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The solution of partial differential equations which describe physical phenomena by the use of coordinate transformations is described. The constraints of the problems are stated in geometric terms and include boundary constraints, uniformity constraints, and internal constraints. Algebraic mesh generations are very satisfactory for these constraints.
The development of multiprocessor simulations from a serial set of ordinary differential equations describing a physical system is described. Degrees of parallelism (i.e., coupling between the equations) and their impact on parallel processing are discussed. The problem of identifying computational parallelism within sets of closely coupled equations that require the exchange of current values of variables is described. A technique is presented for identifying this parallelism and for partitioning the equations for parallel solution on a multiprocessor. An algorithm which packs the equations into a minimum number of processors is also described. The results of the packing algorithm when applied to a turbojet engine model are presented in terms of processor utilization.
This paper deals with the development of multiprocessor simulations from a serial set of ordinary differential equations describing a physical system. The identification of computational parallelism within the model equations is discussed. A technique is presented for identifying this parallelism and for partitioning the equations for parallel solution on a multiprocessor. Next, an algorithm which packs the equations into a minimum number of processors is described. The results of applying the packing algorithm to a turboshaft engine model are presented.
Algorithm developed to identify parallelism in set of coupled ordinary differential equations that describe physical system and to divide set into parallel computational paths, along with parts of solution proceeds independently of others during at least part of time. Path-identifying algorithm creates number of paths consisting of equations that must be computed serially and table that gives dependent and independent arguments and "can start," "can end," and "must end" times of each equation. "Must end" time used subsequently by packing algorithm.
This paper discusses the development and performance of the X-33 Aerospike Engine RealTime Model. This model was developed for the purposes of control law development, six degree-of-freedom trajectory analysis, vehicle system integration testing, and hardware-in-the loop controller verification. The Real-Time Model uses time-step marching solution of non-linear differential equations representing the physical processes involved in the operation of a liquid propellant rocket engine, albeit in a simplified form. These processes include heat transfer, fluid dynamics, combustion, and turbomachine performance. Two engine models are typically employed in order to accurately model maneuvering and the powerpack-out condition where the power section of one engine is used to supply propellants to both engines if one engine malfunctions. The X-33 Real-Time Model is compared to actual hot fire test data and is been found to be in good agreement.
Integration of diverse datasets on the Moon may render some paradigms of lunar science either better-defended or vulnerable. We will consider three paradigms commonly used for understanding the processes of lunar regolith evolution in light of new and accumulated data. Our premise is that all data-sets should converge to a single interpretation if a concept or model is to be accepted as a paradigm. If a convergence is lacking, the paradigm needs fresh scrutiny. SteadyState: Lunar regolith evolution is currently understood in terms of comminution, agglutination, and replenishment as described by McKay and coworkers). Briefly, the model envisages continued micrometeoritic bombardment to comminute exposed soil particles to finer sizes while continued agglutination consumes finer sizes to produce larger constructional particles. Eventually, a balance between these two opposing processes achieves a steady state; soils at steady state maintain their mean grain size (M(sub z)). Episodic higher-energy impacts excavate fresh coarse material from below the soil cover, disturb the steady state, and restart the process to achieve a new steady state. It follows that the thickness of the regolith at any site would control the frequency of replenishment; indeed, the thickness of the regolith at Apollo landing sites was predicted by McKay et al. from the average M(sub z) of local soils. However, replenishment may come also from disintegrating boulders and cobbles at the lunar surface, and rates of comminution and agglutination may depend on the properties of target material. Regression between M(sub z) and I(sub s)/Fe(sup 0) (a measure of maturity or total surface exposure) of Apollo soils at different sites shows the following relations and estimated M(sub z) at a high maturity of I(sub s)/Fe(sup 0)= 100. It is possible that Apollo 12 and 15 sites have the thickest regolith and the Apollo 16 site has the thinnest. It is also possible that Apollo 12 and 15 basalts are comminuted faster than Apollo 16 highland rocks and Apollo 14 and 17 soils are products of mixed parentage. If a soil becomes continually finer as it matures until agglutination catches up, and if comminution is differential-dependent on the physical properties of the constituents, then the composition of the bulk soil has to match the composition of some "fulcrum" grain size fraction, say X Grain size fractions >X and <X will complement each other; their mass balance is the bulk soil. It appears that the 10-20-micron size fraction may be the fulcrum. In general, trace-element chemistry and IR reflectance spectra of this size fraction are closest to that of the bulk soil, regardless of maturity that is surprising. Disaggregated products of regolith breccias may also show similar relationships. If the 10-20 gm is the fulcrum (i.e., X as above) for many soil properties (e.g., major element composition, FMR, solar-wind-implanted elements), then this may be the ultimate mean grain size of lunar soils at steady state. However, different properties of soils may find steady states at different grain size fractions. The steady state of solar-wind-implanted elements, on the other hand, will climb up the grain-size scale as agglutinates transfer surface-correlated components into volume correlated components until a saturation level is reached or the rates of replenishment and implantation become equal. The same will be the case with vapor-deposited reduced metals as they too are incorporated inside constructional particles. Properties that are directly affected by soil-maturation processes will thus have different pathways of achieving steady states. Maturity, i.e., cumulative surface exposure, of lunar soils is best quantified by the amount of nanophase superparamagnetic Fe(sup 0) (np-Fe(sup 0)) normalized to Fe content (=I(sub s)/Fe(sup 0). The majority consensus (paradigm?) for the production of np-Fe(sup 0) is associated with the production of agglutinates. Because large doses of solar-wind H are implanted in all lunar soils upon exposure, any melting (e.g., during agglutinate production) triggers a chemical reduction of Fe-bearing minerals resulting in np-Fe(sup 0) production. The quantity of np-Fe(sup 0) is thus dependent on melting events, (i.e., exposure), and limited by the Fe content of the soil. All freshly produced np-Fe(sup 0) resides in agglutinitic glass, as new TEM images show. Apparently, the correction procedure developed by Lucey et al. to estimate the Fe content of the lunar surface from IR-reflectance spectra depends on accepting the above. However, the process of producing np-Fe(sup 0) may be physical rather than chemical. All np-Fe(sup 0) could be deposits from a vapor produced by micrometeoritic impact on lunar soils. If metal-O bonds in target phases are broken, O being "most volatile" will escape leaving an O-deficient vapor to facilitate the production of np-Fe(sup 0). If so, the quantity of np-Fe(sup 0) is dependent on the vaporizing events, (i.e., exposure), and limited by the efficiency of breaking metal-O bonds and the escape of 0. To the extent that strengths of metal-O bonds are dependent on the local crystal field, production of np-Fe(sup 0) may be limited by the mineral composition of target soils and not by their total Fe content. According to this model, vapor-deposited np-Fe(sup 0) should be found at any retentive sites on lunar soil grains. Indeed, TEM images show np-Fe(sup 0) on plagioclase and ilmenite. Incorporation of such pre-irradiated np-Fe(sup 0)-bearing grains into agglutinates may account for eventual increased emplacement of np-Fe(sup 0) in agglutinates. Such a paradigm shift in understanding the origin of np-Fe(sup 0) will raise questions ranging from the unquestionable use of Is/FeO as the universal maturity parameter of lunar soils to global elemental maps of the Moon from remote-sensing data. Additional information is contained in the original.
Physical parameterizations (or closures) are used as representations of unresolved subgrid processes within weather and global climate models or coarse-scale turbulent models, whose resolutions are too coarse to resolve small-scale processes. These parameterizations are typically grounded on physically based, yet empirical, representations of the underlying small-scale processes. Machine learning-based parameterizations have recently been proposed as an alternative solution and have shown great promise to reduce uncertainties associated with the parameterization of small-scale processes. Yet, those approaches still show some important mismatches that are often attributed to the stochasticity of the considered process. This stochasticity can be due to coarse temporal resolution, unresolved variables, or simply to the inherent chaotic nature of the process. To address these issues, we propose a new type of parameterization (closure), which is built using memory-based neural networks, to account for the non-instantaneous response of the closure and to enhance its stability and prediction accuracy. We apply the proposed memory-based parameterization, with differentiable solver, to the Lorenz ’96 model in the presence of a coarse temporal resolution and show its capacity to predict skillful forecasts over a long time horizon of the resolved variables compared to instantaneous parameterizations. This approach paves the way for the use of memory-based parameterizations for closure problems.
The Differential Equation System Solver (DESS) is a Rust crate implementing fixed-step and adaptive-step solvers and designed especially for modeling physical systems. Seven explicit ordinary differential equation (ODE) solver methods have been added so far: Euler’s, Heun’s, Midpoint, Ralston’s, Classic Runge-Kutta, Bogacki-Shampine, and Cash-Karp. These comprise five fixed-step methods and two adaptive-step methods. Few solver packages are implemented in the Rust ecosystem and none are intended specifically for physical system modeling, so the goal of DESS is to create a Rust ODE solver crate designed to easily specify and model physical systems with modular, configurable solver options. In addition to allowing users to directly input equations to solve, DESS allows users to optionally specify and define relationships between nodes in their system, which the package then translates into a system of equations via the Rust macro system, leading to simpler and more intuitive code.
Here, closure problems are omnipresent when simulating multiscale systems, where some quantities and processes cannot be fully prescribed despite their effects on the simulation's accuracy. Recently, scientific machine learning approaches have been proposed as a way to tackle the closure problem, combining traditional (physics-based) modeling with data-driven (machine-learned) techniques, typically through enriching differential equations with neural networks. This paper reviews the different reduced model forms, distinguished by the degree to which they include known physics, and the different objectives of a priori and a posteriori learning. The importance of adhering to physical laws (such as symmetries and conservation laws) in choosing the reduced model form and choosing the learning method is discussed. The effect of spatial and temporal discretization and recent trends toward discretization-invariant models are reviewed. In addition, we make the connections between closure problems and several other research disciplines: inverse problems, Mori-Zwanzig theory, and multi-fidelity methods. In conclusion, much progress has been made with scientific machine learning approaches for solving closure problems, but many challenges remain. In particular, the generalizability and interpretability of learned models is a major issue that needs to be addressed further.
The physical properties of the upper atmosphere are determined mainly by heat conduction, the heat sources and the barometric law. An analysis of the integro-differential equation which describes these physical processes has been carried out. It is found that heating of the thermosphere due to absorption of the solar extreme ultraviolet (EUV) radiation alone cannot explain the observed diurnal variation of density and temperature, since it would yield a maximum of these properties at about 17h local time, instead of 14h where it is observed. Secondly, if the EUV flux is adjusted to give the observed average temperature, then the diurnal variation in density would be much too large compared with the observed amplitude. Thirdly, it would require an extremely high efficiency for the conversion of EUV radiation into heat, if we compare the required flux with Hinteregger's measurements of the EUV flux. Thus, it is necessary to have another heat source in addition to the heating due to absorption of EUV radiation. If an additional beat source is used, which has a maximum at about 9h local time and a flux of 1 erg cm−1 sec−1, a time-dependent model of the upper atmosphere is obtained that is in good agreement with the observed densities. There is evidence that this additional heat source derives its energy ultimately from the solar corpuscular radiation. In this paper we present the results of calculations for a model in the equatorial and temperature zones of the earth, for those times when the average solar activity corresponds to a solar radiation flux of 200×110−22 Wm−2 (cps)−1 at 10.7-cm wavelength. The physical properties (temperature, density, pressure, scale height, mean molecular weight and the number densities of N2, O2, O, He and H) are given as a function of local time and for the altitudes between 120 km and 2050 km.
Ordinary and partial differential equations (ODEs/PDEs) play a paramount role in analyzing and simulating complex dynamic processes across all corners of science and engineering. In recent years machine learning tools are aspiring to introduce new effective ways of simulating such equations, however existing approaches are not able to reliably return stable and accurate predictions across long temporal horizons. We aim to address this challenge by introducing an effective framework for learning evolution operators that map random initial conditions to associated ODE/PDE solutions within a short time interval. Such operators can be parametrized by deep neural networks that are trained in an entirely self-supervised manner without requiring one to generate any paired input-output observations. Global long-time predictions across a range of initial conditions can be then obtained by iteratively evaluating the trained model using each prediction as the initial condition for the next evaluation step. Here, this introduces a new approach to temporal domain decomposition that is shown to be effective in performing accurate long-time simulations for a wide range of parametric ODE and PDE systems, from wave propagation, to reaction-diffusion dynamics and stiff chemical kinetics, introducing a new way of rapidly emulating non-equilibrium processes in science and engineering.
Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas from one reservoir to another through a two-dimensional slit. The cases considered are for hard vacuum downstream pressure, finite pressure ratios, and isobaric pressure with thermal diffusion, which are not well established in spite of the simplicity of the flow field. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, three kinds of collision sampling techniques, the time counter (TC) method, the null collision (NC) method, and the no time counter (NTC) method, are used.
Two different approaches, the finite-difference method coupled with the discrete-ordinate method (FDDO), and the direct-simulation Monte Carlo (DSMC) method, are used in the analysis of the flow of a rarefied gas expanding through a two-dimensional nozzle and into a surrounding low-density environment. In the FDDO analysis, by employing the discrete-ordinate method, the Boltzmann equation simplified by a model collision integral is transformed to a set of partial differential equations which are continuous in physical space but are point functions in molecular velocity space. The set of partial differential equations are solved by means of a finite-difference approximation. In the DSMC analysis, the variable hard sphere model is used as a molecular model and the no time counter method is employed as a collision sampling technique. The results of both the FDDO and the DSMC methods show good agreement. The FDDO method requires less computational effort than the DSMC method by factors of 10 to 40 in CPU time, depending on the degree of rarefaction.
A new high resolution and genuinely multidimensional numerical method for solving conservation laws is being, developed. It was designed to avoid the limitations of the traditional methods. and was built from round zero with extensive physics considerations. Nevertheless, its foundation is mathmatically simple enough that one can build from it a coherent, robust. efficient and accurate numerical framework. Two basic beliefs that set the new method apart from the established methods are at the core of its development. The first belief is that, in order to capture physics more efficiently and realistically, the modeling, focus should be placed on the original integral form of the physical conservation laws, rather than the differential form. The latter form follows from the integral form under the additional assumption that the physical solution is smooth, an assumption that is difficult to realize numerically in a region of rapid chance. such as a boundary layer or a shock. The second belief is that, with proper modeling of the integral and differential forms themselves, the resulting, numerical solution should automatically be consistent with the properties derived front the integral and differential forms, e.g., the jump conditions across a shock and the properties of characteristics. Therefore a much simpler and more robust method can be developed by not using the above derived properties explicitly.
One of the most promising applications of machine learning in computational physics is to accelerate the solution of partial differential equations (PDEs). The key objective of machine-learning-based PDE solvers is to output a sufficiently accurate solution faster than standard numerical methods, which are used as a baseline comparison. Here, we first perform a systematic review of the ML-for-PDE-solving literature. Out of all of the articles that report using ML to solve a fluid-related PDE and claim to outperform a standard numerical method, we determine that 79% (60/76) make a comparison with a weak baseline. Second, we find evidence that reporting biases are widespread, especially outcome reporting and publication biases. We conclude that ML-for-PDE-solving research is overoptimistic: weak baselines lead to overly positive results, while reporting biases lead to under-reporting of negative results. To a large extent, these issues seem to be caused by factors similar to those of past reproducibility crises: researcher degrees of freedom and a bias towards positive results. We call for bottom-up cultural changes to minimize biased reporting as well as top-down structural reforms to reduce perverse incentives for doing so.
Here we present the PI-CKL-NN method for parameter estimation in differential equation (DE) models given sparse measurements of the parameters and states. In the proposed approach, the space- or time-dependent parameters are approximated by Karhunen-Loeve (KL) expansions that are conditioned on the parameters’ measurements, and the states are approximated by deep neural networks (DNNs). The unknown weights in the KL expansions and DNNs are found my minimizing the cost function that enforces the measurements of the states the DE constraint. Regularization is achieved by adding the l2 norm of the conditional KL coefficients into the loss function. Our approach assumes that the parameter fields are correlated in space or time and enforces the statistical knowledge (the mean and the covariance function) in addition to the DE constraints and measurements as opposed to the physics-informed neural network (PINN) and other similar physics-informed machine learning methods where only DE constraints and data are used for parameter estimation. We use the PI-CKL-NN method for parameter estimation in an ordinary differential equation with an unknown time-dependent parameter and the one- and two-dimensional partial differential diffusion equations with unknown space-dependent diffusion coefficients. We also demonstrate that PI-CKL-NN is more accurate than the PINN method, especially when the observations of the parameters are very sparse
Deep generative neural networks have enabled modeling complex distributions, but incorporating physics knowledge into the neural networks is still challenging and is at the core of current physics-based machine learning research. To this end, we propose a physics generative neural network (PhysGNN), a new class of generative neural networks for learning unknown distributions in a physical system described by partial differential equations (PDE). PhysGNN couples PDE systems with generative neural networks. It is a fully differentiable model that allows back-propagation of gradients through both numerical PDE solvers and generative neural networks, and is trained by minimizing the discrete Wasserstein distance between generated and observed probability distributions of the PDE outputs using the stochastic gradient descent method. Moreover, PhysGNN does not require adversarial training like standard generative neural networks, which offers better stability than adversarial training. We show that PhysGNN can learn complex distributions in stochastic inverse problems, where conventional methods such as maximum likelihood estimation and momentum matching methods may be inapplicable when little knowledge is known about the form of unknown distributions or the physical model is too complex. Furthermore, our method allows physics-based generative neural network training for learning complex distributions in the context of differential equations.
A summary is presented of theoretical and speculative research on the physics of igneous processes involved in asteroid differentiation. Partial melting processes, melt migration, and their products are discussed and explosive volcanism is described. Evidence for the existence of asteroidal magma oceans is considered and processes which may have occurred in these oceans are examined. Synthesis and inferences of asteroid heat sources are discussed under the assumption that asteroids are heated mainly by internal processes and that the role of impact heating is small. Inferences of these results for earth-forming planetesimals are suggested.