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At least 163 records · Page 9

Scaling kinetic Monte-Carlo simulations of grain growth with combined convolutional and graph neural networks

Graph neural networks (GNN) have emerged as a promising machine learning method for microstructure simulations such as grain growth. However, accurate modeling of realistic grain boundary networks requires large simulation cells, which GNN has difficulty scaling up to. To alleviate the computational costs and memory footprint of GNN, we suggest a hybrid architecture combining a convolutional neural network (CNN) based bijective autoencoder to compress the spatial dimensions, and a GNN that evolves the microstructure in the latent space of reduced spatial sizes. Our results demonstrate that the new design significantly reduces computational costs with using fewer message passing layer (from 12 down to 3) compared with GNN alone. The reduction in computational cost becomes more pronounced as the spatial size increases, indicating strong computational scalability. For the largest mesh evaluated (160 3 ), our method reduces memory usage and runtime in inference by 117× and 115×, respectively, compared with GNN-only baseline. More importantly, it shows higher accuracy and stronger spatiotemporal capability than the GNN-only baseline, especially in long-term testing. Such combination of scalability and accuracy is essential for simulating realistic material microstructures over extended time scales. The improvements can be attributed to the bijective autoencoder’s ability to compress information losslessly from spatial domain into a high dimensional feature space, thereby producing more expressive latent features for the GNN to learn from, while also contributing its own spatiotemporal modeling capability. Training data are generated from stochastic grain growth simulations, providing realistic variability for learning robust microstructure evolution. Comprehensive system validation confirms that the model is accurate, robust, and scalable.

36 MATERIALS SCIENCE↗

Lineage frequency time series reveal elevated levels of genetic drift in SARS-CoV-2 transmission in England

Genetic drift in infectious disease transmission results from randomness of transmission and host recovery or death. The strength of genetic drift for SARS-CoV-2 transmission is expected to be high due to high levels of superspreading, and this is expected to substantially impact disease epidemiology and evolution. However, we don’t yet have an understanding of how genetic drift changes over time or across locations. Furthermore, noise that results from data collection can potentially confound estimates of genetic drift. To address this challenge, we develop and validate a method to jointly infer genetic drift and measurement noise from time-series lineage frequency data. Our method is highly scalable to increasingly large genomic datasets, which overcomes a limitation in commonly used phylogenetic methods. We apply this method to over 490,000 SARS-CoV-2 genomic sequences from England collected between March 2020 and December 2021 by the COVID-19 Genomics UK (COG-UK) consortium and separately infer the strength of genetic drift for pre-B.1.177, B.1.177, Alpha, and Delta. We find that even after correcting for measurement noise, the strength of genetic drift is consistently, throughout time, higher than that expected from the observed number of COVID-19 positive individuals in England by 1 to 3 orders of magnitude, which cannot be explained by literature values of superspreading. Our estimates of genetic drift suggest low and time-varying establishment probabilities for new mutations, inform the parametrization of SARS-CoV-2 evolutionary models, and motivate future studies of the potential mechanisms for increased stochasticity in this system.

60 APPLIED LIFE SCIENCES↗

Low-Earth Orbit Trajectory Optimization in the Presence of Atmospheric Uncertainty

The previous 20 to 25 years have seen a tremendous increase in space exploration, and with that an increase in the level of logistics planning needed to ensure mission success. For spacecraft that are designed to be periodically re-supplied, a key logistics consumable is propellant, as it constitutes the greatest up-mass on re-supply vehicles. A trajectory design strategy is therefore desired that minimizes propellant usage in order to ease the demand for propellant re-supply missions. This thesis develops such a strategy in three stages, and uses the International Space Station (ISS) as its testbed, as no other LEO spacecraft is more challenging from a space logistics standpoint. First, the ISS trajectory planning problem is formulated as a constrained burn optimization problem assuming a deterministic atmosphere. The cost function is total ∆v, with constraints imposed on longitude of ascending viii node (LAN) and semi-major axis (SMA) altitude. Analytic derivatives are constructed for both the cost and constraints, which are necessary given the 6-week to 2-year time frames being considered. A gradient-based optimizer is then utilized to find locally-optimal solutions to real-world ISS trajectory planning problems. Second, atmospheric uncertainty is addressed by constructing a probabilistic model of space weather data using Gaussian Processes (GPs). Bayesian inference is performed using the GP model to generate mean and covariance estimates for space weather predictions, whose pedigree is assessed against test data. The predictions are then mapped into atmospheric density via the analytic Jacchia-Roberts density model, and the effect of space weather uncertainty on orbital lifetime is examined. Third, an ISS burn execution uncertainty model is developed. This model, along with the space weather uncertainty model, are deployed in a linear covariance analysis to ascertain their combined effect on LAN and SMA altitude dispersions. The deterministic constraints from the original problem are re-formulated as stochastic constraints, where now the constraint uncertainty interval is required to fall within specified bounds. An updated optimization framework is constructed using the original ∆v cost function along with the stochastic constraints to solve the trajectory optimization problem under atmospheric uncertainty. Finally, the complete architecture is summarized for deployment in an operational setting.

Trajectory Optimization↗

General-Purpose Bayesian Tensor Learning With Automatic Rank Determination and Uncertainty Quantification

A major challenge in many machine learning tasks is that the model expressive power depends on model size. Low-rank tensor methods are an efficient tool for handling the curse of dimensionality in many large-scale machine learning models. The major challenges in training a tensor learning model include how to process the high-volume data, how to determine the tensor rank automatically, and how to estimate the uncertainty of the results. While existing tensor learning focuses on a specific task, this paper proposes a generic Bayesian framework that can be employed to solve a broad class of tensor learning problems such as tensor completion, tensor regression, and tensorized neural networks. We develop a low-rank tensor prior for automatic rank determination in nonlinear problems. Our method is implemented with both stochastic gradient Hamiltonian Monte Carlo (SGHMC) and Stein Variational Gradient Descent (SVGD). We compare the automatic rank determination and uncertainty quantification of these two solvers. We demonstrate that our proposed method can determine the tensor rank automatically and can quantify the uncertainty of the obtained results. We validate our framework on tensor completion tasks and tensorized neural network training tasks.

Bayesian inference↗

Solar seismology. II - The stochastic excitation of the solar p-modes by turbulent convection

We test the hypothesis that the solar p-modes are stabilized by damping due to turbulent viscosity in the convective zone. Starting from the assumption that the modes are stable, we calculate expectation values for the modal energies. We find that the interaction between a p-mode and the turbulent convection is such that the modal energy tends toward equipartition with the kinetic energy of turbulent eddies whose lifetimes are comparable to the modal period. From the calculated values of the modal energies, we compute rms surface velocity amplitudes. Our predicted rms surface velocities range from 0.01 cm/sec for the fundamental radial mode to 0.6 cm/sec for the radial mode whose period is approximately 5 minutes. The predicted surface velocities for the low order p-modes are much smaller than the velocities inferred from recent observations.

Goldreich, P.↗

Adversarial sampling of unknown and high-dimensional conditional distributions

Many engineering problems require the prediction of realization-to-realization variability or a refined description of modeled quantities. In that case, it is necessary to sample elements from unknown high-dimensional spaces with possibly millions of degrees of freedom. While there exist methods able to sample elements from probability density functions (PDF) with known shapes, several approximations need to be made when the distribution is unknown. In this paper the sampling method, as well as the inference of the underlying distribution, are both handled with a data-driven method known as generative adversarial networks (GAN), which trains two competing neural networks to produce a network that can effectively generate samples from the training set distribution. In practice, it is often necessary to draw samples from conditional distributions. When the conditional variables are continuous, only one (if any) data point corresponding to a particular value of a conditioning variable may be available, which is not sufficient to estimate the conditional distribution. This work handles this problem using an a priori estimation of the conditional moments of a PDF. Herein, two approaches, stochastic estimation, and an external neural network are compared for computing these moments; however, any preferred method can be used. The algorithm is demonstrated in the case of the deconvolution of a filtered turbulent flow field. It is shown that all the versions of the proposed algorithm effectively sample the target conditional distribution with minimal impact on the quality of the samples compared to state-of-the-art methods. Additionally, the procedure can be used as a metric for the diversity of samples generated by a conditional GAN (cGAN) conditioned with continuous variables.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

A Magnetized Strongly Turbulent Corona as the Source of Neutrinos from NGC 1068

Abstract The cores of active galactic nuclei are potential accelerators of 10–100 TeV cosmic rays, in turn producing high-energy neutrinos. This picture was confirmed by the compelling evidence of a TeV neutrino signal from the nearby active galaxy NGC 1068, leaving open the question of what is the site and mechanism of cosmic-ray acceleration. One candidate is the magnetized turbulence surrounding the central supermassive black hole. Recent particle-in-cell simulations of magnetized turbulence indicate that stochastic cosmic-ray acceleration is nonresonant, in contrast to the assumptions of previous studies. We show that this has important consequences on a self-consistent theory of neutrino production in the corona, leading to a more rapid cosmic-ray acceleration than previously considered. The turbulent magnetic-field fluctuations needed to explain the neutrino signal are consistent with a magnetically powered corona. We find that strong turbulence, with turbulent magnetic energy density higher than 1% of the rest-mass energy density, naturally explains the normalization of the IceCube neutrino flux, in addition to the neutrino spectral shape. Only a fraction of the protons in the corona, which can be directly inferred from the neutrino signal, are accelerated to high energies. Thus, in this framework, the neutrino signal from NGC 1068 provides a testbed for particle acceleration in magnetized turbulence.

Fiorillo, Damiano F. G. (ORCID:0000000349279850)↗

Assessment of Two Stochastic Cloud Subcolumn Generators Using Observed Fields of Vertically Resolved Cloud Extinction

We evaluate two stochastic subcolumn generators used in GCMs to emulate subgrid cloud variability enabling comparisons with satellite observations and simulations of certain physical processes. Our evaluation necessitated the creation of a reference observational dataset that resolves horizontal and vertical cloud variability. The dataset combines two CloudSat cloud products that resolve two-dimensional cloud optical depth variability of liquid, ice, and mixed phase clouds when blended at ~200 m vertical and ~ 2 km horizontal scales. Upon segmenting the dataset to individual “scenes”, mean profiles of the cloud fields are passed as input to generators that produce scene-level cloud subgrid variability. The assessment of generator performance at the scale of individual scenes and in a mean sense is largely based on inferred joint histograms that partition cloud fraction within predetermined combinations of cloud top pressure –cloud optical thickness ranges. Our main finding is that both generators tend to underestimate optically thin clouds, while one of them also tends to overestimate some cloud types of moderate and high optical thickness. Associated radiative flux errors are also calculated by applying a simple transformation to the cloud fraction histogram errors, and are found to approach values almost as high as 3 Wm-2 for the cloud radiative effect in the shortwave part of the spectrum.

cloud variability↗

Runaway electron seed formation at reactor-relevant temperature

Systematic variation of the pre-disruption core electron temperature (T e ) from 1 to 12 keV using an internal transport barrier scenario reveals a dramatic increase in the production of ‘seed’ runaway electrons (REs), ultimately accessing near-complete conversion of the pre-disruption current into sub-MeV RE current. Injected Ar pellets are observed to ablate more intensely and promptly as T e rises. At high T e , the observed ablation exceeds predictions from published thermal ablation models. Simultaneously, the thermal quench (TQ) is observed to significantly shorten with increasing T e —a surprising result. While the reason for the shorter TQ is not yet understood, candidate mechanisms include: insufficiently accurate thermal ablation models, enhanced ablation driven by the seed RE population, or significant parallel heat transport along stochastic fields. Kinetic modeling that self-consistently treats the plasma cooling via radiation, the induced electric field, and the formation of the seed RE is performed. Including the combined effect of the inherent dependence of hot-tail RE seeding on T e together with the shortened TQ, modeling recovers the progression towards near-complete conversion of the pre-disruption current to RE current as T e rises. Measurement of the HXR spectrum during the early current quench (CQ) reveals a trend of decreasing energy with pre-disruption T e . At the very highest T e (≈ 12 keV), ≈ 100% conversion of the thermal current to runaway current is found. The energy of this peculiar RE beam is inferred to be sub-MeV as it emits vanishingly few MeV hard x-rays (HXRs). These measurements demonstrate novel TQ dynamics as Te is varied and illustrate the limitations of treating the RE seed formation problem without considering the inter-related dependencies of the pellet ablation, radiative energy loss, and resultant variations of the TQ duration. Finally, if the observed shortening of the TQ with increasing T e extends to fusion-grade plasmas, than their propensity to form large quantities of RE seeds at high T e may be far worse than previously thought. Positively, the high T e scenario in DIII-D produces REs so prodigiously that it can serve as a meaningful new platform for demonstrating RE avoidance techniques.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

High-resolution radiation detection using Ni/SiO 2 /n-4H-SiC vertical metal-oxide-semiconductor capacitor

In this article, we demonstrate the radiation detection performance of vertical metal-oxide-semiconductor (MOS) capacitors fabricated on 20 μm thick n-4H-SiC epitaxial layers with the highest energy resolution ever reported. The 100 nm SiO2 layer was achieved on the Si face of n-4H-SiC epilayers using dry oxidation in air. The Ni/SiO2/n-4H-SiC MOS detectors not only demonstrated an excellent energy resolution of 0.42% (ΔE/E 100) for 5.48 MeV alpha particles but also caused a lower enhancement in the electronic noise components of the spectrometer compared with that observed for the best high-resolution Schottky barrier detectors. The MOS detectors also exhibited a high charge collection efficiency (CCE) of 96% at the optimized operating bias despite the presence of the oxide layer. A drift-diffusion model applied to the CCE vs gate bias voltage data revealed a minority (hole) carrier diffusion length of 24 μm. Capacitance mode deep level transient spectroscopy (C-DLTS) scans in the temperature range 84–800 K were carried out to identify the resolution limiting electrically active defects. Interestingly, the C-DLTS spectra revealed both positive and negative peaks, indicating the simultaneous presence of electron (majority) and hole (minority) trap centers. It has been inferred that at the steady-state bias for the C-DLTS measurement, the MOS detector operates in the inversion mode at certain device temperatures, causing holes to populate the minority trap centers and, hence, manifests minority carrier peaks as well.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Probabilistic picture for particle number densities in stretched tips of the branching Brownian motion

Abstract In the framework of a stochastic picture for the one-dimensional branching Brownian motion, we compute the probability density of the number of particles near the rightmost one at a time T , that we take very large, when this extreme particle is conditioned to arrive at a predefined position x T chosen far ahead of its expected position m T . We recover the previously conjectured fact that the typical number density of particles at a distance Δ to the left of the lead particle, when both Δ and are large, is smaller than the mean number density by a factor proportional to , where ζ is a constant that was so far undetermined. Our picture leads to an expression for the probability density of the particle number, from which a value for ζ may be inferred.

Physics↗

Utilizing computer vision and artificial intelligence algorithms to predict and design the mechanical compression response of direct ink write 3D printed foam replacement structures

Additive Manufacturing (AM) of porous polymeric materials, such as foams, recently became a topic of intensive research due their unique combination of low density, impressive mechanical properties, and stress dissipation capabilities. Conventional methods for fabricating foams rely on complex and stochastic processes, making it challenging to achieve precise architectural control of structured porosity. In contrast, AM provides access to a wide range of printable materials, where precise spatial control over structured porosity can be modulated during the fabrication process enabling the production of foam replacement structures (FRS). Current approaches for designing FRS are based on intuitive understanding of their properties or an extensive number of finite element method (FEM) simulations. These approaches, however, are computationally expensive and time consuming. As such, in this work, we present a novel methodology for determining the mechanical compression response of direct ink write (DIW) 3D printed FRS using a simple cross-sectional image. By obtaining measurement data for a relatively small number of samples, an artificial neural network (ANN) was trained, and a computer vision algorithm was used to make inferences about foam compression characteristics from a single cross-sectional image. Finally, a genetic algorithm (GA) was used to solve the inverse design problem, generating the AM printing parameters that an engineer should use to achieve a desired compression response from a DIW printed FRS. The methods developed herein present an avenue for entirely autonomous design and analysis of additively manufactured structures using artificial intelligence.

36 MATERIALS SCIENCE↗

Hierarchical reconstruction of 3D well-connected porous media from 2D exemplars using statistics-informed neural network

The relationships between porous microstructures and transport properties are of fundamental importance in various scientific and engineering applications. Due to the intricacy, stochasticity and heterogeneity of porous media, reliable characterization and modeling of transport properties often require a complete dataset of internal microstructure samples. However, it is often an unbearable cost to acquire sufficient 3D digital microstructures by purely using microscopic imaging systems. Herein this paper presents a machine learning-based technique to hierarchically reconstruct 3D well-connected porous microstructures from one isotropic or several anisotropic low-cost 2D exemplar(s). To compactly characterize the large-scale microstructural features, a Gaussian image pyramid is built for each 2D exemplar. Local morphology patterns are collected from the Gaussian image pyramids, and then they serve as the training data to embed the 2D morphological statistics into feed-forward neural networks at multiple length levels. By using a specially-developed morphology integration scheme, the 3D morphological statistics at different levels can be inferred from the statistics-informed neural networks. Gibbs sampling is adopted to hierarchically reconstruct 3D microstructures by using multi-level 3D morphological statistics, where the large-scale, regional and local morphological patterns are statistically generated and successively added to the same 3D random field. The proposed method is tested on a series of porous media with distinct morphologies, and the statistical equivalence between the reconstructed and the real microstructures is systematically evaluated by comparing morphological descriptors and transport properties. The results demonstrate that the proposed 2D-to-3D microstructure reconstruction method is a universal and efficient approach to generating morphologically and physically realistic samples of porous media.

42 ENGINEERING↗

Estimate of pre-thermal quench non-thermal electron density profile during Ar pellet shutdowns of low-density target plasmas in DIII-D

The radial density profile of pre-thermal quench (pre-TQ) early-time runaway electrons (REs) is estimated by combining electron cyclotron emission (ECE) and soft x-ray (SXR) data during rapid shutdown of low-density (n e ≲ 10 13 cm -3 ) DIII-D target plasmas with cryogenic argon pellet injection. This technique is limited in these experiments to the pre-TQ phase and quickly loses validity during the TQ. Two different cases are studied: a high (10 keV) temperature target and a lower (4 keV) temperature target. The results indicate that early-time, low-energy (~10 keV) REs form ahead of the argon pellet as it enters the plasma, affecting the pellet ablation rate; it is hypothesized that this may be caused by rapid cross-field transport of argon ions ahead of the pellet. Fokker-Planck modeling of the two shots suggests that the RE current is quite significant during the pre-TQ phase (up to 50% of the total current). Comparison between modeled pre-TQ RE current and post-TQ RE current inferred from avalanche theory suggests that RE current increases during the high temperature target TQ but decreases during the low temperature target TQ. Here, the uncertainties in this estimate are large; but, if true, this suggests that TQ loss of REs can be larger than previously estimated in DIII-D.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

A Stochastic Approach to Reconstruct Gamma-Ray-burst Light Curves

Gamma-ray bursts (GRBs), as they are observed at high redshift ( z = 9.4), are vital to cosmological studies and investigating Population III stars. To tackle these studies, we need correlations among relevant GRB variables with the requirement of small uncertainties on their variables. Thus, we must have good coverage of GRB light curves (LCs). However, gaps in the LC hinder the precise determination of GRB properties and are often unavoidable. Therefore, extensive categorization of GRB LCs remains a hurdle. We address LC gaps using a stochastic reconstruction, wherein we fit two preexisting models (the Willingale model; W07; and a broken power law; BPL) to the observed LC, then use the distribution of flux residuals from the original data to generate data to fill in the temporal gaps. We also demonstrate a model-independent LC reconstruction via Gaussian processes. At 10% noise, the uncertainty of the end time of the plateau, its correspondent flux, and the temporal decay index after the plateau decreases by 33.3%, 35.03%, and 43.32% on average for the W07, and by 33.3%, 30.78%, 43.9% for the BPL, respectively. The uncertainty of the slope of the plateau decreases by 14.76% in the BPL. After using the Gaussian process technique, we see similar trends of a decrease in uncertainty for all model parameters for both the W07 and BPL models. These improvements are essential for the application of GRBs as standard candles in cosmology, for the investigation of theoretical models, and for inferring the redshift of GRBs with future machine-learning analyses.

79 ASTRONOMY AND ASTROPHYSICS↗

Uncertainty quantification in scientific machine learning: Methods, metrics, and comparisons

Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. This is because in addition to aleatoric uncertainty associated with noisy data, there is also uncertainty due to limited data, but also due to NN hyperparameters, overparametrization, optimization and sampling errors as well as model misspecification. Although there are some recent works on uncertainty quantification (UQ) in NNs, there is no systematic investigation of suitable methods towards quantifying the total uncertainty effectively and efficiently even for function approximation, and there is even less work on solving partial differential equations and learning operator mappings between infinite-dimensional function spaces using NNs. In this work, we present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as evaluation metrics and post-hoc improvement approaches. Further, to demonstrate the applicability and reliability of our framework, we present an extensive comparative study in which various methods are tested on prototype problems, including problems with mixed input-output data, and stochastic problems in high dimensions. In the Appendix, we include a comprehensive description of all the UQ methods employed. Further, to help facilitate the deployment of UQ in Scientific Machine Learning research and practice, we present and develop in [1] an open-source Python library (github.com/Crunch-UQ4MI/neuraluq), termed NeuralUQ, that is accompanied by an educational tutorial and additional computational experiments.

11 physics-informed neural networks↗

Enabling probabilistic learning on manifolds through double diffusion maps

Here, we present a generative learning framework for probabilistic sampling that extends Probabilistic Learning on Manifolds (PLoM), which is designed to generate statistically consistent realizations of a random vector in a finite-dimensional Euclidean space, informed by a (representative) set of observations. In its original form, PLoM constructs a reduced-order probabilistic model by combining three main components: (a) kernel density estimation to approximate the underlying probability measure, (b) Diffusion Maps to characterize the manifold of the data, and (c) a reduced-order Itô Stochastic Differential Equation (ISDE) to sample from the learned distribution. However, its sampling dynamics are posed in the ambient space and the retained number of reduced coordinates is chosen by projection-reconstruction error. In practice, this often (i) requires more coordinates than the data’s intrinsic dimension to achieve stable sampling and (ii) lacks a smooth, basis-independent lifting back to the data domain; moreover, standard Diffusion Maps emphasize harmonic eigenfunctions and can miss non-harmonic latent structure. We address these limitations by decoupling geometry learning from sampling: a first Diffusion Maps pass identifies non-harmonic coordinates on which we formulate a full-order ISDE directly in the latent space, while Double Diffusion Maps captures multiscale geometric features and Geometric Harmonics (GH) learns a smooth lifting map to the ambient variables that is independent of the particular diffusion basis. This hybrid design preserves the system’s dynamical richness with a compact geometric representation and enables principled out-of-sample inference. The effectiveness and robustness of the proposed method are illustrated through two numerical studies: one based on data generated from two-dimensional Hermite polynomial functions and another based on high-fidelity simulations of a detonation wave in a reactive flow.

Double diffusion maps↗

Preserving nonlinear constraints in variational flow filtering data assimilation

Data assimilation aims to estimate the states of a dynamical system by optimally combining sparse and noisy observations of the physical system with uncertain forecasts produced by a computational model. The states of many dynamical systems of interest obey nonlinear physical constraints, and the corresponding dynamics is confined to a certain sub-manifold of the state space. Standard data assimilation techniques applied to such systems yield posterior states lying outside the manifold, violating the physical constraints. This work focuses on particle flow filters which use stochastic differential equations to evolve state samples from a prior distribution to samples from an observation-informed posterior distribution. The variational Fokker-Planck (VFP)—a generic particle flow filtering framework—is extended to incorporate non-linear, equality state constraints in the analysis. To this end, two algorithmic approaches that modify the VFP stochastic differential equation are discussed: (i) VFPSTAB, to inexactly preserve constraints with the addition of a stabilizing drift term, and (ii) VFPDAE, to exactly preserve constraints by treating the VFP dynamics as a stochastic differential-algebraic equation (SDAE). Additionally, an implicit-explicit time integrator is developed to evolve the VFPDAE dynamics. The strength of the proposed approach for constraint preservation in data assimilation is demonstrated on three test problems: the double pendulum, Korteweg-de-Vries, and the incompressible Navier-Stokes equations.

97 MATHEMATICS AND COMPUTING↗