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DOE OSTI2020

Assessing mean climate change signals in the global CORDEX-CORE ensemble

The new Coordinated Output for Regional Evaluations (CORDEX-CORE) ensemble provides high-resolution, consistent regional climate change projections for the major inhabited areas of the world. It serves as a solid scientific basis for further research related to vulnerability, impact, adaptation and climate services in addition to existing CORDEX simulations. The aim of this study is to investigate and document the climate change information provided by the CORDEX-CORE simulation ensemble, as a part of the World Climate Research Programme (WCRP) CORDEX community. An overview of the annual and monthly mean climate change information in selected regions in different CORDEX domains is presented for temperature and precipitation, providing the foundation for detailed follow-up studies and applications. Initially, two regional climate models (RCMs), REMO and RegCM were used to downscale global climate model output. The driving simulations by AR5 global climate models (AR5-GCMs) were selected to cover the spread of high, medium, and low equilibrium climate sensitivity at a global scale. The CORDEX-CORE ensemble has doubled the spatial resolution compared to the previously existing CORDEX simulations in most of the regions (25 $\,\mathrm {km}$ (0.22 $^{\circ }$ /jats:inline-formula>) versus 50 $\,\mathrm {km}$ (0.44 $^{\circ }$ )) leading to a potentially improved representation of, e.g., physical processes in the RCMs. The analysis focuses on changes in the IPCC physical climate reference regions. The results show a general reasonable representation of the spread of the temperature and precipitation climate change signals of the AR5-GCMs by the CORDEX-CORE simulations in the investigated regions in all CORDEX domains by mostly covering the AR5 interquartile range of climate change signals. The simulated CORDEX-CORE monthly climate change signals mostly follow the AR5-GCMs, although for specific regions they show a different change in the course of the year compared to the AR5-GCMs, especially for RCP8.5, which needs to be investigated further in region specific process studies.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Evaluation of Autoconversion Representation in E3SMv2 Using an Ensemble of Large-Eddy Simulations of Low-Level Warm Clouds

In numerical atmospheric models that treat cloud and rain droplet populations as separate condensate categories, precipitation initiation in warm clouds is often represented by an autoconversion rate (Au), which is the rate of formation of new rain droplets through the collisions of cloud droplets. Being a function of the cloud droplet size distribution (DSD), the local Au is commonly parameterized as a function of DSD moments: cloud droplet number (n c ) and mass (q c ) concentrations. When applied in a large-scale model, the grid-mean Au must also include a correction, or enhancement factor, to account for the horizontal variability of the cloud properties across the model grid. In this study, we evaluate the Au representation in the Energy Exascale Earth System Model version 2 (E3SMv2) climate model using large-eddy simulations (LES), which explicitly resolve cloud droplet spectra, and therefore the local Au, as well as its spatial variability. The analysis of an ensemble of warm low-level cloud cases shows that the E3SMv2 formulation represents the Au reasonably well compared to the horizontally averaged explicitly computed rate from LES. The agreement, however, comes from a combination of an underestimated E3SM-tuned local Au rate and an overestimated subgrid cloud variability enhancement factor. The latter bias is traced to neglecting the horizontal variability of n c and its co-variability with q c in parameterizing the grid-mean Au.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2024

Scaling neural simulations in STACS

Abstract As modern neuroscience tools acquire more details about the brain, the need to move towards biological-scale neural simulations continues to grow. However, effective simulations at scale remain a challenge. Beyond just the tooling required to enable parallel execution, there is also the unique structure of the synaptic interconnectivity, which is globally sparse but has relatively high connection density and non-local interactions per neuron. There are also various practicalities to consider in high performance computing applications, such as the need for serializing neural networks to support potentially long-running simulations that require checkpoint-restart. Although acceleration on neuromorphic hardware is also a possibility, development in this space can be difficult as hardware support tends to vary between platforms and software support for larger scale models also tends to be limited. In this paper, we focus our attention on Simulation Tool for Asynchronous Cortical Streams (STACS), a spiking neural network simulator that leverages the Charm++ parallel programming framework, with the goal of supporting biological-scale simulations as well as interoperability between platforms. Central to these goals is the implementation of scalable data structures suitable for efficiently distributing a network across parallel partitions. Here, we discuss a straightforward extension of a parallel data format with a history of use in graph partitioners, which also serves as a portable intermediate representation for different neuromorphic backends. We perform scaling studies on the Summit supercomputer, examining the capabilities of STACS in terms of network build and storage, partitioning, and execution. We highlight how a suitably partitioned, spatially dependent synaptic structure introduces a communication workload well-suited to the multicast communication supported by Charm++. We evaluate the strong and weak scaling behavior for networks on the order of millions of neurons and billions of synapses, and show that STACS achieves competitive levels of parallel efficiency.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2021

Compressive Neural Representations of Volumetric Scalar Fields

Here, we present an approach for compressing volumetric scalar fields using implicit neural representations. Our approach represents a scalar field as a learned function, wherein a neural network maps a point in the domain to an output scalar value. By setting the number of weights of the neural network to be smaller than the input size, we achieve compressed representations of scalar fields, thus framing compression as a type of function approximation. Combined with carefully quantizing network weights, we show that this approach yields highly compact representations that outperform state-of-the-art volume compression approaches. The conceptual simplicity of our approach enables a number of benefits, such as support for time-varying scalar fields, optimizing to preserve spatial gradients, and random-access field evaluation. We study the impact of network design choices on compression performance, highlighting how simple network architectures are effective for a broad range of volumes.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

ESMs Latent Space Exploration for Uncertainty Quantification and Spatiotemporal Downscaling

This final report for DOE Award DE-SC0023044 presents advances in two key areas of climate modeling: (1) representative climate model selection and (2) Earth System Model (ESM) downscaling using hybrid AI methods. The first section introduces a reordered, three-stage workflow to select representative GCM runs that more effectively balance historical skill with ensemble spread, validated across Texas, Bihar, and New York. The second section introduces two novel super-resolution frameworks, ViSIR and ViFOR, that integrate Vision Transformers with sinusoidal and Fourier-based implicit neural representations. These models achieve state-of-the-art reconstruction accuracy for ESM variables including surface temperature and heat fluxes. The report includes detailed methodology, benchmarks, and results, demonstrating significant gains in uncertainty quantification, spatial fidelity, and scalability for climate-impact studies.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Artificial Intelligence for Enhancing Multiscale Analysis: Buildings Focus

This project aims to develop multi-scale building energy data, potentially improving the representation of the U.S. buildings sector in GCAM-USA, an U.S.-focused human-energy-Earth systems model. Existing building energy datasets are typically limited to national or regional levels, which constrains the ability of models to capture fine-scale human-energy-Earth systems interactions and reduces their relevance for decision-making on issues such as energy security, resilience, and energy planning. By leveraging AI and advanced data integration methods, this work fuses multiple existing datasets to enhance the physical and geographic representation of both residential and commercial building energy use. So far, progress includes processing residential building data, designing the data structure for commercial buildings, and testing AI approaches for integrating datasets and addressing spatial-temporal gaps. This effort can not only advances GCAM-USA’s capability in modeling the buildings sector but also supports broader DOE missions, such as developing digital testbeds, enhancing grid resilience analysis, and improving building–energy system modeling at decision-relevant scales.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗

NASA NTRS1975

VECTAN II - A computer program for the spatial analysis of the vectorcardiogram

This paper presents the operation of a digital computer program, VECTAN II, for the spatial analysis of the vectorcardiogram (VCG). The program incorporates a unique waveform recognition algorithm based on the spatial vector length which has been shown to perform better than previous algorithms. The waveform analysis employed by the program considers the vectorcardiogram as a three dimensional entity rather than as scalar or planar representations. VECTAN II is designed chiefly to measure and quantify the VCG response of normal subjects to a controlled stress by analyzing one VCG complex every five seconds throughout a long experiment. The program has been used to analyze data from the NASA Johnson Space Center Cardiovascular Laboratory, from the pre- and postflight medical examinations of the Apollo 15, 16 and 17 crewmen, and from onboard Skylab experiments.

Golden, D. P., Jr.↗

NASA NTRS1992

Operator vision aids for space teleoperation assembly and servicing

This paper investigates concepts for visual operator aids required for effective telerobotic control. Operator visual aids, as defined here, mean any operational enhancement that improves man-machine control through the visual system. These concepts were derived as part of a study of vision issues for space teleoperation. Extensive literature on teleoperation, robotics, and human factors was surveyed to definitively specify appropriate requirements. This paper presents these visual aids in three general categories of camera/lighting functions, display enhancements, and operator cues. In the area of camera/lighting functions concepts are discussed for: (1) automatic end effector or task tracking; (2) novel camera designs; (3) computer-generated virtual camera views; (4) computer assisted camera/lighting placement; and (5) voice control. In the technology area of display aids, concepts are presented for: (1) zone displays, such as imminent collision or indexing limits; (2) predictive displays for temporal and spatial location; (3) stimulus-response reconciliation displays; (4) graphical display of depth cues such as 2-D symbolic depth, virtual views, and perspective depth; and (5) view enhancements through image processing and symbolic representations. Finally, operator visual cues (e.g., targets) that help identify size, distance, shape, orientation and location are discussed.

Brooks, Thurston L.↗

NASA NTRS2005

Dynamic wake prediction and visualization with uncertainty analysis

A dynamic wake avoidance system utilizes aircraft and atmospheric parameters readily available in flight to model and predict airborne wake vortices in real time. A novel combination of algorithms allows for a relatively simple yet robust wake model to be constructed based on information extracted from a broadcast. The system predicts the location and movement of the wake based on the nominal wake model and correspondingly performs an uncertainty analysis on the wake model to determine a wake hazard zone (no fly zone), which comprises a plurality of wake planes, each moving independently from another. The system selectively adjusts dimensions of each wake plane to minimize spatial and temporal uncertainty, thereby ensuring that the actual wake is within the wake hazard zone. The predicted wake hazard zone is communicated in real time directly to a user via a realistic visual representation. In an example, the wake hazard zone is visualized on a 3-D flight deck display to enable a pilot to visualize or see a neighboring aircraft as well as its wake. The system substantially enhances the pilot's situational awareness and allows for a further safe decrease in spacing, which could alleviate airport and airspace congestion.

Holforty, Wendy L.↗

DOE OSTI2025

Second-generation downscaled earth system model data using generative machine learning

The second-generation Sup3rCC dataset provides high-resolution meteorological data generated through the downscaling of multiple earth system models (ESMs) from the Coupled Model Intercomparison Project Phase 6 (CMIP6). This downscaling is performed through application of a generative machine learning approach called Super-Resolution for Renewable Resource Data (sup3r). This dataset builds on the first-generation Sup3rCC data by applying improved bias correction methods and adding downscaled precipitation to the output variables. As with the first Sup3rCC version, the data still include temperature, wind speed and direction at multiple heights, pressure, three components of downwelling solar radiation, and relative humidity—all at 4-kilometer (km) hourly resolution over the contiguous United States. This is a 25x spatial enhancement and 24x temporal enhancement of the source 100-km daily-average ESM data. This extension of the Sup3rCC dataset includes data from six ESMs from two shared socioeconomic pathways (SSPs) totaling 400 years of data with multiple future projections of changing meteorological conditions. The scenario selection was based on a structured evaluation of historical ESM skill and comprehensive representation of possible trajectories of future climate change in temperature, humidity, precipitation, solar irradiance, and near-surface wind speeds. The inclusion of multiple future projections is intended to enable users to assess key drivers of un 36 certainty and variability. All data are double-bias corrected, resulting in a product that can be used out-of-the-box for energy system analysis with minimal historical bias. The potential applications of Sup3rCC data extend to various topics in renewable energy resource assessment, energy systems modeling, and grid resilience studies. High-resolution future meteorological projections are critical for evaluating the effects of changing meteorological conditions on renewable energy generation, energy demand, and for optimizing energy storage and grid infrastructure. The 4-km hourly resolution of the downscaled data enables understanding of spatial and temporal variability at the scales necessary for energy system operational planning. In addition, the dataset can support risk assessments by providing detailed information on possible future extreme weather events and long-term meteorological variability at scales relevant to energy infrastructure. By offering an enhanced representation of possible future meteorological conditions, the second-generation Sup3rCC dataset enables more precise modeling of energy resilience and adaptation strategies in response to changing meteorological conditions.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1997

Laser-Induced Incandescence in Microgravity

Knowledge of soot concentration is important due to its presence and impact upon a wide range of combustion processes ranging from diffusion to premixed flames, laminar to turbulent processes and homogeneous to heterogeneous combustion. Measurement of soot volume fraction (f(sub v)) is essential to discerning its formation and growth. The presence of soot also affects other physical and chemical properties of combustion thereby affecting studies not directly concerned with either its formation or growth, such as radiative heat transfer, CO oxidation and fuel vaporization or pyrolysis rates. Microgravity offers unique opportunities for studying both soot growth and the effect of soot radiation upon flame structure and spread. Spatial scales and residence time scales are greatly extended in 0-g facilitating soot growth studies. With the varied geometries, short duration microgravity test times and time-varying processes there is a demand for measurement of f(sub v) with high spatial and temporal resolution. Laser-induced incandescence (LII) has advanced f(sub v) measurements in many 1-g combustion processes. To create laser-induced incandescence, a pulsed high intensity laser heats soot to incandescence temperatures. Using appropriate spectral and temporal detection conditions, the resulting incandescence can be selectively detected apart from the non-laser-heated soot and flame gases. Theoretical modelling and experiments have shown that the resulting incandescence is representative of f(sub v). Using an intensified array camera and a laser sheet for excitation, one- and two-dimensionally resolved LII images of f(sub v) have been obtained in 1-g. LII has been characterized and developed at NASA-Lewis for soot volume fraction determination in a wide range of 1-g combustion applications. Broadly grouped, the characterization work has included studies of excitation intensity, excitation wavelength and the optimum temporal and spectral detection conditions to enable an accurate representation of soot volume fraction by LII. Tests for special requirements imposed by different combustion processes have been performed in laminar and turbulent diffusion flames, rich sooting premixed flames, single droplet combustion, and other heterogeneous combustion. These studies demonstrated LII's high sensitivity, temporal and spatial capabilities and its geometric versatility. In contrast to the advantages offered to combustion studies by a microgravity environment, advanced diagnostics, specifically those requiring pulsed laser diagnostics have been limited due to the size, weight and power limitations in a low-gravity environment. Reported here are the first demonstrations of LII performed in a microgravity environment. Examples are shown for laminar and turbulent gas-jet diffusion flames in 0-g.

VanderWal, Randy L.↗

DOE OSTI2020

Unmixing Symmetries

The low-lying spectra of atomic nuclei display diverse behaviors, for example, rotational bands, which can be described phenomenologically by simple symmetry groups such as spatial SU(3). This leads to the idea of dynamical symmetry, where the Hamiltonian commutes with the Casimir operator(s) of a group, and is block diagonal in subspaces defined by the group’s irreducible representations or irreps. Detailed microscopic calculations, however, show these symmetries are in fact often strongly mixed and the wave function fragmented across many irreps. More commonly, the fragmentation across members of a band are similar, which is called a quasidynamical symmetry. In this Letter I explicitly, albeit numerically, construct unitary transformations from a quasidynamical symmetry to a dynamical symmetry, adapting the similarity renormalization group (SRG) in order to transform away the symmetry-mixing parts of the Hamiltonian. In this work, the standard SRG produces unsatisfactory results, forcing the induced dynamical symmetry to be dominated by high-weight irreps irrespective of the original decomposition. Using spectral distribution theory to rederive and diagnose standard SRG, I introduce a new form of SRG. The new SRG transforms a quasidynamical symmetry to a dynamical symmetry, that is, unmixes the mixed symmetries, with intuitively more appealing results.

74 ATOMIC AND MOLECULAR PHYSICS↗

DOE OSTI2025

Concurrent two-way coupling of global and local models across internal boundaries with non-matching discretizations

Coupling local and global models enables efficient simulation of multiscale systems, where global models capture large-scale behavior and local models, with enhanced physics, resolve finer details over a smaller region. Here, this paper presents a mathematically consistent method for coupling physics-based models of varying fidelity across adjacent, non-overlapping subdomains, even when discretizations do not match at the immersed interdomain interfaces. Incompressible Navier-Stokes equations (NSE) constitute the global model while residual-based turbulence model serves as the local high-fidelity model. In addition, a scalar advection-diffusion equation that models the convection of an active scalar field is appended to the turbulence model in the local domain. This scalar field does not have its complement in the global model, giving rise to unequal number of equations at the immersed boundary between local and global models. Interdomain coupling terms are derived via the Variational Multiscale Discontinuous Galerkin (VMDG) method with new developments in scale representation and efficient fine-scale estimation. While transient laminar flows modeled with NSE in the global domain can be resolved with relatively coarse mesh, turbulent flow calculations in the local model require much finer spatial discretizations as well as smaller time-step for appropriately resolving the turbulent flow physics. The proposed framework also accommodates non-matching meshes at the immersed boundaries. Test problems in 2D and 3D numerically showcase the concurrent two-way coupling of unknown fields across the immersed boundaries. The 3D test presents a case with an unequal number of equations, where the scalar field represents the convection of contaminant concentration. This provides more detailed physics in the local region and highlights its application in climate modeling and atmospheric sciences.

Variational Multiscale Discontinuous Galerkin (VMD↗

NASA NTRS1984

Modeling long-term changes in forested landscapes and their relation to the Earth's energy balance

The dynamics of the forested parts of the Earth's surface on time scales from decades to centuries are discussed. A set of computer models developed at Oak Ridge National Laboratory and elsewhere are applied as tools. These models simulate a landscape by duplicating the dynamics of growth, death and birth of each tree living on a 0.10 ha element of the landscape. This spatial unit is generally referred to as a gap in the case of the forest models. The models were tested against and applied to a diverse array of forests and appear to provide a reasonable representation for investigating forest-cover dynamics. Because of the climate linkage, one important test is the reconstruction of paleo-landscapes. Detailed reconstructions of changes in vegetation in response to changes in climate are crucial to understanding the association of the Earth's vegetation and climate and the response of the vegetation to climate change.

Shugart, H. H.↗

NASA NTRS2016

Optimal Reorganization of NASA Earth Science Data for Enhanced Accessibility and Usability for the Hydrology Community

A long-standing "Digital Divide" in data representation exists between the preferred way of data access by the hydrology community and the common way of data archival by earth science data centers. Typically, in hydrology, earth surface features are expressed as discrete spatial objects (e.g., watersheds), and time-varying data are contained in associated time series. Data in earth science archives, although stored as discrete values (of satellite swath pixels or geographical grids), represent continuous spatial fields, one file per time step. This Divide has been an obstacle, specifically, between the Consortium of Universities for the Advancement of Hydrologic Science, Inc. and NASA earth science data systems. In essence, the way data are archived is conceptually orthogonal to the desired method of access. Our recent work has shown an optimal method of bridging the Divide, by enabling operational access to long-time series (e.g., 36 years of hourly data) of selected NASA datasets. These time series, which we have termed "data rods," are pre-generated or generated on-the-fly. This optimal solution was arrived at after extensive investigations of various approaches, including one based on "data curtains." The on-the-fly generation of data rods uses "data cubes," NASA Giovanni, and parallel processing. The optimal reorganization of NASA earth science data has significantly enhanced the access to and use of the data for the hydrology user community.

data rods↗

DOE OSTI2020

Modeling the smoky troposphere of the southeast Atlantic: a comparison to ORACLES airborne observations from September of 2016

The southeast Atlantic is home to well-defined smoke outflow from Africa coinciding vertically with extensive marine boundary-layer cloud decks, both reaching their climatological maxima in spatial extent around September. A framework is put forth for evaluating the performance of a range of global and regional aerosol models against observations made during the NASA ORACLES (ObseRvations of Aerosols above CLouds and their intEractionS) airborne mission in September 2016. The sparse airborne observations are first aggregated into 2o grid boxes and into three vertical layers: the cloud-topped marine boundary layer (MBL), the layer from cloud top to 3 km, and the 3-6 km layer. Aerosol extensive properties simulated for the entire study region for all September suggest that the 2016 ORACLES observations are reasonably representative of the regional monthly average, with systematic deviations of 30% or less. All six models typically place the bottom of the smoke layer at lower altitudes than do the airborne lidar observations by 300-1400 m, whereas model aerosol top heights are within 0-500 m of the observations. All but one of the models that report carbonaceous aerosol masses underestimate the ratio of particulate extinction to the masses, a proxy for mass extinction efficiency, in 3-6 km. Notable findings on individual models include that WRF-CAM5 predicts the mass of black carbon and organic aerosols with minor (~10% or less) biases. GEOS-5 overestimates the carbonaceous particle masses in the MBL by a factor of 3-6. Extinction coefficients in the free troposphere (FT) and above-cloud aerosol optical depth (ACAOD) are 10-30% lower in WRF-CAM5, 30-50% lower in GEOS-5, 10-40% higher in GEOS-Chem, 10-20% higher in EAM-E3SM except for the practically unbiased 3-6 km extinction, and 20-70% lower in the Unified Model, than the airborne in situ, lidar and sunphotometer measurements. ALADIN-Climate also underestimates the ACAOD, by 30%. GEOS-5 and GEOS-Chem predict carbon monoxide in the MBL with small (10% or less) negative biases, despite their overestimates of carbonaceous aerosol masses. Overall, this study highlights a new approach to utilizing airborne aerosol measurements for model diagnosis.

Shinozuka, Yohei↗

NASA NTRS2020

Modeling the smoky troposphere of the southeast Atlantic: a comparison to ORACLES airborne observations from September of 2016

In the southeast Atlantic, well-defined smoke plumes from Africa advect over marine boundary layer cloud decks; both are most extensive around September, when most of the smoke resides in the free troposphere. A framework is put forth for evaluating the performance of a range of global and regional atmospheric composition models against observations made during the NASA ORACLES (ObseRvations of Aerosols above CLouds and their intEractionS) airborne mission in September 2016. A strength of the comparison is a focus on the spatial distribution of a wider range of aerosol composition and optical properties than has been done previously. The sparse airborne observations are aggregated into approximately 2° grid boxes and into three vertical layers: 3–6 km, the layer from cloud top to 3 km, and the cloud-topped marine boundary layer. Simulated aerosol extensive properties suggest that the flight-day observations are reasonably representative of the regional monthly average, with systematic deviations of 30 % or less. Evaluation against observations indicates that all models have strengths and weaknesses, and there is no single model that is superior to all the others in all metrics evaluated. Whereas all six models typically place the top of the smoke layer within 0–500 m of the airborne lidar observations, the models tend to place the smoke layer bottom 300–1400 m lower than the observations. A spatial pattern emerges, in which most models underestimate the mean of most smoke quantities (black carbon, extinction, carbon monoxide) on the diagonal corridor between 16° S, 6° E, and 10° S, 0° E, in the 3–6 km layer, and overestimate them further south, closer to the coast, where less aerosol is present. Model representations of the above-cloud aerosol optical depth differ more widely. Most models overestimate the organic aerosol mass concentrations relative to those of black carbon, and with less skill, indicating model uncertainties in secondary organic aerosol processes. Regional-mean free-tropospheric model ambient single scattering albedos vary widely, between 0.83 and 0.93 compared with in situ dry measurements centered at 0.86, despite minimal impact of humidification on particulate scattering. The modeled ratios of the particulate extinction to the sum of the black carbon and organic aerosol mass concentrations (a mass extinction efficiency proxy) are typically too low and vary too little spatially, with significant inter-model differences. Most models overestimate the carbonaceous mass within the offshore boundary layer. Overall, the diversity in the model biases suggests that different model processes are responsible. The wide range of model optical properties requires further scrutiny because of their importance for radiative effect estimates.

Yohei Shinozuka↗

DOE OSTI2025

Constrained or unconstrained? Neural-network-based equation discovery from data

Throughout many fields, practitioners often rely on differential equations to model systems. Yet, for many applications, the theoretical derivation of such equations and/or the accurate resolution of their solutions may be intractable. Instead, recently developed methods, including those based on parameter estimation, operator subset selection, and neural networks, allow for the data-driven discovery of both ordinary and partial differential equations (PDEs), on a spectrum of interpretability. The success of these strategies is often contingent upon the correct identification of representative equations from noisy observations of state variables and, as importantly and intertwined with that, the mathematical strategies utilized to enforce those equations. Specifically, the latter has been commonly addressed via unconstrained optimization strategies. Representing the PDE as a neural network, we propose to discover the PDE (or the associated operator) by solving a constrained optimization problem and using an intermediate state representation similar to a physics-informed neural network (PINN). The objective function of this constrained optimization problem promotes matching the data, while the constraints require that the discovered PDE is satisfied at a number of spatial collocation points. We present a penalty method and a widely used trust-region barrier method to solve this constrained optimization problem, and we compare these methods on numerical examples. Our results on several example problems demonstrate that the latter constrained method outperforms the penalty method, particularly for higher noise levels or fewer collocation points. This work motivates further exploration into using sophisticated constrained optimization methods in scientific machine learning, as opposed to their commonly used, penalty-method or unconstrained counterparts. For both of these methods, we solve these discovered neural network PDEs with classical methods, such as finite difference methods, as opposed to PINNs-type methods relying on automatic differentiation. Here, we briefly highlight how simultaneously fitting the data while discovering the PDE improves the robustness to noise and other small, yet crucial, implementation details.

Data-driven discovery↗