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DOE OSTI2026

Quantum-Inspired Bayesian Sampling for Uncertainty Quantification and Machine Learning (Final Technical Report)

With increasing simulation and measurement data, machine learning and artificial intelligence have been widely used in computational decision-making of complex engineering systems. The resulting tools, such as uncertainty quantification solvers, reinforcement learning, and physics-informed machine learning, have achieved great success in critical DOE tasks such as material discovery and design, energy system modeling and control, and numerical weather and climate prediction. A core topic in scientific machine learning and artificial intelligence is Bayesian inference: given an observed data set, people want to estimate the posterior distribution of a (possibly large) number of hidden parameters. Due to the flexibility and weak assumptions, Bayesian sampling has been the mainstream Bayesian inference solvers despite the rapid progress of approximate Bayesian inference. Classical Bayesian sampling methods such as Markov-chain Monte Carlo suffer from a low-acceptance rate due to the random walk nature, therefore state-of-the-art techniques use Hamiltonian Monte Carlo and its variants to efficiently draw posterior samples in a high dimension. The key idea of Hamiltonian Monte Carlo and its variants is to simulate the Hamiltonian dynamics of a classical particle with a fixed mass, and their performance significantly degrades when the posterior distribution is highly spiky or has multiple modes. Leveraging the idea of quantum physics, this project has investigated new theory, algorithms and applications of Bayesian inference (especially Bayesian sampling). The main results include: (1) novel quantum-inspired Bayesian sampling methods that can lead to better accuracy for challenging multi-modal or spiky distributions, (2) more scalable machine learning framework leveraging tensor-compressed Bayesian inference, and (3) Bayesian and sampling approaches for verifying the robustness of continuous and binary neural networks.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Time-of-Flight Estimation for Nondestructive Evaluation

Ultrasonic testing (UT) for nondestructive evaluation (NDE) is a critical entity necessary to resolve both the quality and precision questions of complex parts evolving from the classical approaches to the innovative additive manufacturing (AM) process. This modality provides the essential quantitative information for acceptance and potential flaw detectionof apart under investigation. A primary ingredient in UT besides the required precision robotic hardware for theacquisition of high-qualitymeasurement data is the underlying signal processing. It is here that much of the system performance capability resides. In this report,we discuss the basic steps in UT signal processing along with current capabilities that must be achieved in order to satisfy the criticaldemands of the various LLNL programs.In itsbasic foundations, signal processing is essentially the “extraction of critical information (signals) from noisy, uncertain data.” Clearly, in a perfect world the best signal processing is none---just make a reliable, noise-free, uncertainty-free, measurement. Unfortunately, even the best of systems is still straddled within the confines and limitations of physical instrumentation. With this in mind, we discuss the development of signal processing techniques toextract the desired UT information from noisy measurement data. We start with a discussion of a simple homogeneous representation of a “part” under investigation and its insonification by an ultrasonicexcitation from a physics-based perspective. Once developed, we briefly discussthe various choices of excitation signals and their tradeoffs.Next,we discuss a simulation approach employing simple models from the signal processing perspective and then move on to the development of the basic signal processing approach to ultrasonic signal processing all based on estimating the pulse arrival estimation. Starting with simple peak detection techniques, progressing to the processing workhorse so-called “matched-filter” and finally to the more sophisticated “model-based matched filter.”The underlying pre-and post-processing of noisy measurement data along with the application of these techniques are subsequently demonstrated on experimental ultrasonic data.Finally, we present some results for processing “weld” data when the ultrasonic excitation signal is not directly available.

42 ENGINEERING↗

DOE OSTI2024

Investigations on the multiple-sector hard-copper X-band accelerating structures

The development of advanced, high gradient accelerating structures is one of the leading activity of the particle accelerator community. In the technological research of new construction methods for these devices, high-power testing is a critical step for the verification of their viability. Recent experiments showed that accelerating cavities made out of hard copper, fabricated without high-temperature processes, can achieve better performance as compared with soft copper ones. Recently, we have built cavities using Tungsten Inert Gas welding and the high-power experiments confirmed that this joining process is a robust and low-cost alternative to brazing or diffusion bonding. This is a good solution for high-gradient operation, with a gradient of about 150 MV/m in X-band, at a breakdown rate of 10 -3 /pulse/meter using a shaped RF pulse with a 150 ns flat part. We continue the design, construction and high power tests of three-cells standing-wave X-band accelerating cavities fabricated with a split-open geometry, made of hard copper and vacuum-sealed by welding. Our aim is the fabrication of accelerating structures made out of hard copper alloys by using innovative cost-effective technologies. Moreover, our method of multiple-sector structures opens the way to new technological approaches and design methods, for example providing the possibility of placing dampers of the parasitic higher-order modes. Such structures with dampers allow for acceleration of multi-bunch beams, both in standing and in travelling-wave accelerating modality. Here, this paper describes the design of two cavities made of multiple sectors that were fabricated at the Italian Institute for Nuclear Physics in Frascati, Italy (INFN-LNF), assembled by means of clamping and then joined with the Tungsten Inert Gas welding in order to preserve the hardness of the metal. These are three-cell standing-wave X-band accelerating cavities, to be operated in the π-mode. We report the low-power RF tests of the first one, made of two halves, and the RF design of the second one, made of four quadrants. Both structures were built for high-gradient tests at SLAC National Accelerator Laboratory.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗

DOE OSTI2022

Novel reusable animal model for comparative evaluation of in vivo growth and protein-expression of Escherichia coli O157 strains in the bovine rumen

Previously, we had demonstrated that Escherichia coli O157:H7 (O157) strain 86–24 expresses proteins involved in survival rather than virulence in vitro in rumen fluid from dairy cattle limit fed a maintenance diet. Here, we verified if this observation would be true for different O157 strains grown in vitro in rumen fluid from, and in vivo in the rumen of, animals on contrasting maintenance (high fiber) and lactation (high energy-protein) diets usually limit fed to dairy cattle. For the in vivo studies, an economical, novel, reusable and non-terminal rumen-fistulated animal model permitting simultaneous evaluation of multiple bacterial strains in the bovine rumen was developed. All experiments were conducted in duplicate using different animals to account for host-related variations. The O157 strains included, 86–24, EDL933 and the super shed SS-17. E. coli Nal R (#5735), derived from a bovine intestinal commensal E. coli, was included as a control. As expected, diet influenced ruminal pH and volatile fatty acid (VFA) composition. The pH ranged from 6.2–7.0 and total VFA concentrations from 109–141 μM/ml, in animals fed the maintenance diet. In comparison, animals fed the lactation diet had a ruminal pH ranging between 5.18–6.0, and total VFA of 125–219 μM/ml. Strain dependent differences in O157 recovery from the rumen fluid of cattle fed either diet was observed, both in vitro and in vivo, with O157 strains 86–24 and EDL933 demonstrating similar survival patterns. Analysis of the O157 proteomes expressed in the rumen fluid/rumen verified previous observations of adaptive responses. Any difference in the adaptive response was mainly influenced by the animal’s diet and growth conditions (in vitro and in vivo) and not the O157 strain. These new insights into the O157 responses could help formulate modalities to control O157 across strains in cattle at all stages of husbandry.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2020

Modal analysis of the behavior of inertial particles in turbulence subjected to Stokes drag

The clustering of small heavy inertial particles subjected to Stokes drag in turbulence, which is the concern of present study, is known to be minimal at small and large Stokes number and substantial at St = $\textit{O}$(1). This nonmonotonic trend, which has been shown computationally and experimentally, is yet to be explained analytically. In this study, we obtain an analytical expression for the Lyapunov exponents that quantitatively predicts this trend. The sum of the exponents, which is the normalized rate of change of the signed volume of a small cloud of particles, is correctly predicted to be negative and positive at small and large Stokes numbers, respectively, asymptoting to $\textit{τQ}$ as $\textit{τ}$ → 0 and $τ^{–1/2}|Q|^{1/4}$ as $\textit{τ} → ∞$, where $\textit{τ}$ is the particle relaxation time and $\textit{Q(τ)}$ is the difference between the norm of the rotation- and strain-rate tensors computed along the particle trajectory, which remains negative at all $\textit{τ}$ in turbulent flows. Additionally, the trajectory crossing is predicted only in hyperbolic flows where Q < 0 for sufficiently inertial particles with a $\textit{τ}$ that scales with $|Q|^{–1/2}$. Following the onset of crossovers, a transition from clustering to dispersion is predicted correctly. We show these behaviors are not unique to three-dimensional isotropic turbulence and can be reproduced closely by a one-dimensional mono-harmonic flow, which appears as a fundamental canonical problem in the study of particle clustering. Analysis of this one-dimensional canonical flow shows that the rate of clustering, quantified as the product of the Lyapunov exponent and particle relaxation time, is bounded by –1/2, behaving with extreme nonlinearity in the hyperbolic flows and always remaining positive in the elliptic flows. Finally, this model problem also confirms that dispersion in a hyperbolic flow will occur only if particles' trajectory cross. These findings, which stem from our analysis, are corroborated by the direct numerical simulations.

42 ENGINEERING↗

DOE OSTI2021

Parsed synthesis of pyocyanin via co-culture enables context-dependent intercellular redox communication

Abstract Background Microbial co-cultures and consortia are of interest in cell-based molecular production and even as “smart” therapeutics in that one can take advantage of division of labor and specialization to expand both the range of available functions and mechanisms for control. The development of tools that enable coordination and modulation of consortia will be crucial for future application of multi-population cultures. In particular, these systems would benefit from an expanded toolset that enables orthogonal inter-strain communication. Results We created a co-culture for the synthesis of a redox-active phenazine signaling molecule, pyocyanin (PYO), by dividing its synthesis into the generation of its intermediate, phenazine carboxylic acid (PCA) from the first strain, followed by consumption of PCA and generation of PYO in a second strain. Interestingly, both PCA and PYO can be used to actuate gene expression in cells engineered with the soxRS oxidative stress regulon, although importantly this signaling activity was found to depend on growth media. That is, like other signaling motifs in bacterial systems, the signaling activity is context dependent. We then used this co-culture’s phenazine signals in a tri-culture to modulate gene expression and production of three model products: quorum sensing molecule autoinducer-1 and two fluorescent marker proteins, eGFP and DsRed. We also showed how these redox-based signals could be intermingled with other quorum-sensing (QS) signals which are more commonly used in synthetic biology, to control complex behaviors. To provide control over product synthesis in the tri-cultures, we also showed how a QS-induced growth control module could guide metabolic flux in one population and at the same time guide overall tri-culture function. Specifically, we showed that phenazine signal recognition, enabled through the oxidative stress response regulon soxRS, was dependent on media composition such that signal propagation within our parsed synthetic system could guide different desired outcomes based on the prevailing environment. In doing so, we expanded the range of signaling molecules available for coordination and the modes by which they can be utilized to influence overall function of a multi-population culture. Conclusions Our results show that redox-based signaling can be intermingled with other quorum sensing signaling in ways that enable user-defined control of microbial consortia yielding various outcomes defined by culture medium. Further, we demonstrated the utility of our previously designed growth control module in influencing signal propagation and metabolic activity is unimpeded by orthogonal redox-based signaling. By exploring novel multi-modal strategies for guiding communication and consortia outcome, the concepts introduced here may prove to be useful for coordination of multiple populations within complex microbial systems.

Chun, Kayla↗

DOE OSTI2024

Machine-Learning-Based Mapping and Modeling of Solar Energy with Ultra-High Spatiotemporal Granularity

Despite the rapid growth of solar energy, we still lack a dynamic, high-fidelity database that tracks the spatiotemporal variations of solar PVs and their associated infrastructures across different places at a spatially resolved scale. The absence of such data presents a barrier to various applications such as solar PV growth projection, solar energy integration, solar incentive design, and climate risk assessment. In this project, we aim to bridge this gap by developing AI-based algorithms to extract granular information about solar PV installations and their associated infrastructures (i.e., distribution grids) from widely available unstructured data like remote sensing images and street views. As a result, we have built the Solar Energy Atlas, a fine-grained, large-scale geospatial overlay of distributed solar PVs and distribution grids. On top of it, we have advanced the understanding of solar adoption and distribution grid vulnerability to climate-induced extremes. Our major contributions can be summarized as follow: (1) By developing new AI algorithms, we have built the most comprehensive solar PV spatiotemporal database covering the entire US. This is the first time we obtained the exact GPS locations, size, subtype, and installation year information for rooftop solar PVs across the US. This database can be used for solar PV growth projection, solar energy integration, solar energy policy analysis and design, and spatially-resolved climate risk assessment. (2) Leveraging this database, we have uncovered the socioeconomic driving factors that are correlated with earlier onset of solar adoption and higher saturated adoption levels. We have identified the heterogeneity in the effects of different types of financial incentives on solar adoption and provided implications for tailoring incentive design based on local income levels to promote equitable solar adoption. (3) We have developed a distribution grid GIS mapping algorithm which can obtain granular geospatial and topology information about distribution grids using multi-modal open data, reducing the dependency on hard-to-obtain smart meter data of conventional approaches. It shows effectiveness in both the U.S. and Sub-Saharan Africa. Using this algorithm, we have uncovered the non-uniform vulnerability of distribution grids to wildfires in California in the aspects of undergrounding protection and Distributed Energy Resources (DER) preparedness. This has provided important implications for improving the affordability and equity of grid adaptation approaches. (3) We have made our produced database publicly available and provided user-friendly interface to enable various stakeholders and the general public to interact with the data. We have also integrated the produced data into the Data Commons platform to enable the public to access the data and correlate it with other location-specific characteristics simply using natural language as queries. The impact of our project is three-fold: (1) New algorithms for mapping solar PVs and distribution grids across space and time, which are open source to facilitate researchers and industry; (2) New databases of solar PVs and distribution grids that have been made publicly available for engineering, social, and policy applications; (3) New understandings and actionable insights on the potential approaches to promoting solar adoption and reducing energy infrastructure vulnerabilities. In this report, we start by discussing the project background and motivation (section 5), followed by the overview of project objectives (section 6). Results and discussion for each task are presented in section 7. Significant accomplishments are summarized in section 8. This report will be concluded by discussing the paths forwards (section 9), products (section 10), and team roles (section 11).

14 SOLAR ENERGY↗

DOE OSTI2020

Global System Reduction Order Modeling for Localized Feature Inclusion

The development of reduced-order models remains an active research area, despite advances in computational resources. Here we develop a novel order-reduction approach that is designed to incorporate isolated regions that contain, for example, nonlinearitites or accumulating damage. The approach is designed to use global modes of the overall system response, which are then naturally coupled to the response in the isolated region of interest. Two examples are provided to demonstrate both the accuracy and the computational efficiency of the proposed approach. The performance of this approach is compared to the exact response corresponding to a finite element simulation for the chosen problems. In addition, the accuracy and computational efficiency are shown relative to a standard Galerkin reduction based on the linear normal modes. It is found that the proposed reduction offer computational efficiency comparable to a Galerkin reduction, but more accurately represents the response of the system when both are compared to the finite element simulation.

42 ENGINEERING↗

DOE OSTI2024

Optimal Stopping Ages for Colorectal Cancer Screening

Importance Prior studies have shown that the benefits, harms, and costs of colorectal cancer (CRC) screening at older ages are associated with a patient’s sex, health, and screening history. However, these studies were hypothetical exercises and not directly informed by data on CRC risk. Objective To identify the optimal stopping ages for CRC screening by sex, comorbidity, and screening history from a cost-effectiveness perspective. Design, Setting, and Participants This economic evaluation first validated the MISCAN-Colon (Microsimulation Screening Analysis–Colon) model against community-based CRC incidence and mortality rates for 2 subcohorts of the PRECISE (Optimizing Colorectal Cancer Screening Precision and Outcomes in Community-Based Populations) cohort. Subsequently, different CRC screening scenarios were simulated in older individuals. Cohorts of US adults aged 76 to 90 years varied by sex and comorbidity status (none, low, moderate, or severe). Statistical and sensitivity analyses were performed from March 2023 to May 2024. Exposures CRC screening histories including fecal immunochemical test (FIT) or colonoscopy, such as a negative colonoscopy result from 10, 15, 20, 25, or 30 years before the index age; 1 to 5 negative FIT results within 5 years of the index age, with different patterns of recency; or a combination of negative colonoscopy and negative FIT results. Main Outcomes and Measures The main outcomes included estimated lifetime clinical outcomes, incremental costs, and quality-adjusted life-years gained (QALYG) associated with 1 additional FIT or colonoscopy. Optimal stopping age for screening, defined as the oldest age for which the incremental cost-effectiveness ratio was still below the willingness-to-pay threshold of $\$$100 000 per QALYG, was evaluated. Results The first of the 2 PRECISE subcohorts used in validating the simulation model included 25 974 adults (15 060 females [58.0%]; 54.7% aged 76 to 80 years) with a negative colonoscopy result 10 years before the index date. The second subcohort consisted of 118 269 adults (67 058 females [56.7%]; 90.5% aged 76 to 80 years) with a negative FIT result 1 year before the index date. Older age, male sex, higher comorbidity levels, and recent CRC screenings were associated with reduced incremental benefit and cost-effectiveness of additional screening. For the reference cohort of 76-year-old females without comorbidities and a negative colonoscopy result 10 years before the index age, 1 additional colonoscopy cost $\$$38 226 per QALYG. For cohorts with otherwise equivalent characteristics, associated costs increased to $\$$1 689 945 per QALYG for females at age 90 years without comorbidities and a negative colonoscopy results 10 years before the index age, $\$$51 604 per QALYG for males at age 76 years without comorbidities and a negative colonoscopy result 10 years before the index age, and $\$$108 480 per QALYG for females at age 76 years with severe comorbidities and a negative colonoscopy result 10 years before the index age and decreased to $\$$16 870 per QALYG for females without comorbidities and a negative colonoscopy result 30 years before the index age. The optimal stopping ages across different cohorts ranged from younger than 76 to 86 years for colonoscopy and younger than 76 to 88 years for FIT. Conclusions and Relevance In this economic evaluation, age, sex, screening history, comorbidity, and future screening modality were associated with the clinical outcomes, cost-effectiveness, and optimal stopping age for CRC screening. These results can inform guideline development and patient-directed informed decision-making.

Harlass, Matthias [Erasmus Erasmus University Medi↗

DOE OSTI2021

Evaluating MPI resource usage summary statistics

The Message Passing Interface (MPI) remains the dominant programming model for scientific applications running on today’s high-performance computing (HPC) systems. This dominance stems from MPI’s powerful semantics for inter-process communication that has enabled scientists to write applications for simulating important physical phenomena. MPI does not, however, specify how messages and synchronization should be carried out. Those details are typically dependent on low-level architecture details and the message characteristics of the application. Therefore, analyzing an application’s MPI resource usage is critical to tuning MPI’s performance on a particular platform. The result of this analysis is typically a discussion of the mean message sizes, queue search lengths and message arrival times for a workload or set of workloads. While a discussion of the arithmetic mean in MPI resource usage might be the most intuitive summary statistic, it is not always the most accurate in terms of representing the underlying data. In this paper, we analyze MPI resource usage for a number of key MPI workloads using an existing MPI trace collector and discrete-event simulator. Our analysis demonstrates that the average, while easy and efficient to calculate, is a useful metric for characterizing latency and bandwidth measurements, but may not be a good representation of application message sizes, match list search depths, or MPI inter-operation times. Additionally, we show that the median and mode are superior choices in many cases. We also observe that the arithmetic mean is not the best representation of central tendency for data that are drawn from distributions that are multi-modal or have heavy tails. Furthermore, the results and analysis of our work provide valuable guidance on how we, as a community, should discuss and analyze MPI resource usage data for scientific applications.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2026

New Norms or Old Habits: Evaluating Interlinked Trajectories of Online Shopping and Work Commute Post-Pandemic

The COVID-19 pandemic has significantly shifted travel behaviors, with major changes observed in online shopping and travel to work. Despite considerable research into pandemic-induced changes in travel behavior, it remains uncertain whether these new patterns have persisted or reverted to pre-pandemic norms. This study addresses this uncertainty by evaluating whether shifts in online shopping and work travel during the pandemic have become permanently ingrained in individuals' daily routine. Leveraging data from the 2022 National Household Travel Survey, a bivariate ordered probit model is employed to analyze changes in online shopping and work travel - whether they have increased, decreased, or remained stable compared to pre-pandemic levels across different population segments. The analysis finds that the pandemic did not significantly alter online shopping for home delivery and travel to work for the majority of society. However, a substantial portion of respondents reported increased online shopping for home delivery and reduced travel to work compared to pre-pandemic levels, with online shopping trends appearing more permanent. Segment-wise analysis and model results indicate heterogeneity in behavioral shifts with females engaging more in online shopping, while zero-vehicle households are traveling less to work, compared to pre-pandemic levels. Additionally, increase in online shopping frequency is significantly and negatively correlated with decrease in traveling to work. These findings highlight the need for improved digital infrastructure, flexible work policies, and integrated transportation solutions tailored to evolving demographic and socioeconomic needs in the post-pandemic era. Additionally, the study calls for integrating passenger and freight movement in a single framework rather than treating them in silos.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

DOE OSTI2021

Burst-mode dual-comb spectroscopy

We introduce a new modality of dual-comb spectroscopy that enables a simplified and powerful new approach for time-resolved measurements with increased acquisition rates. This "burst mode" form of dual-comb spectroscopy (BM-DCS) relies on the multiplexing of each probe pulse into a short train of pulses. With this approach we demonstrate a time-resolved series of absorption-based spectroscopic measurements of a laser-induced plasma using only a single laser ablation shot and identify 22 Nd lines not previously reported in the literature. The transmission spectra spanned 3.1 THz and were acquired at an effective acquisition rate of 25 kHz with 40 ms time-resolution. This simple modification to 100 MHz level dual-comb systems provides a flexible approach for studying transient and low duty-cycle events such as laser-induced plasmas, combustion, and explosive reactions.

(280.5395) Plasma Diagnostics, broadband laser spe↗

DOE OSTI2021

Striving to translate shale physics across ten orders of magnitude: What have we learned?

Shales will play an important role in the successful transition of energy from fossil-based resources to renewables in the coming decades. Aside from being a significant source of low-carbon intensity fuels, like natural gas, they also serve as geologic seals of subsurface formations that may be used to isolate nuclear waste, sequester CO 2 , or store intermittent energy (e.g., solar hydrogen). Despite their importance, shales pose significant engineering and environmental challenges due to their nanoporous structure and extreme heterogeneity that spans at least ~10 orders of magnitude in spatial scale. Two challenges inhibit a system-level understanding: (1) the physics of fluid flow and phase behavior in shales are poorly understood due to the dominant molecular interactions between minerals and fluids under confinement, and (2) the apparent lack of scale separation that prevents a reliable (closed) description of the physics at any single scale of observation. In this review, we focus on the latter issue and discuss scale translation, which in its broadest sense is transforming data or simulations from one spatiotemporal scale to another. While effective scale translation is not exclusive to shales, but all geologic porous media, the need for it is especially acute in shales given their high degree of heterogeneity. Classical theories like homogenization, while indispensable, fail when scales are not separated. Other methods, like numerical upscaling, scale-translate in only one direction: small to large, but not the reverse, called downscaling. However, the confluence of advances in three areas are bringing challenging problems such as shales within reach: increased computational power and scalable algorithms; high-resolution imaging and multi-modal data acquisition; and machine learning to process massive amounts of data. While these advances equip geoscientists with a wide array of experimental and computational tools, no individual tool can probe the entire gamut of heterogeneity in shales. Their effective use, therefore, requires an ability to bridge between various data types obtained at different scales. The aim of this review is to present a coherent account of computational and experimental methods that may be used to achieve just that, i.e., to perform scale translation. We provide a broader definition of scale translation, one that transcends classical homogenization and upscaling methods, but is consistent with them and accommodates notions like downscaling and data translation. After a brief introduction to homogenization, we review hybrid methods, numerical upscaling and its recent extensions, multiscale computing, high-resolution imaging, and machine learning. We place particular emphasis on multiscale computing and propose an algorithmic framework to bridge between the pore (micro) and Darcy (macro) scales. Throughout the paper, we draw comparisons between the various methods and highlight their (often hidden) similarities, differences, benefits, and pitfalls. We finally conclude with two case studies on shales that exemplify some of the methods presented.

58 GEOSCIENCES↗

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↗