this study, we extend the DBB framework, originally designed for isothermal conditions, to address non-isothermal problems by incorporating an energy conservation equation. The resulting solver, hybridBiotThermalInterFoam, enables simulations of coupled multiphase fluid flow, heat transfer, and solid deformation in hybrid-scale systems containing both solid-free regions and ductile porous domains. The new solver is validated through comparisons with analytical solutions and, also, against established heat transfer solvers chtMultiRegionFoam and compressibleInterFoam. Further, a series of 2D and 3D case studies, including two-phase heat transfer in solid-free, static, or deformable porous media, highlights the solver's capacity to simulate complex flow dynamics and heat transport in systems involving high mobility ratios, viscous fingering, and fracture propagation. Our results establish the feasibility of incorporating thermal effects in simulations of a wide variety of energy geotechnics and environmental applications, including enhanced hydrocarbon recovery, soil remediation, and enhanced geothermal energy systems.

04 OIL SHALES AND TAR SANDS↗

NASA NTRSDate not supplied

Validation of a Two-Phase CFD Model for Autogenous Pressurization and Expulsion

This paper presents a two-phase computational fluid dynamics (CFD) model for simulating autogenous pressurization and expulsion in a cryogenic liquid hydrogen storage tank, utilizing a volume-of-fluid (VOF) approach combined with the kinetics-based Schrage equation to accurately capture the phase boundary and interfacial heat, mass, and momentum transfer between liquid and vapor phases. The model is validated against experimental data from NASA's K-site tank facility, specifically experimental case 225, which involved pressurization and controlled expulsion of liquid hydrogen. Various turbulence models are evaluated to assess their influence on model accuracy. The CFD simulations successfully replicate key thermodynamic behaviors observed during the experiments, including pressure evolution, temperature profiles, and phase-change dynamics at the vapor-liquid interface. The predicted tank pressures, temperatures, and pressurant requirements agree with experimental data, with pressurant mass predictions within 16\% of observed values. This study highlights the importance of selecting appropriate turbulence models to accurately simulate complex flow and heat transfer phenomena during tank pressurization and expulsion. By enhancing the accuracy and reliability of CFD models for liquid hydrogen under cryogenic conditions, this research contributes to developing efficient cryogenic propellant management strategies for future space missions.

Computational Fluid Dynamics↗

DOE OSTI2025

BlueCRAB Domain Overlapping Coupling: Theory & Verification

Coupling low- and high-fidelity codes is a useful way to model complex engineering systems. Thanks to the high-fidelity code, complex phenomena can be resolved in areas of the system where this is required, yet the efficiency of the lower-fidelity code is still retained in modeling the rest of the system. This document details the theory and implementation of coupling two different thermal-hydraulic codes: the system thermal-hydraulics (STH) code System Analysis Module (SAM) and the coarse-mesh computational fluid dynamics (CFD) code Pronghorn. Both applications are included in the Comprehensive Reactor Analysis Bundle (BlueCRAB) code suite and are based on the Multiphysics Object-Oriented Simulation Environment (MOOSE) framework. The domain overlapping (DO) coupling approach was adopted, as it offers proven advantages over more conventional domain decomposition methods. In the DO coupling, SAM provides Pronghorn with boundary conditions that depend on the system-level simulation of the entire plant. In return, the overlapping coupled SAM components–termed “surrogate components”–receive friction factors and source terms computed online based on the Pronghorn simulation. The framework is designed to be generic and enable coupling regardless of geometry and the number of inlet/outlet boundaries in the DO coupled domain. The developed method leads to consistent pressure drops, enthalpies, and scalar concentrations when comparing the coupled SAM and Pronghorn simulations. This document presents the DO coupling approach, along with two verification and two demonstration cases. The proposed problems explore different physical aspects relevant to nuclear reactor analysis, including buoyancy-driven flows, complex flow patterns, and multiple inlets and outlets. Periodically, new versions of this “BlueCRAB Domain Overlapping Coupling: Theory & Verification” report will be issued to reflect future developments and verification tests.

22 - GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2020

Improving Gas Reactor Design with Complex Non-Standard Reaction Mechanisms in a Reactive Flow Model

Fluorinated gases are critical to a number of high-technology industries including semiconductor manufacturing and related processes such as flat panel display, LED, and photovoltaic manufacturing. These fluorinated gases have very high global warming potentials (GWPs) and, unfortunately are chemically very stable, in some cases lasting for thousands of years in the atmosphere. For environmental reasons, it is imperative that the release of these gases be minimized, but existing methods to destroy them requires a significant amount of energy. The goal of this project was to develop a more energy efficient method of destroying these high GWP gases using computational fluid dynamics, but the computational requirements are significant. In this project, we bring together resources available at the national labs, including high-performance computing hardware and modern computational methods, in order to explore an industrially important problem with key environmental and economic impact. In this project, we examined the state-of-the-art models for the physical processes involved, and made subtle changes to arrive at alternative formulations considerably better suited for computation and study. We examined several representative scenarios and developed a set of analyses techniques and tools that could be used to explore the systems in much greater detail. This exploratory project has demonstrated to ALZETA that HPC can indeed provide a unique view into their complex engineering problem - one that is simply inaccessible by any other means. The tools allow us to explore the details of the chemical process, and allow one to construct "what-if" scenarios to address shortcomings of the physical device in its current operating modes. However, as with many sources of completely new information, many new questions are illuminated with this capability. It will take considerable time, effort and experience before we learn to probe the solutions, pose operating scenarios and confirm modifications to the device that lead to real engineering advances.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2021

SECARB-USA: Needs Assessment Framework for Storage Complexes (Task 2.1.b)

A team of SECARB storage experts examined 63 formations at 39 sites that were selected to represent the range of diversity of newly assessed, as well as well-advanced, storage prospects in the SECARB region. We inventoried the data needs triggered by the requirements to obtain a Class VI UIC storage permit, the data needs that results from the requirement to create geocellular fluid flow models to support that permit, and by pragmatic and best practice inputs such as public acceptance, regulatory readiness, and pore space leasing. We anchored both the needs inventory and the processes for and cost of meeting the needs with data from 9 sites in in the SECARB area that have advanced far in characterization. Results show that the total cost of characterization prior to obtaining a permit is convergent, because the permit and modeling requirements drive projects to obtain the same types of data for all cases. The high cost data that control cost are 1) drilling, coring, core-testing, logging, sampling and testing a characterization well and 2) collection of a 3-D seismic survey to map reservoir and confining system properties over the area of the plume or the area of elevated pressure. In 5 of our case study sites we determined that one or both of these costs could be avoided because the needs are met by available data.

54 ENVIRONMENTAL SCIENCES↗

NASA NTRS1989

Representation and display of vector field topology in fluid flow data sets

The visualization of physical processes in general and of vector fields in particular is discussed. An approach to visualizing flow topology that is based on the physics and mathematics underlying the physical phenomenon is presented. It involves determining critical points in the flow where the velocity vector vanishes. The critical points, connected by principal lines or planes, determine the topology of the flow. The complexity of the data is reduced without sacrificing the quantitative nature of the data set. By reducing the original vector field to a set of critical points and their connections, a representation of the topology of a two-dimensional vector field that is much smaller than the original data set but retains with full precision the information pertinent to the flow topology is obtained. This representation can be displayed as a set of points and tangent curves or as a graph. Analysis (including algorithms), display, interaction, and implementation aspects are discussed.

Helman, James↗

DOE OSTI2023

CoolPINNs: A physics-informed neural network modeling of active cooling in vascular systems

Emerging technologies like hypersonic aircraft, space exploration vehicles, and batteries avail fluid circulation in embedded microvasculatures for efficient thermal regulation. Modeling is vital during the design and operational phases of these engineered systems. However, many challenges exist in developing a modeling framework. What is lacking is an accurate framework that (i) captures sharp jumps in the thermal flux across complex vasculature layouts, (ii) deals with oblique derivatives (involving tangential and normal components), (iii) handles nonlinearity because of radiative heat transfer, (iv) provides a high-speed forecast for real-time monitoring, and (v) facilitates robust inverse modeling. Here, this paper addresses these challenges by availing the power of physics-informed neural networks (PINNs). We develop a fast, reliable, and accurate Scientific Machine Learning (SciML) framework for vascular-based thermal regulation—called CoolPINNs: a PINNs-based modeling framework for active cooling. The proposed mesh-less framework elegantly overcomes all the mentioned challenges. The significance of the reported research is multi-fold. First, the framework is valuable for real-time monitoring of thermal regulatory systems because of rapid forecasting. Second, researchers can address complex thermoregulation designs since the approach is meshless. Finally, the framework facilitates systematic parameter identification and inverse modeling studies, perhaps the most significant utility of the current framework.

97 MATHEMATICS AND COMPUTING↗

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↗