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DOE OSTI2026

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

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

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

Order conditions for nonlinearly partitioned Runge-Kutta methods

Recently, a new class of nonlinearly partitioned Runge–Kutta (NPRK) methods was proposed for nonlinearly partitioned systems of autonomous ordinary differential equations y' = F(y, y). The target class of problems are those in which different scales, stiffnesses, or physics are coupled in a nonlinear way, wherein the desired partition cannot be written in a classical additive or component-wise fashion. Here we use a rooted-tree analysis to derive full-order conditions for NPRKM methods, where M denotes the number of nonlinear partitions. Due to the nonlinear coupling and thereby the mixed product differentials, it turns out that the standard node-colored rooted tree analysis used in analyzing ODE integrators does not naturally apply. Instead we develop a new edge-colored rooted-tree framework to address the nonlinear coupling. The resulting order conditions are enumerated, are provided directly for up to fourth order with M = 2 and third order with M = 3, and are related to existing order conditions of additive and partitioned RK methods. We conclude with an example that shows how the nonlinear order conditions can be used to obtain an embedded estimate of the state-dependent nonlinear coupling strength in a dynamical system.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Effect of a Fine-Scale Layered Structure of the Atmosphere on Infrasound Signals from Fragmenting Meteoroids

We investigate the influence of a fine-scale (FS) layered structure in the atmosphere on the propagation of infrasound signals generated by fragmenting meteoroids. Using a pseudo-differential parabolic equation (PPE) approach, we model broadband acoustic signals from point sources at altitudes of 35–100 km. The presence of FS fluctuations in the stratosphere (37–45 km) and the lower thermosphere (100–120 km) modifies ray trajectories, causing multiple arrivals and prolonged signal durations at ground stations. In particular, meteoroids fragmenting at 80–100 km can produce two distinct thermospheric arrivals beyond 150 km range, while meteoroids descending to 50 km or below yield weak, long-lived arrivals within the acoustic shadow zone via antiguiding propagation and diffraction. Comparison with observed infrasound data confirms that FS-layered inhomogeneities can account for multi-arrival “N-waves,” broadening potential interpretations of meteoroid signals. The results also apply to other atmospheric-entry objects, such as sample return capsules, emphasizing how FS structure impacts shock wave propagation. In conclusion, our findings advance understanding of wavefield evolution in a layered atmosphere and have broad relevance for global infrasound monitoring of diverse phenomena (e.g., re-entry capsules, rocket launches, and large-scale explosions).

Aeroacoustics↗
DOE OSTI2026

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

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

Allen-Cahn equations↗
DOE OSTI2026

Multiscale Modeling and Experimental Insights into High-Temperature Soil Biodegradation Dynamics of Semi-Crystalline Poly(Lactic Acid) Nonwoven Fabrics

This study investigates the biodegradation of semi-crystalline poly(lactic acid) (PLA) nonwovens (NWs) in soil at 58 °C using both experimental and mathematical modeling approaches. The model utilizes a system of parabolic diffusion-reaction partial differential equations (PDEs) to elucidate chemical transformations over time and in space. It accounts for phenomena such as the diffusion of water and lactic acid monomers through the polymer matrix and into the surrounding soil, along with their microbial breakdown. It also accounts for the initial PLA crystallinity and predicts its evolution in time. The model is solved numerically for a single filament, and the results were used to shed light on PLA NW transformations observed in soil over a 180-day incubation period. Various characterization techniques, including scanning electron microscopy (SEM), differential scanning calorimetry (DSC), and Raman spectroscopy, were employed to assess morphological changes, crystallinity, and molecular changes in the NWs throughout the experiment. By comparing the experimental data with the model predictions, the hydrolysis rate coefficient was found to be 3.37 × 10 -7 s -1 , while the rate of microbial degradation of lactic acid monomers was faster, of the order of 9.63 × 10 -7 s -1 . The findings highlight the significant role of crystallinity in the biodegradation process. The PLA degradation ceases when no amorphous material remains, and the crystallinity reaches 0.8, as observed in the experiments by day 120. Furthermore, this research contributes to a deeper understanding of PLA biodegradation dynamics and offers insights for effectively managing biodegradable materials in environmental settings.

Diffusion−reaction modeling↗
DOE OSTI2025

PETSc/TAO developments for GPU-based early exascale systems

The Portable Extensible Toolkit for Scientific Computation (PETSc) library provides scalable solvers for nonlinear time-dependent differential and algebraic equations and for numerical optimization via the Toolkit for Advanced Optimization (TAO). PETSc is used in dozens of scientific fields and is an important building block for many simulation codes. During the U.S. Department of Energy’s Exascale Computing Project, the PETSc team has made substantial efforts to enable efficient utilization of the massive fine-grain parallelism present within exascale compute nodes and to enable performance portability across exascale architectures. We recap some of the challenges that designers of numerical libraries face in such an endeavor, and then discuss the many developments we have made, which include the addition of new GPU backends, features supporting efficient on-device matrix assembly, better support for asynchronicity and GPU kernel concurrency, and new communication infrastructure. In conclusion, we evaluate the performance of these developments on some pre-exascale systems as well as the early exascale systems Frontier and Aurora, using compute kernel, communication layer, solver, and mini-application benchmark studies, and then close with a few observations drawn from our experiences on the tension between portable performance and other goals of numerical libraries.

Exascale Computing Project (ECP)↗
DOE OSTI2025

High‐Resolution National‐Scale Water Modeling Is Enhanced by Multiscale Differentiable Physics‐Informed Machine Learning

Abstract The National Water Model (NWM) is a key tool for flood forecasting, planning, and water management. Key challenges facing the NWM include calibration and parameter regionalization when confronted with big data. We present two novel versions of high‐resolution (∼37 km 2 ) differentiable models (a type of hybrid model): one with implicit, unit‐hydrograph‐style routing and another with explicit Muskingum‐Cunge routing in the river network. The former predicts streamflow at basin outlets whereas the latter presents a discretized product that seamlessly covers rivers in the conterminous United States (CONUS). Both versions use neural networks to provide a multiscale parameterization and process‐based equations to provide a structural backbone, which were trained simultaneously (“end‐to‐end”) on 2,807 basins across the CONUS and evaluated on 4,997 basins. Both versions show great potential to elevate future NWM performance for extensively calibrated as well as ungauged sites: the median daily Nash‐Sutcliffe efficiency of all 4,997 basins is improved to around 0.68 from 0.48 of NWM3.0. As they resolve spatial heterogeneity, both versions greatly improved simulations in the western CONUS and also in the Prairie Pothole Region, a long‐standing modeling challenge. The Muskingum‐Cunge version further improved performance for basins >10,000 km 2 . Overall, our results show how neural‐network‐based parameterizations can improve NWM performance for providing operational flood predictions while maintaining interpretability and multivariate outputs. The modeling system supports the Basic Model Interface (BMI), which allows seamless integration with the next‐generation NWM. We also provide a CONUS‐scale hydrologic data set for further evaluation and use.

Song, Yalan [Civil and Environmental Engineering T↗
DOE OSTI2026

Parallel-in-Time Solution of Allen-Cahn Equations by Integrating Operator Learning into the Parareal Method

While recent advances in deep learning have shown promising efficiency gains in solving time-dependent partial differential equations (PDEs), matching the accuracy of conventional numerical solvers still remains a challenge. One strategy to improve the accuracy of deep learning-based solutions for time-dependent PDEs is to use the learned model as the coarse propagator in the Parareal method and a traditional numerical method as the fine solver. However, successful integration of deep learning into the Parareal method requires consistency between the coarse and fine solvers, particularly for PDEs exhibiting rapid changes such as sharp transitions. Here, to ensure this consistency, we propose using convolutional neural networks (CNNs) to learn the fully discrete time-stepping operator defined by the same numerical scheme employed as the fine solver. We demonstrate the effectiveness of the proposed method in solving the classical and mass-conservative Allen–Cahn (AC) equations. Through iterative updates in the Parareal algorithm, our approach achieves a significant computational speedup compared to traditional fine solvers while converging to high-accuracy solutions. Our results highlight that the proposed hybrid Parareal algorithm effectively accelerates simulations, particularly when implemented on multiple GPUs, and converges to the desired accuracy in only a few iterations. Another advantage of our method is that the CNN model is trained on trajectory-based data generated from random initial conditions, such that the trained model can be used to solve the AC equations with various initial conditions without retraining. This work demonstrates the potential of integrating neural network methods into parallel-in-time frameworks for efficient and accurate simulations of time-dependent PDEs.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

General field evaluation in high-order meshes on GPUs

Robust and scalable function evaluation at any arbitrary point in the finite/spectral element mesh is required for querying the partial differential equation solution at points of interest, comparison of solution between different meshes, and Lagrangian particle tracking. This is a challenging problem, particularly for high-order unstructured meshes partitioned in parallel with MPI, as it requires identifying the element that overlaps a given point and computing the corresponding reference space coordinates. Here, we present a robust and efficient technique for general field evaluation in large-scale high-order meshes with quadrilaterals and hexahedra. In the proposed method, a combination of globally partitioned and processor-local maps are used to first determine a list of candidate MPI ranks, and then locally candidate elements that could contain a given point. Next, element-wise bounding boxes further reduce the list of candidate elements. Finally, Newton’s method with trust region is used to determine the overlapping element and corresponding reference space coordinates. Since GPU-based architectures have become popular for accelerating computational analyses using meshes with tensor-product elements, specialized kernels have been developed to utilize the proposed methodology on GPUs. The method is also extended to enable general field evaluation on surface meshes. The paper concludes by demonstrating the use of the proposed method in various applications ranging from mesh-to-mesh transfer during r-adaptivity to Lagrangian particle tracking.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Karhunen–Loève deep learning method for surrogate modeling and approximate Bayesian parameter estimation

We evaluate the performance of the Karhunen-Loève Deep Neural Network (KL-DNN) framework for surrogate modeling and approximate Bayesian parameter estimation in partial differential equation models. In the surrogate model, the Karhunen-Loève (KL) expansions are used for the dimensionality reduction of the number of unknown parameters and variables, and a deep neural network is employed to relate the reduced space of parameters to that of the state variables. The KL-DNN surrogate model is used to formulate a maximum-a-posteriori-like least-squares problem, which is randomized to draw samples of the posterior distribution of the parameters. We test the proposed framework for a hypothetical unconfined aquifer via comparison with the forward MODFLOW and inverse PEST++ iterative ensemble smoother (IES) solutions as well as the state-of-the-art Fourier neural operator (FNO) and deep operator networks (DeepONets) operator learning surrogate models. Our results show that the KL-DNN surrogate model outperforms FNO and DeepONet for forward predictions. For solving inverse problems, the randomized algorithm provides the same or more accurate Bayesian predictions of the parameters than IES as evidenced by the higher log-predictive probability of both the estimated parameter field and the forecast hydraulic head. The posterior mean obtained from the randomized algorithm is closer to the reference parameter field than that obtained with FNO as the maximum a posteriori estimate.

Approximate Bayesian inference↗
NASA NTRS2014

Very Large Rain Drops from 2D Video Disdrometers and Concomitant Polarimetric Radar Observations

Drop size distribution (DSD) measurements using ground-based disdrometers (point measurements) have often been used to derive equations to relate radar observations to the integral rainfall parameters (Atlas et al. 1999, Bringi et al., 2003, Kozu et al., 2006, Tokay and Short, 1996, Ajayi and Owolabi, 1987, Battan, 1973). Disdrometers such as JWD, MRR and several others have a major limitation in measuring drops with equi-volume diameters (D(sub eq)) larger than 5 mm because they often rely on the velocity-diameter relationship which plateaus beyond this diameter range (Atlas et al., 1973, Gunn & Kinzer, 1949). Other disdrometers such as Parsivel also lack accuracy beyond this diameter range. The 2D video disdrometer (2DVD: Schönhuber et al., 2008) on the other hand gives drop-shape contours and velocities for each individual drop/hydrometeor falling through its sensor area; this provides a unique opportunity to study the role of very-large drops on radar measurements in particular those with polarimetric radar capability where DSDs with a significant component of very large drops may require special consideration given that the differential reflectivity and other polarimetric radar parameters including attenuation-correction methods will be sensitive to the concentrations of these large drops. A recent study on the occurrence of large drops by Gatlin et al. (2014) has compiled a large and diverse set of measurements made with the 2D video disdrometers from many locations around the globe. Some of the largest drops found in this study were 9 mm D(sub eq) and larger, and in this paper, we report on three such events, with maximum D(sub eq's) of 9.0, 9.1 and 9.7 mm, which occurred in Colorado, Northern Alabama, and Oklahoma, respectively. Detailed examination of the 2DVD data - in terms of shapes and fall velocities - has confirmed that these are fully-melted hydrometeors, although for the last case in Oklahoma, a bigger and non-fully-melted hydrometeor was also observed. All three events were also captured by polarimetric radars, namely the S-band CHILL radar operated by Colorado State University (Brunkow et al., 2000), the C-band ARMOR radar (Petersen et al., 2007) operated by University of Alabama in Huntsville, and NEXRADKVNX, operated by the US National Weather Service, respectively. For the last event, several other radar observations were also made, including two X-band radars operated by the US Dept. of Energy. Analyses of 2DVD data in conjunction with the corresponding radar observations are presented, along with some discussion on sampling issues related to the measurements of such large rain drops. The latter is addressed using maximum diameter D(sub max) measurements from 1-minute DSDs using two collocated 2DVDs for 37 events in Huntsville.

Thurai, Merhala↗
NASA NTRS2000

Influence of Humidity on the Aerosol Scattering Coefficient and Its Effect on the Upwelling Radiance During ACE-2

Aerosol scattering coefficients (sigma(sub sp)) have been measured over the ocean at different relative humidities (RH) as a function of attitude in the region surrounding the Canary Islands during the Second Aerosol Characterization Experiment (ACE-2) in June and July 1997. The data were collected by the University of Washington passive humidigraph (UWPH) mounted on the Pelican research aircraft. Concurrently, particle size distributions absorption coefficients and aerosol optical depth were measured throughout 17 flights. A parameterization of sigma(sub sp) as a function of RH was utilized to assess the impact of aerosol hydration on the upwelling radiance (normalized to the solar constant and cosine of zenith angle). The top of the atmosphere radiance signal was simulated at wavelengths corresponding to visible and near-infrared bands or the EOS-AM ("Terra") detectors, MODIS and MISR. The UWPH measured (sigma(sub sp)) at 2 RHs, one below and the other above ambient conditions. Ambient (sigma(sub sp)) was obtained by interpolation of these 2 measurements. The data were stratified in terms of 3 types of aerosols: Saharan dust, clean marine (marine boundary layer background) and polluted marine aerosols (i.e., 2- or 1-day old polluted aerosols advected from Europe). An empirical relation for the dependence of (sigma(sub sp)) on RH, defined by (sigma(sub sp))(RH) = k. ((1 - RH/100)(exp -gamma), was used with the hygroscopic exponent gamma derived from the data. The following gamma values were obtained for the 3 aerosol types: gamma(dust) = 0.23 +/- 0.05, gamma(clean marine) = 0.69 +/- 0.06 and gamma(polluted marine) = 0.57 + 0.06. Based on the measured (gamma)(s), the above equation was utilized to derive aerosol models with different hygroscopicities. The satellite simulation signal code 6S was used to compute the upwelling radiance corresponding to each of those aerosol models at several ambient humidities. For the pre-launch estimated precision of the sensors and the assumed viewing geometry of the instrument, the simulations suggest that the spectral and angular dependence of the reflectance measured by MISR is not sufficient to distinguish aerosol models with various different combinations of values ror dry composition. y and ambient RH. A similar behavior is observed for MODIS at visible wavelengths. However, the 2100 nm band of MODIS appears to be able to differentiate between at least some aerosol models with different aerosol hygroscopicity given the MODIS calibration error requirements. This result suggests the possibility of retrieval of aerosol hygroscopicity by MODIS.

Gasso, B. S.↗
DOE OSTI2026

Physics-aware adaptive checkpointing with shadow systems for nonlinear PDE simulations

Large-scale simulations of nonlinear partial differential equations (PDEs) that exhibit strongly transient behavior and pattern-forming dynamics produce enormous amounts of data, which, even with modern storage systems, cannot be stored for later curation. Current I/O strategies either write dense time series of snapshots, which is often prohibitive in I/O and storage, or store a few checkpoints that enable restart but incur expensive recomputation cost and provide no control over post-restart error growth, especially when lossy compression is used. Moreover, most, if not all, existing strategies take no account of the actual physical state of the system. Here, we present a simple physics-aware I/O framework in which a low-cost shadow system adaptively triggers lossy checkpoints when the shadow system deviates from the fine-scale simulation. The shadow system can be a coarsened replica of the fine-scale simulation that evolves concurrently. This means that checkpoints are taken based on the physical state of the system: fewer checkpoints are triggered when the system is quiescent while more are taken when the system undergoes a rapid change. This type of behavior is observed in many systems such as Brusselator and FitzHugh–Nagumo. We illustrate that our framework maintains stable restarts, keeps fine-scale restart errors bounded by shadow errors, and reconstructs the time history with significantly lower error and storage than interpolating fixed-interval snapshots, with low-cost shadow replay and modest online synchronization overhead.

Gong, Qian [ORNL] (ORCID:0000000235704142)↗
NASA NTRS2000

Influence of Humidity On the Aerosol Scattering Coefficient and Its Effect on the Upwelling Radiance During ACE-2

Aerosol scattering coefficients (sigma(sub sp)) have been measured over the ocean at different relative humidities (RH) as a function of altitude in the region surrounding the Canary Islands during the Second Aerosol Characterization Experiment (ACE-2) in June and July 1997. The data were collected by the University of Washington passive humidigraph (UWPH) mounted on the Pelican research aircraft. Concurrently, particle size distributions, absorption coefficients and aerosol optical depth were measured throughout 17 flights. A parameterization of sigma(sub sp) as a function of RH was utilized to assess the impact of aerosol hydration on the upwelling radiance (normalized to the solar constant and cosine of zenith angle). The top of the atmosphere radiance signal was simulated at wavelengths corresponding to visible and near-infrared bands of the EOS (Earth Observing System) AM-1 (Terra) detectors, MODIS (Moderate Resolution Imaging Spectroradiometer) and MISR (Multi-angle Imaging Spectroradiometer). The UWPH measured sigma(sub sp) at two RHs, one below and the other above ambient conditions. Ambient sigma(sub sp) was obtained by interpolation of these two measurements. The data were stratified in terms of three types of aerosols: Saharan dust, clean marine (marine boundary layer background) and polluted marine aerosols (i.e., two- or one-day old polluted aerosols advected from Europe). An empirical relation for the dependence of sigma(sub sp) on RH, defined by sigma(sub sp)(RH) = k.(1 - RH/100)(sup gamma), was used with the hygroscopic exponent gamma derived from the data. The following gamma values were obtained for the 3 aerosol types: gamma(dust) = 0.23 +/- 0.05, gamma(clean marine) = 0.69 +/- 0.06 and gamma(polluted marine) = 0.57 +/- 0.06. Based on the measured gammas, the above equation was utilized to derive aerosol models with different hygroscopicities. The satellite simulation signal code 6S was used to compute the upwelling radiance corresponding to each of those aerosol models at several ambient humidities. For the prelaunch estimated precision of the sensors and the assumed viewing geometry of the instrument, the simulations suggest that the spectral and angular dependence of the reflectance measured by MISR is not sufficient to distinguish aerosol models with various different combinations of values for dry composition, gamma and ambient RH. A similar behavior is observed for MODIS at visible wavelengths. However, the 2100 nm band of MODIS appears to be able to differentiate between at least same aerosol models with different aerosol hygroscopicity given the MODIS calibration error requirements. This result suggests the possibility of retrieval of aerosol hygroscopicity by MODIS.

Gasso, S.↗
DOE OSTI2025

Understanding latent timescales in neural ordinary differential equation models of advection-dominated dynamical systems

The neural ordinary differential equation (ODE) framework has shown considerable promise in recent years in developing highly accelerated surrogate models for complex physical systems characterized by partial differential equations (PDEs). For PDE-based systems, state-of-the-art neural ODE strategies leverage a two-step procedure to achieve this acceleration: a nonlinear dimensionality reduction step provided by an autoencoder, and a time integration step provided by a neural-network based model for the resultant latent space dynamics (the neural ODE). This work explores the applicability of such autoencoder-based neural ODE strategies for PDEs in which advection terms play a critical role. More specifically, alongside predictive demonstrations, physical insight into the sources of model acceleration (i.e., how the neural ODE achieves its acceleration) is the scope of the current study. Such investigations are performed by quantifying the effects of both autoencoder and neural ODE components on latent system time-scales using eigenvalue analysis of dynamical system Jacobians. To this end, the sensitivity of various critical training parameters – de-coupled versus end-to-end training, latent space dimensionality, and the role of training trajectory length, for example – to both model accuracy and the discovered latent system timescales is quantified. Furthermore, this work specifically uncovers the key role played by the training trajectory length (the number of rollout steps in the loss function during training) on the latent system timescales: larger trajectory lengths correlate with an increase in limiting neural ODE time-scales, and optimal neural ODEs are found to recover the largest time-scales of the full-order (ground-truth) system. Demonstrations are performed across fundamentally different unsteady fluid dynamics configurations influenced by advection: (1) the Kuramoto–Sivashinsky equations (2) Hydrogen-Air channel detonations (the compressible reacting Navier–Stokes equations with detailed chemistry), and (3) 2D Atmospheric flow.

Advection-dominated dynamical systems↗
DOE OSTI2026

Taylor approximation variance reduction for approximation errors in PDE-constrained Bayesian inverse problems

In numerous applications, surrogate models are used as a replacement for accurate parameter-to-observable mappings when solving large-scale inverse problems governed by partial differential equations (PDEs). The surrogate model may be a computationally cheaper alternative to the accurate parameter-to-observable mappings and/or may ignore additional unknowns or sources of uncertainty. The Bayesian approximation error (BAE) approach provides a means to account for the induced uncertainties and approximation errors, i.e. the errors between the accurate parameter-to-observable mapping and the surrogate. The statistics of these errors are, however, in general unknown a priori, and are thus calculated using Monte Carlo sampling. Although the sampling is typically carried out offline, i.e. before considering the data, the process can still represent a computational bottleneck. In this work, we develop a scalable computational approach for reducing the costs associated with the sampling stage of the BAE approach. Specifically, we consider the Taylor expansion of the accurate and surrogate forward models with respect to the uncertain parameter fields either as a control variate for variance reduction or as a means to directly and efficiently approximate the mean and covariance of the approximation errors. We propose efficient methods for evaluating the expressions for the mean and covariance of the Taylor approximations based on linear(-ized) PDE solves. Furthermore, the proposed approach is independent of the dimension of the uncertain parameter, depending instead on the intrinsic dimension of the data, ensuring scalability to high-dimensional problems. The potential benefits of the proposed approach are demonstrated for two high-dimensional inverse problems governed by PDE examples, namely for the estimation of a distributed Robin boundary coefficient in a linear diffusion problem, and for a coefficient estimation problem governed by a nonlinear diffusion problem.

Bayesian approximation error↗
DOE OSTI2024

Forecasting high-dimensional spatio-temporal systems from sparse measurements

This paper introduces a new neural network architecture designed to forecast high-dimensional spatio-temporal data using only sparse measurements. The architecture uses a two-stage end-to-end framework that combines neural ordinary differential equations (NODEs) with vision transformers. Initially, our approach models the underlying dynamics of complex systems within a low-dimensional space; and then it reconstructs the corresponding high-dimensional spatial fields. Many traditional methods involve decoding high-dimensional spatial fields before modeling the dynamics, while some other methods use an encoder to transition from high-dimensional observations to a latent space for dynamic modeling. In contrast, our approach directly uses sparse measurements to model the dynamics, bypassing the need for an encoder. This direct approach simplifies the modeling process, reduces computational complexity, and enhances the efficiency and scalability of the method for large datasets. We demonstrate the effectiveness of our framework through applications to various spatio-temporal systems, including fluid flows and global weather patterns. Although sparse measurements have limitations, our experiments reveal that they are sufficient to forecast system dynamics accurately over long time horizons. Our results also indicate that the performance of our proposed method remains robust across different sensor placement strategies, with further improvements as the number of sensors increases. This robustness underscores the flexibility of our architecture, particularly in real-world scenarios where sensor data is often sparse and unevenly distributed.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

FORCE Update 2024

The Framework for Optimization of Resources and Economics (FORCE) tool suite is the U.S. Department of Energy’s Nuclear Integrated Energy Systems (IES) Program flagship tool suite for technoeconomic IES analysis of IES. This tool suite is useful for analysis designed to evaluate and improve the technoeconomics of energy production systems, particularly for systems including nuclear technology. In this report, we document the development activity for the FORCE tool suite to extend its capabilities as performed during fiscal year 2024. In addition to reliability and accessibility, capability is one of the three standards guiding the development of the FORCE tool suite and the software codes that are its constituent parts. Extending the capabilities of the FORCE tool suite allows analysis both within the IES program as well as industry, university, and laboratory partners to perform analysis with more accuracy, insight, and impactful narrative. Four areas of capability development were the focus of activity this year: economic parameter uncertainty quantification, multiresolution analysis, components-to-optimization workflow automation, and statespace construction workflows for real-time optimal control. In economic parameter uncertainty quantification, the ability of HERON to capture risk due to scenarios (weather and energy demand uncertainty) was expanded to also include uncertainties in financial parameters such as capital cost or operation and maintenance costs. By including these sources of uncertainty, which are sometimes very large compared with scenario uncertainty, HERON is better able to capture the risk posed by investment in various IES technology. Because of this, analysts can also consider the reduction in risks that can be realized by choice of some technologies. In multiresolution analysis, development activity extended on work completed previously. In fiscal year 2023, methods for decomposing time series signals, such as demand, solar and wind availability, and price profiles, were analyzed and down-selected to those most effective at splitting signals into different resolutions. These resolutions allow considering the influence of different energy demand and supply behaviors across different time scales. For example, energy demand might be divided into seasonal, weekly, and hourly profiles. In fiscal year 2024, this preliminary work was extended and implemented within the Risk Analysis Virtual Environment (RAVEN) risk and uncertainty analysis platform, which is used throughout the FORCE framework. This development of the “multi-resolution time series analysis” (MR-TSA) module in RAVEN allows training synthetic history generators on complex time series. These synthetic history generators can then be used in HERON for generating scenarios that represent possible market and weather scenarios that can be analyzed on different time scales. We envision completing this work in the future, implementing multiresolution dispatch optimization strategies that can make the most beneficial use of these stratified time histories. In components-to-optimization workflow development, workflows for translating user inputs of components into algorithms for algebraic optimization were selected and implemented. Similar algorithms within the Holistic Energy Resource Optimization Network (HERON) were separated from the main code base of HERON and gathered with the components-to-optimization workflows in the new Dispatch Optimization Variable Engine (DOVE) software library. This modularization allows FORCE users to analyze dispatch optimization and energy system duty cycles independently of HERON, which previously was a burdensome task. Additionally, these dispatch optimization algorithms, set up in an independent library, can now be used across all software applications within FORCE, especially including the real-time optimal control software Optimization of Real-time Capacity Allocation (ORCA). Allowing FORCE software to share dispatch optimization algorithms within a single library allows for improved software maintenance and reliability. In statespace characterization workflow development, alternative workflows for optimizing dispatch with additional technical accuracy was the focus, particularly to improve the real-time optimization decision making in ORCA. Using algorithms and workflows initially developed for the Feasible Actuator Range Modifier (FARM), workflows for determining the statespace representation of IES were identified and demonstrated. The resulting dispatch optimization required a more robust optimization algorithm than that originally used in HERON (and moved to DOVE), which required adding an alternate workflow to DOVE that can more accurately match the behavior of physical systems using a partial differential equation representation. In conclusion, capability developments in the FORCE tool suite in fiscal year 2024 have improved the ability of the FORCE tool suite to perform

29 ENERGY PLANNING, POLICY, AND ECONOMY↗