Simulating molecules and atomic systems at quantum accuracy is a grand challenge for science in the 21st century. Quantum-accurate simulations would enable the design of new medicines and the discovery of new materials. The defining problem in this challenge is that quantum calculations on large molecules, like proteins or DNA, are fundamentally impossible with current algorithms. In this work, we explore a range of different methods that aim to make large, quantum-accurate simulations possible. We show that using advanced classical models, we can accurately simulate ion channels, an important biomolecular system. We show how advanced classical models can be implemented in an exascale-ready software package. Lastly, we show how machine learning can learn the laws of quantum mechanics from data and enable quantum electronic structure calculations on thousands of atoms, a feat that is impossible for current algorithms. Altogether, this work shows that combining advances in physics models, computing, and machine learning, we are moving closer to the reality of accurately simulating our molecular world.
This report details work that was completed to address the Fiscal Year 2022 Advanced Science and Technology (AS&T) Laboratory Directed Research and Development (LDRD) call for “AI-enhanced Co-Design of Next Generation Microelectronics.” This project required concurrent contributions from the fields of 1) materials science, 2) devices and circuits, 3) physics of computing, and 4) algorithms and system architectures. During this project, we developed AI-enhanced circuit design methods that relied on reinforcement learning and evolutionary algorithms. The AI-enhanced design methods were tested on neuromorphic circuit design problems that have real-world applications related to Sandia’s mission needs. The developed methods enable the design of circuits, including circuits that are built from emerging devices, and they were also extended to enable novel device discovery. We expect that these AI-enhanced design methods will accelerate progress towards developing next-generation, high-performance neuromorphic computing systems.
Applied Mathematical Modelling
In this work we infer the underlying distribution on pore radius in human cortical bone samples using ultrasonic attenuation data. We first discuss how to formulate polydisperse attenuation models using a probabilistic approach and the Waterman Truell model for scattering attenuation. We then compare the Independent Scattering Approximation and the higher-order Waterman Truell models’ forward predictions for total attenuation in polydisperse samples. Following this, we formulate an inverse problem under the Prohorov Metric Framework coupled with variational regularization to stabilize this inverse problem. We then use experimental attenuation data taken from human cadaver samples and solve inverse problems resulting in nonparametric estimates of the probability density function on pore radius. We compare these estimates to the “true” microstructure of the bone samples determined via microCT imaging. We find that our methodology allows us to reliably estimate the underlying microstructure of the bone from attenuation data.
This report documents the progress made in simulating the HERMES-III Magnetically Insulated Transmission Line (MITL) and courtyard with EMPIRE and ITS. This study focuses on the shots that were taken during the months of June and July of 2019 performed with the new MITL extension. There were a few shots where there was dose mapping of the courtyard, 11132, 11133, 11134, 11135, 11136, and 11146. This report focuses on these shots because there was full data return from the MITL electrical diagnostics and the radiation dose sensors in the courtyard. The comparison starts with improving the processing of the incoming voltage into the EMPIRE simulation from the experiment. The currents are then compared at several location along the MITL. The simulation results of the electrons impacting the anode are shown. The electron impact energy and angle is then handed off to ITS which calculates the dose on the faceplate and locations in the courtyard and they are compared to experimental measurements. ITS also calculates the photons and electrons that are injected into the courtyard, these quantities are then used by EMPIRE to calculated the photon and electron transport in the courtyard. The details for the algorithms used to perform the courtyard simulations are presented as well as qualitative comparisons of the electric field, magnetic field, and the conductivity in the courtyard. Because of the computational burden of these calculations the pressure was reduce in the courtyard to reduce the computational load. The computation performance is presented along with suggestion on how to improve both the computational performance as well as the algorithmic performance. Some of the algorithmic changed would reduce the accuracy of the models and detail comparison of these changes are left for a future study. As well as, list of code improvements there is also a list of suggested experimental improvements to improve the quality of the data return.
This report documents the Resilience Enhancements through Deep Learning Yields (REDLY) project, a three-year effort to improve electrical grid resilience by developing scalable methods for system operators to protect the grid against threats leading to interrupted service or physical damage. The computational complexity and uncertain nature of current real-world contingency analysis presents significant barriers to automated, real-time monitoring. While there has been a significant push to explore the use of accurate, high-performance machine learning (ML) model surrogates to address this gap, their reliability is unclear when deployed in high-consequence applications such as power grid systems. Contemporary optimization techniques used to validate surrogate performance can exploit ML model prediction errors, which necessitates the verification of worst-case performance for the models.
Predictive design of REHEDS experiments with radiation-hydrodynamic simulations requires knowledge of material properties (e.g. equations of state (EOS), transport coefficients, and radiation physics). Interpreting experimental results requires accurate models of diagnostic observables (e.g. detailed emission, absorption, and scattering spectra). In conditions of Local Thermodynamic Equilibrium (LTE), these material properties and observables can be pre-computed with relatively high accuracy and subsequently tabulated on simple temperature-density grids for fast look-up by simulations. When radiation and electron temperatures fall out of equilibrium, however, non-LTE effects can profoundly change material properties and diagnostic signatures. Accurately and efficiently incorporating these non-LTE effects has been a longstanding challenge for simulations. At present, most simulations include non-LTE effects by invoking highly simplified inline models. These inline non-LTE models are both much slower than table look-up and significantly less accurate than the detailed models used to populate LTE tables and diagnose experimental data through post-processing or inversion. Because inline non-LTE models are slow, designers avoid them whenever possible, which leads to known inaccuracies from using tabular LTE. Because inline models are simple, they are inconsistent with tabular data from detailed models, leading to ill-known inaccuracies, and they cannot generate detailed synthetic diagnostics suitable for direct comparisons with experimental data. This project addresses the challenge of generating and utilizing efficient, accurate, and consistent non-equilibrium material data along three complementary but relatively independent research lines. First, we have developed a relatively fast and accurate non-LTE average-atom model based on density functional theory (DFT) that provides a complete set of EOS, transport, and radiative data, and have rigorously tested it against more sophisticated first-principles multi-atom DFT models, including time-dependent DFT. Next, we have developed a tabular scheme and interpolation methods that compactly capture non-LTE effects for use in simulations and have implemented these tables in the GORGON magneto-hydrodynamic (MHD) code. Finally, we have developed post-processing tools that use detailed tabulated non-LTE data to directly predict experimental observables from simulation output.
When modeling complex physical systems with advanced dynamics, such as shocks and singularities, many classic methods for solving partial differential equations can return inaccurate or unusable results. One way to resolve these complex dynamics is through r-adaptive refinement methods, in which a fixed number of mesh points are shifted to areas of high interest. The mesh refinement map can be found through the solution of the Monge-Ampére equation, a highly nonlinear partial differential equation. Due to its nonlinearity, the numerical solution of the Monge-Ampére equation is nontrivial and has previously required computationally expensive methods. In this report, we detail our novel optimization-based, multigrid-enabled solver for a low-order finite element approximation of the Monge-Ampére equation. This fast and scalable solver makes r-adaptive meshing more readily available for problems related to large-scale optimal design. Beyond mesh adaptivity, our report discusses additional applications where our fast solver for the Monge-Ampére equation could be easily applied.
Plasma physics simulations are vital for a host of Sandia mission concerns, for fundamental science, and for clean energy in the form of fusion power. Sandia's most mature plasma physics simulation capabilities come in the form of particle-in-cell (PIC) models and magnetohydrodynamics (MHD) models. MHD models for a plasma work well in denser plasma regimes when there is enough material that the plasma approximates a fluid. PIC models, on the other hand, work well in lower-density regimes, in which there is not too much to simulate; error in PIC scales as the square root of the number of particles, making high-accuracy simulations expensive. Real-world applications, however, almost always involve a transition region between the high-density regimes where MHD is appropriate, and the low-density regimes for PIC. In such a transition region, a direct discretization of Vlasov is appropriate. Such discretizations come with their own computational costs, however; the phase-space mesh for Vlasov can involve up to six dimensions (seven if time is included), and to apply appropriate homogeneous boundary conditions in velocity space requires meshing a substantial padding region to ensure that the distribution remains sufficiently close to zero at the velocity boundaries. Moreover, for collisional plasmas, the right-hand side of the Vlasov equation is a collision operator, which is non-local in velocity space, and which may dominate the cost of the Vlasov solver. The present LDRD project endeavors to develop modern, foundational tools for the development of continuum-kinetic Vlasov solvers, using the discontinuous Petrov-Galerkin (DPG) methodology, for discretization of Vlasov, and machine-learning (ML) models to enable efficient evaluation of collision operators. DPG affords several key advantages. First, it has a built-in, robust error indicator, allowing us to adapt the mesh in a very natural way, enabling a coarse velocity-space mesh near the homogeneous boundaries, and a fine mesh where the solution has fine features. Second, it is an inherently high-order, high-intensity method, requiring extra local computations to determine so-called optimal test functions, which makes it particularly suited to modern hardware in which floating-point throughput is increasing at a faster rate than memory bandwidth. Finally, DPG is a residual-minimizing method, which enables high-accuracy computation: in typical cases, the method delivers something very close to the $L^2$ projection of the exact solution. Meanwhile, the ML-based collision model we adopt affords a cost structure that scales as the square root of a standard direct evaluation. Moreover, we design our model to conserve mass, momentum, and energy by construction, and our approach to training is highly flexible, in that it can incorporate not only synthetic data from direct-simulation Monte Carlo (DSMC) codes, but also experimental data. We have developed two DPG formulations for Vlasov-Poisson: a time-marching, backward-Euler discretization and a space-time discretization. We have conducted a number of numerical experiments to verify the approach in a 1D1V setting. In this report, we detail these formulations and experiments. We also summarize some new theoretical results developed as part of this project (published as papers previously): some new analysis of DPG for the convection-reaction problem (of which the Vlasov equation is an instance), a new exponential integrator for DPG, and some numerical exploration of various DPG-based time-marching approaches to the heat equation. As part of this work, we have contributed extensively to the Camellia open-source library; we also describe the new capabilities and their usage. We have also developed a well-documented methodology for single-species collision operators, which we applied to argon and demonstrated with numerical experiments. We summarize those results here, as well as describing at a high level a design extending the methodology to multi-species operators. We have released a new open-source library, MLC, under a BSD license; we include a summary of its capabilities as well.
Analog computing has been widely proposed to improve the energy efficiency of multiple important workloads including neural network operations, and other linear algebra kernels. To properly evaluate analog computing and explore more complex workloads such as systems consisting of multiple analog data paths, system level simulations are required. Moreover, prior work on system architectures for analog computing often rely on custom simulators creating signficant additional design effort and complicating comparisons between different systems. To remedy these issues, this report describes the design and implementation of a flexible tile-based analog accelerator element for the Structural Simulation Toolkit (SST). The element focuses on heavily on the tile controller—an often neglected aspect of prior work—that is sufficiently versatile to simulate a wide range of different tile operations including neural network layers, signal processing kernels, and generic linear algebra operations without major constraints. The tile model also interoperates with existing SST memory and network models to reduce the overall development load and enable future simulation of heterogeneous systems with both conventional digital logic and analog compute tiles. Finally, both the tile and array models are designed to easily support future extensions as new analog operations and applications that can benefit from analog computing are developed.
Journal of Advances in Modeling Earth Systems
For decades, Arctic temperatures have increased twice as fast as average global temperatures. As a first step towards quantifying parametric uncertainty in Arctic climate, we performed a variance-based global sensitivity analysis (GSA) using a fully-coupled, ultra-low resolution (ULR) configuration of version 1 of the U.S. Department of Energy’s Energy Exascale Earth System Model (E3SMv1). Specifically, we quantified the sensitivity of six quantities of interest (QOIs), which characterize changes in Arctic climate over a 75 year period, to uncertainties in nine model parameters spanning the sea ice, atmosphere and ocean components of E3SMv1. Sensitivity indices for each QOI were computed with a Gaussian process emulator using 139 random realizations of the random parameters and fixed pre-industrial forcing. Uncertainties in the atmospheric parameters in the CLUBB (Cloud Layers Unified by Binormals) scheme were found to have the most impact on sea ice status and the larger Arctic climate. Our results demonstrate the importance of conducting sensitivity analyses with fully coupled climate models. The ULR configuration makes such studies computationally feasible today due to its low computational cost. When advances in computational power and modeling algorithms enable the tractable use of higher-resolution models, our results will provide a baseline that can quantify the impact of model resolution on the accuracy of sensitivity indices. Moreover, the confidence intervals provided by our study, which we used to quantify the impact of the number of model evaluations on the accuracy of sensitivity estimates, have the potential to inform the computational resources needed for future sensitivity studies.
Journal of Computational Physics
Kinetic gas dynamics in rarefied and moderate-density regimes have complex behavior associated with collisional processes. These processes are generally defined by convolution integrals over a high-dimensional space (as in the Boltzmann operator), or require evaluating complex auxiliary variables (as in Rosenbluth potentials in Fokker-Planck operators) that are challenging to implement and computationally expensive to evaluate. In this work, we develop a data-driven neural network model that augments a simple and inexpensive BGK collision operator with a machine-learned correction term, which improves the fidelity of the simple operator with a small overhead to overall runtime. The composite collision operator has a tunable fidelity and, in this work, is trained using and tested against a direct-simulation Monte-Carlo (DSMC) collision operator.
Physics of Plasmas
An approach to numerically modeling relativistic magnetrons, in which the electrons are represented with a relativistic fluid, is described. A principal effect in the operation of a magnetron is space-charge-limited (SCL) emission of electrons from the cathode. We have developed an approximate SCL emission boundary condition for the fluid electron model. This boundary condition prescribes the flux of electrons as a function of the normal component of the electric field on the boundary. We show the results of a benchmarking activity that applies the fluid SCL boundary condition to the one-dimensional Child–Langmuir diode problem and a canonical two-dimensional diode problem. Simulation results for a two-dimensional A6 magnetron are then presented. Computed bunching of the electron cloud occurs and coincides with significant microwave power generation. Numerical convergence of the solution is considered. Sharp gradients in the solution quantities at the diocotron resonance, spanning an interval of three to four grid cells in the most well-resolved case, are present and likely affect convergence.
Computer Methods in Applied Mechanics and Engineering
In this article, we present a general methodology to combine the Discontinuous Petrov–Galerkin (DPG) method in space and time in the context of methods of lines for transient advection–reaction problems. We first introduce a semidiscretization in space with a DPG method redefining the ideas of optimal testing and practicality of the method in this context. Then, we apply the recently developed DPG-based time-marching scheme, which is of exponential-type, to the resulting system of Ordinary Differential Equations (ODEs). Further, we also discuss how to efficiently compute the action of the exponential of the matrix coming from the space semidiscretization without assembling the full matrix. Finally, we verify the proposed method for 1D+time advection–reaction problems showing optimal convergence rates for smooth solutions and more stable results for linear conservation laws comparing to the classical exponential integrators.
The Spent Fuel and Waste Science and Technology (SFWST) Campaign of the U.S. Department of Energy (DOE) Office of Nuclear Energy (NE), Office of Fuel Cycle Technology (FCT) is conducting research and development (R&D) on geologic disposal of spent nuclear fuel (SNF) and high-level nuclear waste (HLW). Two high priorities for SFWST disposal R&D are design concept development and disposal system modeling. These priorities are directly addressed in the SFWST Geologic Disposal Safety Assessment (GDSA) control account, which is charged with developing a geologic repository system modeling and analysis capability, and the associated software, GDSA Framework, for evaluating disposal system performance for nuclear waste in geologic media. GDSA Framework is supported by SFWST Campaign and its predecessor the Used Fuel Disposition (UFD) campaign.
Geoscientific Model Development (Online)
Abstract. Advection of trace species, or tracers, also called tracer transport, in models of the atmosphere and other physical domains is an important and potentially computationally expensive part of a model's dynamical core. Semi-Lagrangian (SL) advection methods are efficient because they permit a time step much larger than the advective stability limit for explicit Eulerian methods without requiring the solution of a globally coupled system of equations as implicit Eulerian methods do. Thus, to reduce the computational expense of tracer transport, dynamical cores often use SL methods to advect tracers. The class of interpolation semi-Lagrangian (ISL) methods contains potentially extremely efficient SL methods. We describe a finite-element ISL transport method that we call the interpolation semi-Lagrangian element-based transport (Islet) method, such as for use with atmosphere models discretized using the spectral element method. The Islet method uses three grids that share an element grid: a dynamics grid supporting, for example, the Gauss–Legendre–Lobatto basis of degree three; a physics parameterizations grid with a configurable number of finite-volume subcells per element; and a tracer grid supporting use of Islet bases with particular basis again configurable. This method provides extremely accurate tracer transport and excellent diagnostic values in a number of verification problems.
Combustion Science and Technology
Tabulated chemistry models are widely used to simulate large-scale turbulent fires in applications including energy generation and fire safety. Tabulation via piecewise Cartesian interpolation suffers from the curse-of-dimensionality, leading to a prohibitive exponential growth in parameters and memory usage as more dimensions are considered. Artificial neural networks (ANNs) have attracted attention for constructing surrogates for chemistry models due to their ability to perform high-dimensional approximation. However, due to well-known pathologies regarding the realization of suboptimal local minima during training, in practice they do not converge and provide unreliable accuracy. Partition of unity networks (POUnets) are a recently introduced family of ANNs which preserve notions of convergence while performing high-dimensional approximation, discovering a mesh-free partition of space which may be used to perform optimal polynomial approximation. In this work, we assess their performance with respect to accuracy and model complexity in reconstructing unstructured flamelet data representative of nonadiabatic pool fire models. Our results show that POUnets can provide the desirable accuracy of classical spline-based interpolants with the low memory footprint of traditional ANNs while converging faster to significantly lower errors than ANNs. For example, we observe POUnets obtaining target accuracies in two dimensions with 40 to 50 times less memory and roughly double the compression in three dimensions. We also address the practical matter of efficiently training accurate POUnets by studying convergence over key hyperparameters, the impact of partition/basis formulation, and the sensitivity to initialization.
PyApprox is a Python-based one-stop-shop for probabilistic analysis of scientific numerical models. Easy to use and extendable tools are provided for constructing surrogates, sensitivity analysis, Bayesian inference, experimental design, and forward uncertainty quantification. The algorithms implemented represent the most popular methods for model analysis developed over the past two decades, including recent advances in multi-fidelity approaches that use multiple model discretizations and/or simplified physics to significantly reduce the computational cost of various types of analyses. Simple interfaces are provided for the most commonly-used algorithms to limit a user’s need to tune the various hyper-parameters of each algorithm. However, more advanced work flows that require customization of hyper-parameters is also supported. An extensive set of Benchmarks from the literature is also provided to facilitate the easy comparison of different algorithms for a wide range of model analyses. This paper introduces PyApprox and its various features, and presents results demonstrating the utility of PyApprox on a benchmark problem modeling the advection of a tracer in ground water.
Numerical Linear Algebra with Applications
We present a polynomial preconditioner for solving large systems of linear equations. The polynomial is derived from the minimum residual polynomial (the GMRES polynomial) and is more straightforward to compute and implement than many previous polynomial preconditioners. Our current implementation of this polynomial using its roots is naturally more stable than previous methods of computing the same polynomial. We implement further stability control using added roots, and this allows for high degree polynomials. We discuss the effectiveness and challenges of root-adding and give an additional check for stability. In this article, we study the polynomial preconditioner applied to GMRES; however it could be used with any Krylov solver. This polynomial preconditioning algorithm can dramatically improve convergence for some problems, especially for difficult problems, and can reduce dot products by an even greater margin.
In many recent applications, particularly in the field of atom-centered descriptors for interatomic potentials, tensor products of spherical harmonics have been used to characterize complex atomic environments. When coupled with a radial basis, the atomic cluster expansion (ACE) basis is obtained. However, symmetrization with respect to both rotation and permutation results in an overcomplete set of ACE descriptors with linear dependencies occurring within blocks of functions corresponding to particular generalized Wigner symbols. All practical applications of ACE employ semi-numerical constructions to generate a complete, fully independent basis. While computationally tractable, the resultant basis cannot be expressed analytically, is susceptible to numerical instability, and thus has limited reproducibility. Here we present a procedure for generating explicit analytic expressions for a complete and independent set of ACE descriptors. The procedure uses a coupling scheme that is maximally symmetric w.r.t. permutation of the atoms, exposing the permutational symmetries of the generalized Wigner symbols, and yields a permutation-adapted rotationally and permutationally invariant basis (PA-RPI ACE). Theoretical support for the approach is presented, as well as numerical evidence of completeness and independence. A summary of explicit enumeration of PA-RPI functions up to rank 6 and polynomial degree 32 is provided. The PA-RPI blocks corresponding to particular generalized Wigner symbols may be either larger or smaller than the corresponding blocks in the simpler rotationally invariant basis. Finally, we demonstrate that basis functions of high polynomial degree persist under strong regularization, indicating the importance of not restricting the maximum degree of basis functions in ACE models a priori.
The causal structure of a simulation is a major determinant of both its character and behavior, yet most methods we use to compare simulations focus only on simulation outputs. We introduce a method that combines graphical representation with information theoretic metrics to quantitatively compare the causal structures of models. The method applies to agent-based simulations as well as system dynamics models and facilitates comparison within and between types. Comparing models based on their causal structures can illuminate differences in assumptions made by the models, allowing modelers to (1) better situate their models in the context of existing work, including highlighting novelty, (2) explicitly compare conceptual theory and assumptions to simulated theory and assumptions, and (3) investigate potential causal drivers of divergent behavior between models. We demonstrate the method by comparing two epidemiology models at different levels of aggregation.
Journal of Applied Physics
The selective amorphization of SiGe in Si/SiGe nanostructures via a 1 MeV Si + implant was investigated, resulting in single-crystal Si nanowires (NWs) and quantum dots (QDs) encapsulated in amorphous SiGe fins and pillars, respectively. The Si NWs and QDs are formed during high-temperature dry oxidation of single-crystal Si/SiGe heterostructure fins and pillars, during which Ge diffuses along the nanostructure sidewalls and encapsulates the Si layers. The fins and pillars were then subjected to a 3 × 10 15 ions/cm 2 1 MeV Si + implant, resulting in the amorphization of SiGe, while leaving the encapsulated Si crystalline for larger, 65-nm wide NWs and QDs. Interestingly, the 26-nm diameter Si QDs amorphize, while the 28-nm wide NWs remain crystalline during the same high energy ion implant. This result suggests that the Si/SiGe pillars have a lower threshold for Si-induced amorphization compared to their Si/SiGe fin counterparts. However, Monte Carlo simulations of ion implantation into the Si/SiGe nanostructures reveal similar predicted levels of displacements per cm 3 . Molecular dynamics simulations suggest that the total stress magnitude in Si QDs encapsulated in crystalline SiGe is higher than the total stress magnitude in Si NWs, which may lead to greater crystalline instability in the QDs during ion implant. The potential lower amorphization threshold of QDs compared to NWs is of special importance to applications that require robust QD devices in a variety of radiation environments.
IEEE International Conference on Plasma Science (ICOPS)
A semi-analytic fluid model has been developed for characterizing relativistic electron emission across a warm diode gap. Here we demonstrate the use of this model in (i) verifying multi-fluid codes in modeling compressible relativistic electron flows (the EMPIRE-Fluid code is used as an example; see also Ref. 1), (ii) elucidating key physics mechanisms characterizing the influence of compressibility and relativistic injection speed of the electron flow, and (iii) characterizing the regimes over which a fluid model recovers physically reasonable solutions.
The objective of this milestone was to finish integrating GenTen tensor software with combustion application Pele using the Ascent in situ analysis software, partnering with the ALPINE and Pele teams. Also, to demonstrate the usage of the tensor analysis as part of a combustion simulation.
International Journal for Numerical Methods in Engineering
We present an adaptive algorithm for constructing surrogate models of multi-disciplinary systems composed of a set of coupled components. With this goal we introduce “coupling” variables with a priori unknown distributions that allow surrogates of each component to be built independently. Once built, the surrogates of the components are combined to form an integrated-surrogate that can be used to predict system-level quantities of interest at a fraction of the cost of the original model. The error in the integrated-surrogate is greedily minimized using an experimental design procedure that allocates the amount of training data, used to construct each component-surrogate, based on the contribution of those surrogates to the error of the integrated-surrogate. The multi-fidelity procedure presented is a generalization of multi-index stochastic collocation that can leverage ensembles of models of varying cost and accuracy, for one or more components, to reduce the computational cost of constructing the integrated-surrogate. Extensive numerical results demonstrate that, for a fixed computational budget, our algorithm is able to produce surrogates that are orders of magnitude more accurate than methods that treat the integrated system as a black-box.
Physics-informed machine learning (PIML) has emerged as a promising new approach for simulating complex physical and biological systems that are governed by complex multiscale processes for which some data are also available. In some instances, the objective is to discover part of the hidden physics from the available data, and PIML has been shown to be particularly effective for such problems for which conventional methods may fail. Unlike commercial machine learning where training of deep neural networks requires big data, in PIML big data are not available. Instead, we can train such networks from additional information obtained by employing the physical laws and evaluating them at random points in the space-time domain. Such PIML integrates multimodality and multifidelity data with mathematical models, and implements them using neural networks or graph networks. Here, we review some of the prevailing trends in embedding physics into machine learning, using physics-informed neural networks (PINNs) based primarily on feed-forward neural networks and automatic differentiation. For more complex systems or systems of systems and unstructured data, graph neural networks (GNNs) present some distinct advantages, and here we review how physics-informed learning can be accomplished with GNNs based on graph exterior calculus to construct differential operators; we refer to these architectures as physics-informed graph networks (PIGNs). We present representative examples for both forward and inverse problems and discuss what advances are needed to scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering problems.