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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

A Machine Learning Framework for Modeling Ensemble Properties of Atomically Disordered Materials

Atomic disorder can strongly influence material properties such as charge transport, optical response, and catalytic activity. However, efficiently modeling these disorder effects remains challenging for first-principles methods due to the cost of sampling large configurational spaces and computing complex physical quantities. Recent advances of machine learning techniques, particularly graph neural networks (GNNs), has enabled the efficient and accurate predictions of complex material properties, offering promising tools for studying disordered systems. In this work, we present a general machine-learning-assisted computational framework that integrates equivariant GNNs with Monte Carlo simulations to compute the thermodynamic and ensemble-averaged functional properties of disordered materials. Using the surface-termination-disordered MXene monolayer Ti 3 C 2 T 2–x as a representative system, we find that electrical conductivity exhibits an emergent peak near the order–disorder phase transition temperature due to the interplay between electron scattering and doping. In contrast, optical conductivity remains largely insensitive to local atomic disorder and reflects the global surface chemical composition. These results highlight the role of atomic disorder in affecting material properties and demonstrate the potential of our approach for statistically modeling disorder effects in a wide range of materials such as high-entropy alloys and spin liquids.

MXene↗
NASA NTRS1992

Neural network architectures to analyze OPAD data

A prototype Optical Plume Anomaly Detection (OPAD) system is now installed on the space shuttle main engine (SSME) Technology Test Bed (TTB) at MSFC. The OPAD system requirements dictate the need for fast, efficient data processing techniques. To address this need of the OPAD system, a study was conducted into how artificial neural networks could be used to assist in the analysis of plume spectral data.

Whitaker, Kevin W.↗
NASA NTRS1994

On-line, adaptive state estimator for active noise control

Dynamic characteristics of airframe structures are expected to vary as aircraft flight conditions change. Accurate knowledge of the changing dynamic characteristics is crucial to enhancing the performance of the active noise control system using feedback control. This research investigates the development of an adaptive, on-line state estimator using a neural network concept to conduct active noise control. In this research, an algorithm has been developed that can be used to estimate displacement and velocity responses at any locations on the structure from a limited number of acceleration measurements and input force information. The algorithm employs band-pass filters to extract from the measurement signal the frequency contents corresponding to a desired mode. The filtered signal is then used to train a neural network which consists of a linear neuron with three weights. The structure of the neural network is designed as simple as possible to increase the sampling frequency as much as possible. The weights obtained through neural network training are then used to construct the transfer function of a mode in z-domain and to identify modal properties of each mode. By using the identified transfer function and interpolating the mode shape obtained at sensor locations, the displacement and velocity responses are estimated with reasonable accuracy at any locations on the structure. The accuracy of the response estimates depends on the number of modes incorporated in the estimates and the number of sensors employed to conduct mode shape interpolation. Computer simulation demonstrates that the algorithm is capable of adapting to the varying dynamic characteristics of structural properties. Experimental implementation of the algorithm on a DSP (digital signal processing) board for a plate structure is underway. The algorithm is expected to reach the sampling frequency range of about 10 kHz to 20 kHz which needs to be maintained for a typical active noise control application.

Lim, Tae W.↗
NASA NTRS1997

Neural Network Research on Tilt-Rotor Hover Performance

Results from a neural network study on XV-15 tilt-rotor hover performance are presented. Two XV-15 test data bases, acquired during separate tests conducted in the National Full-Scale Aerodynamics Complex at NASA Ames, were used in this study. The two XV-15 test data bases used were: the 1997 80- by 120-Foot Wind Tunnel test data base and the 1984 Outdoor Aerodynamic Research Facility (OARF) test data base. The objectives associated with the 80- by 120-Foot Wind Tunnel test data were to conduct data quality checks, obtain neural network representations, and demonstrate sensitivity to test conditions. The objectives associated with the OARF outdoor test data were to obtain a 'zero wind' neural network representation, and formulate and implement a neural network based wind correction procedure. An additional objective, common to both test data bases, was to conduct error comparisons. The conclusions from the present study were as follows: Compared to measured parameters (i.e., collective pitch angle), derived parameters (i.e., thrust coefficient) were found preferable as neural network inputs. For the 80- by 120-Foot Wind Tunnel test data base, the neural network based figure of merit representations did not extrapolate rotor stall. Using the OARF outdoor test database, a neural network based wind correction procedure was formulated and successfully implemented. Based on an error comparison, the present neural network based wind correction procedure was found to be more accurate compared to the existing, momentum-theory-based wind correction procedure. Also, a separate error comparison showed that the 80- by 120-Foot Wind Tunnel hover performance data were comparable to non-wind-corrected outdoor test data.

Kozapalli, S.↗
NASA NTRS2004

Optical Calibration Process Developed for Neural-Network-Based Optical Nondestructive Evaluation Method

A completely optical calibration process has been developed at Glenn for calibrating a neural-network-based nondestructive evaluation (NDE) method. The NDE method itself detects very small changes in the characteristic patterns or vibration mode shapes of vibrating structures as discussed in many references. The mode shapes or characteristic patterns are recorded using television or electronic holography and change when a structure experiences, for example, cracking, debonds, or variations in fastener properties. An artificial neural network can be trained to be very sensitive to changes in the mode shapes, but quantifying or calibrating that sensitivity in a consistent, meaningful, and deliverable manner has been challenging. The standard calibration approach has been difficult to implement, where the response to damage of the trained neural network is compared with the responses of vibration-measurement sensors. In particular, the vibration-measurement sensors are intrusive, insufficiently sensitive, and not numerous enough. In response to these difficulties, a completely optical alternative to the standard calibration approach was proposed and tested successfully. Specifically, the vibration mode to be monitored for structural damage was intentionally contaminated with known amounts of another mode, and the response of the trained neural network was measured as a function of the peak-to-peak amplitude of the contaminating mode. The neural network calibration technique essentially uses the vibration mode shapes of the undamaged structure as standards against which the changed mode shapes are compared. The published response of the network can be made nearly independent of the contaminating mode, if enough vibration modes are used to train the net. The sensitivity of the neural network can be adjusted for the environment in which the test is to be conducted. The response of a neural network trained with measured vibration patterns for use on a vibration isolation table in the presence of various sources of laboratory noise is shown. The output of the neural network is called the degradable classification index. The curve was generated by a simultaneous comparison of means, and it shows a peak-to-peak sensitivity of about 100 nm. The following graph uses model generated data from a compressor blade to show that much higher sensitivities are possible when the environment can be controlled better. The peak-to-peak sensitivity here is about 20 nm. The training procedure was modified for the second graph, and the data were subjected to an intensity-dependent transformation called folding. All the measurements for this approach to calibration were optical. The peak-to-peak amplitudes of the vibration modes were measured using heterodyne interferometry, and the modes themselves were recorded using television (electronic) holography.

Decker, Arthur J.↗
DOE OSTI2024

IRIS-GNN: Leveraging Graph Neural Networks for Scheduling on Truly Heterogeneous Runtime Systems

The diversity of accelerators in computer systems poses significant challenges for software developers, such as managing vendor-specific compiler toolchains, code fragmentation requiring different kernel implementations, and performance portability issues. To address these, the Intelligent Runtime System (IRIS) was developed. IRIS works across various systems, from smartphones to supercomputers, enabling automatic performance scaling based on available accelerators. It introduces abstract tasks for seamless execution transitions between accelerators while ensuring memory consistency and task dependencies. Although IRIS simplifies system details, optimal dynamic scheduling still requires user input to understand workload structures. To address this, we introduce a new scheduling policy for IRIS, termed IRIS-GNN, which is the first IRIS hybrid policy that operates in conjunction with the dynamic policies. This policy employs a Graph-Neural Network (GNN) to conduct Graph Classification of any task graphs submitted to IRIS. This GNN analyzes the structure and attributes of the task graph, categorizing it as either locality, concurrency, or mixed. This classification subsequently guides the selection of the dynamic policy used by IRIS. We provide a comparison of the performance of IRIS-GNN against the complete spectrum of IRIS’s dynamic policies, assess the overhead introduced by the GNN within this scheduling framework, and ultimately explore its practical application in real-world scenarios.

Johnston, Beau↗
DOE OSTI2025

Application of physics-informed neural networks (PINNs) solution to coupled thermal and hydraulic processes in silty sands

Abstract The accurate modeling of water and heat transport in soils is crucial for both geo-environmental and geothermal engineering. Traditional modeling methods are problematic because they require well-defined boundaries and initial conditions. Recently, physics-informed neural networks (PINNs), which incorporate partial differential equations (PDEs) to solve forward and inverse problems, have attracted increasing attention in machine learning research. In this study, we applied PINNs to tackle hydraulic and thermal transport coupling forward problems in silty sands. A fully connected deep neural network was utilized for training. This neural network model leverages automatic differentiation to apply the governing equations as constraints, based on the mathematical approximations established by the neural network itself. We conducted forward problems and compared the solutions derived from PINNs with those from Finite Element Method (FEM) simulations. The forward problem results demonstrate the PINNs model’s capability in predicting hydraulic transport, heat transport, and thermal–hydraulic coupling in silty sands under various boundary conditions. The PINNs exhibited great performance in simulating the thermal–hydraulic coupling problem. The accuracy of the PINNs solutions shows its potential for simulation in geotechnical engineering.

Feng, Yuan↗
DOE OSTI2025

Uncertainty Quantification for Neutron Shield Using Convolutional Neural Networks

Uncertainty quantification from radiation transport calculations was conducted using a Bayesian inference approach. A surrogate model, using a convolutional neural network, was employed to emulate the neutron fluence, which was simulated with a Monte Carlo radiation transport model. This allowed for a computationally cheap approach to evaluate input parameters and to sample their corresponding posterior probability distributions. Experimental data from the literature were employed to perform uncertainty quantification studies for concrete shields. As a result, the method is a nonintrusive approach that enables studies with multiple input parameters and can be applied to any radiation transport model.

Bayesian inference↗
DOE OSTI2023

Peri-Net-Pro: the neural processes with quantified uncertainty for crack patterns

Abstract This paper develops a deep learning tool based on neural processes (NPs) called the Peri-Net-Pro, to predict the crack patterns in a moving disk and classifies them according to the classification modes with quantified uncertainties. In particular, image classification and regression studies are conducted by means of convolutional neural networks (CNNs) and NPs. First, the amount and quality of the data are enhanced by using peridynamics to theoretically compensate for the problems of the finite element method (FEM) in generating crack pattern images. Second, case studies are conducted with the prototype microelastic brittle (PMB), linear peridynamic solid (LPS), and viscoelastic solid (VES) models obtained by using the peridynamic theory. The case studies are performed to classify the images by using CNNs and determine the suitability of the PMB, LBS, and VES models. Finally, a regression analysis is performed on the crack pattern images with NPs to predict the crack patterns. The regression analysis results confirm that the variance decreases when the number of epochs increases by using the NPs. The training results gradually improve, and the variance ranges decrease to less than 0.035. The main finding of this study is that the NPs enable accurate predictions, even with missing or insufficient training data. The results demonstrate that if the context points are set to the 10th, 100th, 300th, and 784th, the training information is deliberately omitted for the context points of the 10th, 100th, and 300th, and the predictions are different when the context points are significantly lower. However, the comparison of the results of the 100th and 784th context points shows that the predicted results are similar because of the Gaussian processes in the NPs. Therefore, if the NPs are employed for training, the missing information of the training data can be supplemented to predict the results.

Mathematics↗
NASA NTRS1990

Porous-Floating-Gate Field-Effect Transistor

Porous-floating-gate, "vertical" field-effect transistor proposed as programmable analog memory device especially suitable for use in electronic neural networks. Analog value of electrical conductance of device represents synaptic weight (strength of synaptic connection) repeatedly modified by application of suitable writing or erasing voltage. Suited for hardware implementations of massively parallel neural-network architectures for two important reasons: vertical transistor structure requires only two external electrodes, and use of tailored amorphous semiconductors provides choice of very wide range of low conductivity values, dictated by overall power dissipation requirements in massively parallel neural-network circuits.

Thakoor, Anilkumar P.↗
DOE OSTI2026

A hybrid Penman-Monteith and machine learning model for simulating evapotranspiration and its components

Integrating physical processes with machine learning has advanced evapotranspiration (ET) simulation, yet most hybrid models fail to partition total ET into its components: soil evaporation (E) and vegetation transpiration (T). This study introduces Residual Neural Network–Penman–Monteith (RNN-PM), a novel hybrid dual-source ET model designed to overcome this limitation. The model synergizes the physically-based Penman–Monteith framework with three specialized residual neural networks trained to estimate key conductance parameters (canopy conductance, soil surface conductance, and aerodynamic conductance). Furthermore this explicit parameterization allows for the direct partitioning of total ET. Validation at National Ecological Observatory Network (NEON) flux sites using high-frequency partitioned E and T shows that RNN-PM reliably reproduces ET and the transpiration fraction (T/ET). For ET, the model achieves an average Kling–Gupta efficiency (KGE) of 0.89 and a root-mean-square error (RMSE) of 0.55 mm/day; for T/ET, the KGE is 0.87 with an RMSE of 0.06. Furthermore, RNN-PM demonstrates robust generalization, accurately simulating ET and its components well beyond the initial training dataset, even under extreme climatic conditions. This study extended the analysis by comparing the RNN-PM model with seven established dual-source ET models. The results indicate that RNN-PM outperforms both conventional machine learning models and purely physical process-based models in simulating ET components in most cases. Among the purely physical process-based dual-source models, those based on surface temperature decomposition showed improved performance as the leaf area index (LAI) decreased when evaluated against high-frequency ET component datasets. In contrast, the performance of conductance-based dual-source models declined with decreasing LAI. Although purely machine learning-based models can produce relatively accurate simulations of ET components, they often exhibit limited generalization capability, an issue that the RNN-PM model effectively overcomes. Ultimately, the RNN-PM model represents a significant advance in simulating ET components, offering a novel and scalable approach for improving the representation of land–atmosphere interactions in Earth system models.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1996

Design of Power System Architectures for Small Spacecraft Systems

The objective of this research is to perform a trade study on several candidate power system architectures for small spacecrafts to be used in NASA's new millennium program. Three initial candidate architectures have been proposed by NASA and two other candidate architectures have been proposed by Howard University. Howard University is currently conducting the necessary analysis, synthesis, and simulation needed to perform the trade studies and arrive at the optimal power system architecture. Statistical, sensitivity and tolerant studies has been performed on the systems. It is concluded from present studies that certain components such as the series regulators, buck-boost converters and power converters can be minimized while retaining the desired functionality of the overall architecture. This in conjunction with battery scalability studies and system efficiency studies have enabled us to develop more economic architectures. Future studies will include artificial neural networks and fuzzy logic to analyze the performance of the systems. Fault simulation studies and fault diagnosis studies using EMTP and artificial neural networks will also be conducted.

Momoh, James A.↗
DOE OSTI2020

The organic redox transistor for neuromorphic computing

Inspired by the in-memory computing architectures of biological systems, neuromorphic computing using crossbar arrays of artificial synapses based on non-volatile memory (NVM) devices with variable conductances has emerged as a new paradigm to enable massively parallel and ultra-low power computing hardware for data centric applications. Although inference has been demonstrated successfully using crossbars based on a variety of NMV technologies, efficient learning and scaling to large arrays (>10 6 elements) remains a challenge due to the synaptic elements' non-ideal electrical characteristics which degrades ANN accuracy. A further challenge is that in the conductive state memristors draw large currents >μA resulting in significant voltage drops in the interconnect wires and increased probability of failure in scaled arrays. We suggest the organic polymer redox transistor (RT) is an alternate approach that could solve many of these challenges, enabling both inference and parallel outer product updates, as recently demonstrated by Fuller et al. An RT consists of redox-active channel and gate electrodes in contact with a liquid or solid electrolyte. lon insertion through the electrolyte controls the channel electronic conductivity, while electron transfer through an external circuit maintains overall charge neutrality. Unlike a rechargeable battery, in the RT the voltage built-up across the electrolyte is kept to a minimum (typically <100 mV) by using the same material for the gate and channel. Elimination of the voltage offset simplifies integration of the RT into programmable arrays by enabling the use of various selectors. RTs based on inorganic and organic materials have been recently demonstrated with conductance tuning occurring at potentials of just a few mV and hundreds to thousands of linearly and symmetrically programmable conductance states, enabling near ideal accuracy in neural network simulations. Introduced in the 1980's, redox transistors with metallic gate electrodes and organic channel materials, also known as organic electrochemical transistors (OECTs), have been explored for a variety of applications such as chem- and bio-sensing, neural interfaces, and low cost printed circuits. A typical channel material for OECTs is the conducting polymer poly(3,4-ethylenedioxythiophene) doped with poly(styrene sulfonate) (PEDOT:PSS). PEDOT is a p-type semiconducting polymer with mobile positively charged polarons that hop chain-to-chain.

Talin, Albert Alec↗
DOE OSTI2020

A gene desert required for regulatory control of pleiotropic Shox2 expression and embryonic survival

ABSTRACT The Shox2 homeodomain transcriptional regulator is known for its critical functions during mouse embryogenesis, enabling accurate development of limbs, craniofacial structures, neural populations and the cardiac conduction system. At the genomic level, the Shox2 gene is flanked by an extensive gene desert, a continuous non-coding genomic region spanning over 500 kilobases that contains a multitude of evolutionarily conserved elements with predicted cis -regulatory activities. However, the transcriptional enhancer potential of the vast majority of these elements in combination with the biological necessity of the gene desert have not yet been explored. Using transgenic reporter assays in mouse embryos to validate an extensive set of stringent epigenomic enhancer predictions, we identify several novel gene desert enhancers with distinct tissue-specific activities in Shox2 expressing tissues. 4C-seq chromatin conformation capture further uncovers a repertoire of gene desert enhancers with overlapping activities in the proximal limb, in a compartment essential for Shox2 -mediated stylopod formation. Leveraging CRISPR/Cas9 to delete the gene desert region contained in the Shox2 topologically associated domain (TAD), we demonstrate that this complex cis -regulatory platform is essential for embryonic survival and required for control of region-specific Shox2 expression in multiple developing tissues. While transcription of Shox2 in the embryonic limb is only moderately affected by gene desert loss, Shox2 expression in craniofacial and cardiac domains is nearly abolished. In particular, Shox2 transcripts in the sinus venosus (SV) encompassing the sinoatrial node (SAN) were depleted in embryos lacking the gene desert, likely accounting for the embryonic lethality due to Shox2 -dependency of the SAN pacemaker. Finally, we discover a 1.5kb SV enhancer within the deleted gene desert region, which may act as a genomic module controlling the development of the cardiac conduction system. In summary, our results identify a gene desert indispensable for pleiotropic patterning and highlight the importance of these extensive regulatory landscapes for embryonic development and viability.

Abassah-Oppong, Samuel↗
DOE OSTI2026

Knowledge gaps for neuromorphic ionic computing

BACKGROUND Neuromorphic computing, inspired by the human brain’s ability to process information efficiently, represents a transformative approach to computation. In this Review, we explore the emerging field of neuromorphic ionic computing, which leverages ionic conduction and coupling to mimic neural processes, and identify critical knowledge gaps that must be addressed to realize its full potential. A central theme of the discussion is energy efficiency, a challenge that is both a limitation and an opportunity for this technology. Although complementary metal-oxide semiconductor (CMOS)–based neuromorphic technologies have made strides in scaling to billions of neurons and are increasingly applied in artificial intelligence and numerical computing, they remain orders of magnitude behind the human brain in terms of connectivity and energy efficiency. Neuromorphic ionic computing promises to overcome these limitations by leveraging the distinct architectural and operational principles of the brain. Our brains achieve this energy efficiency by combining several key features: using the same network elements to store and process information; using an incredibly complex and massively interconnected three-dimensional (3D) network of locally active elements that enables sparsity, robustness in the presence of noise, adaptation, and life-long learning; computing at comparatively low voltage and frequency; and last, taking advantage of a plethora of ions and small molecules as information carriers. Here, we propose that ionic computing systems can take advantage of similar features to achieve substantial gains in energy efficiency. ADVANCES Since the first reports of neuromorphic ionic behavior in nanofluidic channels, we have witnessed an explosion of reports that used ionic devices to produce synaptomimetic behaviors. However, achieving the goals of ionic computing requires not only implementation of much more sophisticated device functionality but also overcoming fundamental barriers in materials science, device architecture, and system integration. Current ionic devices, even those incorporating state-of-the-art materials, still suffer from limited functionality and stability, which restrict their performance and increase energy demands. Developing new materials with enhanced ionic properties is essential to overcome these limitations. Similarly, the design of neuromorphic devices must evolve to leverage the particular advantages of ionic processes. Existing architectures often follow a single-information-carrier logic of conventional electronics or are constructed of mesoscale fluidics, failing to capitalize on the energy-efficient mechanisms inherent to ionic systems or implement the multiple-information-carrier paradigm. Current neuromorphic chips focus on large-scale networks of analog memory elements based on mechanisms such as charge trap (flash), filamentary, phase change, or spin, which are built on top of a network of artificial CMOS neurons. Although such prototype networks have achieved impressive performance, it is difficult to envision how they can implement the key features such as massive connectivity, sophisticated plasticity, adaptability, sparsity, and “multichromatic” computing. Although small-scale devices have demonstrated promising results, integrating them, maintaining energy efficiency, and implementing temperature control as systems grow in complexity and size to computationally relevant scale remain major hurdles. Furthermore, interfacing neuromorphic ionic devices with existing computing technologies presents technical and conceptual challenges that will require innovative approaches that combine insights from neuroscience, materials science, and engineering. OUTLOOK Despite these challenges, the potential impact of neuromorphic ionic computing is profound with potential applications ranging from artificial intelligence to robotics and beyond. We also argue that neuromorphic ionic computing systems should not, at least in the beginning, compete with CMOS technologies but rather should focus on applications that require extreme energy efficiency with chemical and/or biological compatibility, such as biomedical applications (for example, brain-computer interfaces), environmental monitoring, and agricultural and food applications. Ultimately, this Review highlights the crucial role of interdisciplinary collaboration in advancing the field. Neuromorphic ionic computing is not merely a technological innovation; it represents a substantial step toward sustainable computation, aligning with the growing demand for energy-conscious solutions in a world that is increasingly reliant on data and computation.

Neuromorphic↗
NASA NTRS1990

Investigation of automated task learning, decomposition and scheduling

The details and results of research conducted in the application of neural networks to task planning and decomposition are presented. Task planning and decomposition are operations that humans perform in a reasonably efficient manner. Without the use of good heuristics and usually much human interaction, automatic planners and decomposers generally do not perform well due to the intractable nature of the problems under consideration. The human-like performance of neural networks has shown promise for generating acceptable solutions to intractable problems such as planning and decomposition. This was the primary reasoning behind attempting the study. The basis for the work is the use of state machines to model tasks. State machine models provide a useful means for examining the structure of tasks since many formal techniques have been developed for their analysis and synthesis. It is the approach to integrate the strong algebraic foundations of state machines with the heretofore trial-and-error approach to neural network synthesis.

Livingston, David L.↗
DOE OSTI2020

Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref. Freno and Carlberg (2019) to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) method constructs a regression model that maps features – which comprise error indicators that are derived from standard a posteriori error-quantification techniques – to a random variable for the approximate-solution error at each time instance. The proposed framework considers a wide range of candidate features, regression methods, and additive noise models. We consider primarily recursive regression techniques developed for time-series modeling, including both classical time-series models (e.g., autoregressive models) and recurrent neural networks (RNNs), but also analyze standard non-recursive regression techniques (e.g., feed-forward neural networks) for comparative purposes. Finally, numerical experiments conducted on multiple benchmark problems illustrate that the long short-term memory (LSTM) neural network, which is a type of RNN, outperforms other methods and yields substantial improvements in error predictions over traditional approaches.

42 ENGINEERING↗
DOE OSTI2021

Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

Advection-dispersion equations (ADEs) are commonly used to describe transport phenomena in porous media. Even though mature discretization-based numerical methods for ADEs exist, some challenges still remain, especially when it comes to solving advection-dominated forward ADEs and diffusion-dominated backward ADEs. The latter problem usually arises in the source identification context and leads to numerically unstable grid-based solutions that require a form of regularization or should be treated as an inverse problem that is computationally more expensive because it requires solving the forward problem multiple times. In this study, we propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving coupled ADE and Darcy flow equations with space-dependent hydraulic conductivity. In this approach, the hydraulic conductivity, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that the conductivity field is given by its values on a grid, and we use these values to train the conductivity DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one- and two-dimensional forward ADE problems, where its performance for various P\'{e}clet numbers ($Pe$) is compared with the analytical and numerical solutions. We find that the PINN method is accurate with errors of less than 1\% and outperforms some conventional discretization-based methods for $Pe$ larger that 100. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5\% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50\% in the considered cases) the accuracy of the PINN solution of the backward ADE.

He, Qizhi↗