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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS1986

An efficient method for solving the steady Euler equations

An efficient numerical procedure for solving a set of nonlinear partial differential equations is given, specifically for the steady Euler equations. Solutions of the equations were obtained by Newton's linearization procedure, commonly used to solve the roots of nonlinear algebraic equations. In application of the same procedure for solving a set of differential equations we give a theorem showing that a quadratic convergence rate can be achieved. While the domain of quadratic convergence depends on the problems studied and is unknown a priori, we show that firstand second-order derivatives of flux vectors determine whether the condition for quadratic convergence is satisfied. The first derivatives enter as an implicit operator for yielding new iterates and the second derivatives indicates smoothness of the flows considered. Consequently flows involving shocks are expected to require larger number of iterations. First-order upwind discretization in conjunction with the Steger-Warming flux-vector splitting is employed on the implicit operator and a diagonal dominant matrix results. However the explicit operator is represented by first- and seond-order upwind differencings, using both Steger-Warming's and van Leer's splittings. We discuss treatment of boundary conditions and solution procedures for solving the resulting block matrix system. With a set of test problems for one- and two-dimensional flows, we show detailed study as to the efficiency, accuracy, and convergence of the present method.

Liou, M. S.↗
NASA NTRS1994

Understanding the determinants of problem-solving behavior in a complex environment

It is often argued that problem-solving behavior in a complex environment is determined as much by the features of the environment as by the goals of the problem solver. This article explores a technique to determine the extent to which measured features of a complex environment influence problem-solving behavior observed within that environment. In this study, the technique is used to determine how complex flight deck and air traffic control environment influences the strategies used by airline pilots when controlling the flight path of a modern jetliner. Data collected aboard 16 commercial flights are used to measure selected features of the task environment. A record of the pilots' problem-solving behavior is analyzed to determine to what extent behavior is adapted to the environmental features that were measured. The results suggest that the measured features of the environment account for as much as half of the variability in the pilots' problem-solving behavior and provide estimates on the probable effects of each environmental feature.

Casner, Stephen A.↗
NASA NTRS2009

Workflow Agents vs. Expert Systems: Problem Solving Methods in Work Systems Design

During the 1980s, a community of artificial intelligence researchers became interested in formalizing problem solving methods as part of an effort called "second generation expert systems" (2nd GES). How do the motivations and results of this research relate to building tools for the workplace today? We provide an historical review of how the theory of expertise has developed, a progress report on a tool for designing and implementing model-based automation (Brahms), and a concrete example how we apply 2nd GES concepts today in an agent-based system for space flight operations (OCAMS). Brahms incorporates an ontology for modeling work practices, what people are doing in the course of a day, characterized as "activities." OCAMS was developed using a simulation-to-implementation methodology, in which a prototype tool was embedded in a simulation of future work practices. OCAMS uses model-based methods to interactively plan its actions and keep track of the work to be done. The problem solving methods of practice are interactive, employing reasoning for and through action in the real world. Analogously, it is as if a medical expert system were charged not just with interpreting culture results, but actually interacting with a patient. Our perspective shifts from building a "problem solving" (expert) system to building an actor in the world. The reusable components in work system designs include entire "problem solvers" (e.g., a planning subsystem), interoperability frameworks, and workflow agents that use and revise models dynamically in a network of people and tools. Consequently, the research focus shifts so "problem solving methods" include ways of knowing that models do not fit the world, and ways of interacting with other agents and people to gain or verify information and (ultimately) adapt rules and procedures to resolve problematic situations.

Clancey, William J.↗
NASA NTRS2011

Technique for Solving Electrically Small to Large Structures for Broadband Applications

Fast iterative algorithms are often used for solving Method of Moments (MoM) systems, having a large number of unknowns, to determine current distribution and other parameters. The most commonly used fast methods include the fast multipole method (FMM), the precorrected fast Fourier transform (PFFT), and low-rank QR compression methods. These methods reduce the O(N) memory and time requirements to O(N log N) by compressing the dense MoM system so as to exploit the physics of Green s Function interactions. FFT-based techniques for solving such problems are efficient for spacefilling and uniform structures, but their performance substantially degrades for non-uniformly distributed structures due to the inherent need to employ a uniform global grid. FMM or QR techniques are better suited than FFT techniques; however, neither the FMM nor the QR technique can be used at all frequencies. This method has been developed to efficiently solve for a desired parameter of a system or device that can include both electrically large FMM elements, and electrically small QR elements. The system or device is set up as an oct-tree structure that can include regions of both the FMM type and the QR type. The system is enclosed with a cube at a 0- th level, splitting the cube at the 0-th level into eight child cubes. This forms cubes at a 1st level, recursively repeating the splitting process for cubes at successive levels until a desired number of levels is created. For each cube that is thus formed, neighbor lists and interaction lists are maintained. An iterative solver is then used to determine a first matrix vector product for any electrically large elements as well as a second matrix vector product for any electrically small elements that are included in the structure. These matrix vector products for the electrically large and small elements are combined, and a net delta for a combination of the matrix vector products is determined. The iteration continues until a net delta is obtained that is within the predefined limits. The matrix vector products that were last obtained are used to solve for the desired parameter. The solution for the desired parameter is then presented to a user in a tangible form; for example, on a display.

Jandhyala, Vikram↗
NASA NTRSDate not supplied

Students Solving Problems for ISS and Beyond: Inspiring the Next-Generation

Students Solving Problems for ISS and Beyond: Inspiring the Next-Generation Session Title: Students Solving Problems for ISS and Beyond: Inspiring the Next-Generation Session Description: NASA High school students United with NASA to Create Hardware (HUNCH) mission is to empower and inspire students through a Project-Based Learning program where 7-12 grade students learn 21st century skills and can launch their careers through participation in the design and fabrication of real-world valued products for NASA. With six different tracks consisting of Design & Prototype, Culinary Challenge, Softgoods, Precision Machining, Software, and Video Challenge, students are given the opportunity to create solutions for the International Space Station (ISS), the Moon, and beyond. Many projects are requested by the Crew to help ease living conditions, giving students the opportunity to make an impact on the lives of Astronauts. Other projects come directly from NASA and its partners. Join our session to learn how NASA is working with teachers across the country to mentor the next generation of scientists and engineers to solve some of NASA’s greatest challenges. Learning Outcomes: 1. Describe the NASA HUNCH Program, including the program objectives, goals, and strategic partnerships for middle school and high school outreach and advocacy efforts. 2. Identify the innovative strategies NASA is using to work with middle and high school students to solve real-world problems 3. Use the knowledge gained to inspire the next generation of scientists and engineers Session Track: Advocacy & Outreach Specialized Focus Area: Women in Government and Military Learning Level: Foundational Session Format: Listen & Learn Speaker Qualifications: 1. Deboshri Sadhukhan • Current Job Title: Deputy Project Manager • Topic Experience (years of experience related to proposed topic): 1-5 years • Biography: Deboshri Sadhukhan is an engineer for NASA Glenn Research Center (GRC). She has worked on numerous projects — from International Space Station (ISS) fluid technologies, to Orion European Service Module propulsion, to planetary science missions and other game-changing technologies. Her roles have ranged from Project Manager to System Safety Lead. She currently serves as a GRC Regional Mentor for the High school students United with NASA to Create Hardware (HUNCH) program. She also serves as Deputy Project Manager for an ISS payload and Safety & Mission Assurance Lead for a Radioisotope Power Systems project. In these roles, she oversees each phase of a project from beginning to end and provides leadership to increase the reliability, maintainability and system safety of hardware and personnel throughout the system life cycle. She holds a Bachelor of Science in Electrical Engineering from The University of Akron. 2. Nancy Hall • Current Job Title: Project Manager • Topic Experience: 20+ years • Biography: Nancy Rabel Hall earned a B.S. degree in Space Sciences from Florida Institute of Technology and a M.S. degree in Mechanical Engineering from the University of Toledo. She has been at NASA Glenn for over 30 years. She has led several International Space Station experiments that studied how the behavior of fluids and fluid systems behave differently in microgravity as compared to here on Earth. She is also the High school students United with NASA to Create Hardware (HUNCH) project manager, a program that allows students to design and fabricate hardware and softgoods for NASA as well as participate in a culinary and video challenge. She enjoys talking to the public and students about the work being done at NASA as well as showing students how math and science can be fun. She is an amateur radio operator, enjoys playing golf, and reading science fiction and fantasy books.

HUNCH↗
NASA NTRSDate not supplied

Content and Representation of Information Needed to Support Time-Constrained Problem Solving

NASA’s current mission-operations paradigm originated with Project Mercury and endured with minimum evolution through the Apollo Program, Space Shuttle Program, and ISS missions. At its foundation is a near-complete real-time dependence on a ground team to manage the combined state of the mission, vehicle, and crew. Utilizing many engineers and operators with broad and deep expertise; large, distributed datasets including extensive telemetry; and expansive analytical and computing power, this ground team has served as the safety net for crewed spaceflight missions over the past 60 years. This approach must change to address challenges associated with missions beyond low Earth orbit (BLEO), including infrequent resupply, reduced ability to evacuate, and delayed communications that prohibit real-time operational support. We anticipate that a necessary part of this change will be increased independence for the crew, as roles and responsibilities traditionally performed by ground teams move on board the vehicle. While many risks are associated with Earth-independent operations, one particular concern is ensuring that the crew will have adequate onboard support to perform urgent problem solving when communication with the ground is delayed or intermittent. A key resource that enables the ground team to respond to anomalies quickly and effectively is the extraordinary expertise and experience it possesses. It is comprised of 80+ experts on at any given time, with a combined 600+ years of system-specific experience across 22 unique console disciplines. A small crew will face the unprecedented challenge of independently responding to anomalies that have historically been handled by a team 20 times their size. Another important resource upon which the ground heavily relies to support procedure execution and anomaly response is data. The amount of telemetry data that each flight controller monitors is extensive. In addition, as the ground team works to further assess impacts, trouble shoot, identify workarounds, and oversee procedure execution, it accesses and synthesizes engineering and procedure information, as well as system build, test, and configuration documentation. It is not feasible nor useful to put all these data onboard as crews become more Earth independent. Each member of a small Mars mission small crew will have multiple roles beyond monitoring telemetry and data gathering, and multiple roles within anomaly resolution processes, thereby limiting their capacity for copious amounts of information. Moreover, while access is necessary, it alone is insufficient. Information will need to be compiled, refined, and represented appropriately to support the crew’s reduced attention and expertise. This work seeks to understand the content and representation of information needed to support time-constrained problem solving and decision making by the crew without real-time ground support. To build this understanding, we first surveyed the literature, focusing on how expert problem solvers construct and manipulate their mental models. Next, we interviewed expert problem solvers in spaceflight and analogous domains and surveyed industry solutions for data presentation. Finally, we analyzed current spaceflight operations by investigating flight controller anomaly resolution processes during ISS training simulations and real operational events. These methods led to creating a problem-solving framework that details common themes and features of attending to, assessing, analyzing, and acting on problems in complex, time-constrained domains. Using this framework and the results of our analysis, we identified conceptual data representations needed for crew-led problem-solving. Preliminary onboard user interface concepts to meet identified needs will be presented.

anomaly response↗
DOE OSTI2025

DS-TIDE: Harnessing Dynamical Systems for Efficient Time-Independent Differential Equation Solving

Time-Independent Differential Equations (TIDEs) are central to modeling equilibrium behavior across a wide range of scientific and engineering domains, from electrostatics to porous media flow. Conventional numerical solvers offer reliable solutions but incur significant computational costs due to fine-grained discretization and iterative procedures. Machine learning-based approaches address this by replacing iterative solving processes with one-time inference; however, their sophisticated models require extensive training resources that often exceed those of traditional solvers. Consequently, designing a TIDE solver that achieves high accuracy, broad applicability, and exceptional computational efficiency remains a fundamental challenge. In this paper, we propose DS-TIDE, a novel hardware solver that is inspired by, and subsequently leverages, the intrinsic connection between Dynamical Systems (DS) and Differential Equations (DEs) to efficiently and accurately solve TIDEs. DS-TIDE employs a CMOS-compatible DS-based processor, whose physical states evolve under carefully designed DE-driven dynamics and naturally converge to equilibrium -- the solution of the target TIDE -- within ~1µs on a ~1-watt DS-TIDE processor. To enhance expressivity, DS-TIDE incorporates Heterogeneous Dynamics with Temporal Layering (HDTL), which solves TIDEs through a three-stage DS evolution -- conditioning, solving, and decoding -- each governed by specialized dynamics. The entire evolution process is analogous to an infinitely deep neural network temporally unrolled, offering the system the capability of representing complex equations. Furthermore, DS-TIDE is equipped with an on-device DS-DE Auto-Alignment mechanism that dynamically adapts intrinsic hardware dynamics within milliseconds, effectively aligning the system’s dynamics to diverse target DEs. Experimental results across TIDEs from a wide range of scientific and engineering domains demonstrate that DS-TIDE achieves ~10^3× speedup, ~10^5× energy savings, and competitive or superior accuracy compared to state-of-the-art numerical and ML-based solvers.

Liu, Chuan↗
DOE OSTI2026

Solutions for Lasting, Viable Energy Infrastructure Technologies (SOLVE IT) Prize Final Technical Report

This is a final technical report for the American-Made The Solutions for Lasting, Viable Energy Infrastructure Technologies (SOLVE IT) Prize, funded by the Infrastructure Investments and Jobs Act through the Technology Commercialization Fund, administered by the U.S. Department of Energy (DOE) Office of Technology Commercialization (OTC) in collaboration with the Office of Clean Energy Demonstrations (OCED) and the Office of Energy Efficiency and Renewable Energy (EERE) with support from the National Laboratory of the Rockies (NLR). The SOLVE IT Prize aimed to enable local organizations to identify and implement innovative energy solutions in a way that works for their unique needs and challenges. The competition awarded $3,740,000 to winning teams across two phases. The prize was designed to support local stakeholders and organizations as they identified and implemented innovative energy solutions. In doing so, the SOLVE IT Prize looked to promote the commercialization of promising energy technologies that will lead to reliable, affordable energy across the U.S.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2020

Solving Stochastic Inverse Problems for Property–Structure Linkages Using Data-Consistent Inversion and Machine Learning

Determining process–structure–property linkages is one of the key objectives in material science, and uncertainty quantification plays a critical role in understanding both process–structure and structure–property linkages. In this work, we seek to learn a distribution of microstructure parameters that are consistent in the sense that the forward propagation of this distribution through a crystal plasticity finite element model matches a target distribution on materials properties. This stochastic inversion formulation infers a distribution of acceptable/consistent microstructures, as opposed to a deterministic solution, which expands the range of feasible designs in a probabilistic manner. Furthermore, to solve this stochastic inverse problem, we employ a recently developed uncertainty quantification framework based on push-forward probability measures, which combines techniques from measure theory and Bayes’ rule to define a unique and numerically stable solution. This approach requires making an initial prediction using an initial guess for the distribution on model inputs and solving a stochastic forward problem. To reduce the computational burden in solving both stochastic forward and stochastic inverse problems, we combine this approach with a machine learning Bayesian regression model based on Gaussian processes and demonstrate the proposed methodology on two representative case studies in structure–property linkages.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Parallel computing for power system climate resiliency: Solving a large-scale stochastic capacity expansion problem with mpi-sppy

Here we propose a nodal stochastic generation and transmission expansion planning model that incorporates the output from high-resolution global climate models through load and generation availability scenarios. We implement our model in Pyomo and perform computational studies on a realistically-sized test case of the California electric grid in a high performance computing environment. We propose model reformulations and algorithm tuning to efficiently solve this large problem using a variant of the Progressive Hedging Algorithm. We utilize the parallelization capabilities and overall versatility of mpi-sppy, exploiting its hub-and-spoke architecture to concurrently obtain inner and outer bounds on an optimal expansion plan. Initial results show that instances with 360 representative days on a system with over 8,000 buses can be solved to within 5% of optimality in under 4 h of wall clock time, a first step towards solving a large-scale power system expansion planning problem across a wide range of climate-informed operational scenarios.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

An integrated coupling model for solving multiscale fluid-fluid coupling problems in SAM code

In this study, an integrated coupling method has been developed for solving multiscale fluid-fluid coupling problems in plant-scale safety analysis models in SAM (System Analysis Module) code. In this method, a higher-fidelity multi-dimensional (3D) flow module is used for reactor components of complex flow features (e.g., reactor core) and a lumped parameter one-dimensional (1D) flow module for plant-scale flow loops (e.g., primary loop pipe network), respectively. In this method, the 3D fluid equation/domain and 1D fluid equation/domain are tightly coupled at the residual level and solved simultaneously using the Newton’s method to overcome the convergence issues typically seen in existing approaches like separate domain approach, where the 3D fluid equation and 1D fluid equation are solved separately. Extensive and successful code verifications and demonstrations have been performed for this newly developed method. This new modeling approach significantly simplify the work flow in developing high-fidelity plant-scale safety analysis model, e.g. for pool-type reactors and pebble-bed reactors.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2025

Optical neural engine for solving scientific partial differential equations

Abstract Solving partial differential equations (PDEs) is the cornerstone of scientific research and development. Data-driven machine learning (ML) approaches are emerging to accelerate time-consuming and computation-intensive numerical simulations of PDEs. Although optical systems offer high-throughput and energy-efficient ML hardware, their demonstration for solving PDEs is limited. Here, we present an optical neural engine (ONE) architecture combining diffractive optical neural networks for Fourier space processing and optical crossbar structures for real space processing to solve time-dependent and time-independent PDEs in diverse disciplines, including Darcy flow equation, the magnetostatic Poisson’s equation in demagnetization, the Navier-Stokes equation in incompressible fluid, Maxwell’s equations in nanophotonic metasurfaces, and coupled PDEs in a multiphysics system. We numerically and experimentally demonstrate the capability of the ONE architecture, which not only leverages the advantages of high-performance dual-space processing for outperforming traditional PDE solvers and being comparable with state-of-the-art ML models but also can be implemented using optical computing hardware with unique features of low-energy and highly parallel constant-time processing irrespective of model scales and real-time reconfigurability for tackling multiple tasks with the same architecture. The demonstrated architecture offers a versatile and powerful platform for large-scale scientific and engineering computations.

Tang, Yingheng (ORCID:0009000153622546)↗
DOE OSTI2023

Solving independent set problems with photonic quantum circuits

An independent set (IS) is a set of vertices in a graph such that no edge connects any two vertices. In adiabatic quantum computation [E. Farhi, et al ., Science 292, 472–475 (2001); A. Das, B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061–1081 (2008)], a given graph G ( V , E ) can be naturally mapped onto a many-body Hamiltonian H IS G ( V , E ) , with edges E being the two-body interactions between adjacent vertices V . Thus, solving the IS problem is equivalent to finding all the computational basis ground states of H IS G ( V , E ) . Very recently, non-Abelian adiabatic mixing (NAAM) has been proposed to address this task, exploiting an emergent non-Abelian gauge symmetry of H IS G ( V , E ) [B. Wu, H. Yu, F. Wilczek, Phys. Rev. A 101, 012318 (2020)]. Here, we solve a representative IS problem G ( 8 , 7 ) by simulating the NAAM digitally using a linear optical quantum network, consisting of three C-Phase gates, four deterministic two-qubit gate arrays (DGA), and ten single rotation gates. The maximum IS has been successfully identified with sufficient Trotterization steps and a carefully chosen evolution path. Remarkably, we find IS with a total probability of 0.875(16), among which the nontrivial ones have a considerable weight of about 31.4%. Our experiment demonstrates the potential advantage of NAAM for solving IS-equivalent problems.

Science & Technology - Other Topics↗
DOE OSTI2022

HyKKT: a hybrid direct-iterative method for solving KKT linear systems

Here, we propose a solution strategy for the large indefinite linear systems arising in interior methods for nonlinear optimization. The method is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for sparse indefinite systems is the LBLT factorization where L is a lower triangular matrix and B is 1×1 or 2×2 block diagonal. However, this requires pivoting, which substantially increases communication cost and degrades performance on GPUs. Our approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solver on the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach on large optimal power flow problems and show that it can efficiently utilize GPUs and outperform LBL T factorization of the full system.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Overview of Large Helical Device experiments of basic plasma physics for solving crucial issues in reaching burning plasma conditions

Recently, experiments on basic plasma physics issues for solving future problems in fusion energy have been performed on a Large Helical Device. There are several problems to be solved in future devices for fusion energy. Emerging issues in burning plasma are: alpha-channeling (ion heating by alpha particles), turbulence and transport in electron dominant heating helium ash exhaust, reduction of the divertor heat load. To solve these problems, understanding the basic plasma physics of (1) wave–particle interaction through (inverse) Landau damping, (2) characteristics of electron-scale (high-k) turbulence, (3) ion mixing and the isotope effect, and (4) turbulence spreading and detachment, is necessary. This overview discusses the experimental studies on these issues and turbulent transport in multi-ion plasma and other issues in the appendix.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2025

Solving key challenges in collider physics with foundation models

Foundation models are neural networks that are capable of simultaneously solving many problems. Large language foundation models like ChatGPT have revolutionized many aspects of daily life, but their impact for science is not yet clear. In this paper, we use a new foundation model for hadronic jets to solve three key challenges in collider physics. In particular, we show how experiments can (1) save significant computing power when developing reconstruction algorithms, (2) perform a complete uncertainty quantification for high-dimensional measurements, and (3) search for new physics with model agnostic methods using low-level inputs. In each case, there are significant computational or methodological challenges with current methods that limit the science potential of deep learning algorithms. By solving each problem, we take jet foundation models beyond proof-of-principle studies and into the toolkit of practitioners.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

Quantum-inspired method for solving the Vlasov-Poisson equations

Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equation are often infeasible due to high-resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equations with exponential speed-up, provided that the solution can be compressed and efficiently represented as a MPS within some tolerable error threshold. Here, in this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equations for systems with one coordinate in space and one coordinate in velocity, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher-dimensional problems.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks

One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs), especially with arbitrary initial data. We address this problem by taking advantage of recent advances in scientific machine learning and the spectral dynamically orthogonal (DO) and borthogonal (BO) methods for representing stochastic processes. The recently introduced DO/BO methods reduce the SPDE to solving a system of deterministic PDEs and a system of stochastic ordinary differential equations. Specifically, we propose two new physics-informed neural networks (PINNs) for solving time-dependent SPDEs, namely the neural network (NN)-DO/BO methods. The proposed methods incorporate the DO/BO constraints into the loss function (along with the modal decomposition of the SPDE) with an implicit form instead of generating explicit expressions for the temporal derivatives of the DO/BO modes. Hence, the NN-DO/BO methods can overcome some of the drawbacks of the original DO/BO methods. For example, we do not need the assumption that the covariance matrix of the random coefficients is invertible as in the original DO method, and we can remove the assumption of no eigenvalue crossing as in the original BO method. Moreover, the NN-DO/BO methods can be used to solve time-dependent stochastic inverse problems with the same formulation and same computational complexity as for forward problems. Furthermore, we demonstrate the capability of the proposed methods via several numerical examples, namely: (1) A linear stochastic advection equation with deterministic initial condition: we obtain good results with the proposed methods, while the original DO/BO methods cannot be applied directly in this case. (2) Long-time integration of the stochastic Burgers' equation: we show the good performance of NN-DO/BO methods, especially the effectiveness of the NN-BO approach for such problems with many eigenvalue crossings during the whole time evolution, while the original BO method fails. (3) Nonlinear reaction diffusion equation: we consider both the forward problem and the inverse problems, including very noisy initial point values, to investigate the flexibility of the NN-DO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NN-DO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems, but future work should address the efficiency issue of PINNs for forward problems.

97 MATHEMATICS AND COMPUTING↗