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At least 811 records · Page 45

Physics-informed neural networks (PINNs) for fluid mechanics: a review

Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier–Stokes equations (NSE), we still cannot incorporate seamlessly noisy data into existing algorithms, mesh-generation is complex, and we cannot tackle high-dimensional problems governed by parametrized NSE. Moreover, solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes. Here, we review flow physics-informed learning, integrating seamlessly data and mathematical models, and implement them using physics-informed neural networks (PINNs). Here, we demonstrate the effectiveness of PINNs for inverse problems related to three-dimensional wake flows, supersonic flows, and biomedical flows.

42 ENGINEERING↗

Optimal decision trees for categorical data via integer programming

Decision trees have been a very popular class of predictive models for decades due to their interpretability and good performance on categorical features. However, they are not always robust and tend to overfit the data. Additionally, if allowed to grow large, they lose interpretability. In this paper, we present a mixed integer programming formulation to construct optimal decision trees of a prespecified size. We take the special structure of categorical features into account and allow combinatorial decisions (based on subsets of values of features) at each node. Our approach can also handle numerical features via thresholding. Here we show that very good accuracy can be achieved with small trees using moderately-sized training sets. The optimization problems we solve are tractable with modern solvers.

97 MATHEMATICS AND COMPUTING↗

Scalable implementation of polynomial filtering for density functional theory calculation in PARSEC

In this work, we present an efficient implementation of polynomial filtering methods in PARSEC, a real-space pseudopotential based Kohn–Sham density functional theory solver. The implementation described here improves upon a Chebyshev-filtered subspace iteration algorithm used in the previous version of PARSEC. We present a hybrid polynomial filtering scheme that combines Chebyshev-filtered subspace iteration and a spectrum slicing method that partitions the spectrum into several spectral slices and uses bandpass-filtered subspace iteration to compute approximate eigenpairs within each interior slice simultaneously. We describe a procedure to partition a spectrum and construct polynomial filters. We also discuss a number of practical issues such as the use of appropriate data layouts for carrying out the computation on a two-dimensional process grid and how to achieve good load balance by allocating an appropriate number of process groups to each spectral slice. Numerical examples are presented to demonstrate the effectiveness of the hybrid polynomial filtering method as well as the superior parallel scalability of spectrum slicing in comparison to that of Chebyshev-filtered subspace iteration.

97 MATHEMATICS AND COMPUTING↗

Code-verification techniques for the method-of-moments implementation of the combined-field integral equation

Code verification plays an important role in establishing the credibility of computational simulations by assessing the correctness of the implementation of the underlying numerical methods. In computational electromagnetics, the numerical solution to integral equations incurs multiple interacting sources of numerical error, as well as other challenges, which render traditional code-verification approaches ineffective. In this paper, we provide approaches to separately measure the numerical errors arising from these different error sources for the method-of-moments implementation of the combined-field integral equation. Here, we demonstrate the effectiveness of these approaches for cases with and without coding errors.

97 MATHEMATICS AND COMPUTING↗

Learning physics-based reduced-order models from data using nonlinear manifolds

Here we present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

97 MATHEMATICS AND COMPUTING↗

Resource frugal optimizer for quantum machine learning

Quantum-enhanced data science, also known as quantum machine learning (QML), is of growing interest as an application of near-term quantum computers. Variational QML algorithms have the potential to solve practical problems on real hardware, particularly when involving quantum data. However, training these algorithms can be challenging and calls for tailored optimization procedures. Specifically, QML applications can require a large shot-count overhead due to the large datasets involved. In this work, we advocate for simultaneous random sampling over both the dataset as well as the measurement operators that define the loss function. We consider a highly general loss function that encompasses many QML applications, and we show how to construct an unbiased estimator of its gradient. This allows us to propose a shot-frugal gradient descent optimizer called Refoqus (REsource Frugal Optimizer for QUantum Stochastic gradient descent). Our numerics indicate that Refoqus can save several orders of magnitude in shot cost, even relative to optimizers that sample over measurement operators alone.

97 MATHEMATICS AND COMPUTING↗

Regional Relative Risk, a Physics-Based Metric for Characterizing Airborne Infectious Disease Transmission

Airborne infectious disease transmission events occur over a wide range of spatial scales and can be an important means of disease transmission. Physics- and biology-based models can assist in predicting airborne transmission events, overall disease incidence, and disease control strategy efficacy. We describe a new theory that extends current approaches for the case in which an individual is infected by a single airborne particle, including the scenario in which numerous infectious particles are present in the air but only one causes infection. A single infectious particle can contain more than one pathogenic microorganism and be physically larger than the pathogen itself. This approach allows robust relative risk estimates even when there is wide variation in (i) individual exposures and (ii) the individual response to that exposure (the pathogen dose-response function can take any mathematical form and vary by individual). Based on this theory, we propose the regional relative risk—a new metric, distinct from the traditional relative risk metric, that compares the risk between two regions. In theory, these regions can range from individual rooms to large geographic areas. In this paper, we apply the regional relative risk metric to outdoor disease transmission events over spatial scales ranging from 50 m to 20 km, demonstrating that in many common cases minimal input information is required to use the metric. Also, we demonstrate that the model predictions are consistent with data from prior outbreaks. Future efforts could apply and validate this theory for other spatial scales, such as transmission within indoor environments. This work provides context for (i) the initial stages of an airborne disease outbreak and (ii) larger-scale disease spread, including unexpected low-probability disease “sparks” that potentially affect remote populations, a key practical issue in controlling airborne disease outbreaks.

54 ENVIRONMENTAL SCIENCES↗

Multiscale Control of Generic Second Order Traffic Models by Driver-Assist Vehicles

We study the derivation of generic high order macroscopic traffic models from a follow-the-leader particle description via a kinetic approach. First, we recover a third order traffic model as the hydrodynamic limit of an Enskog-type kinetic equation. Next, we introduce in the vehicle interactions a binary control modeling the automatic feedback provided by driver-assist vehicles and we upscale such a new particle description by means of another Enskog-based hydrodynamic limit. The resulting macroscopic model is now a generic second order model (GSOM), which contains in turn a control term inherited from the microscopic interactions. We show that such a control may be chosen so as to optimize global traffic trends, such as the vehicle flux or the road congestion, constrained by the GSOM dynamics. By means of numerical simulations, we investigate the effect of this control hierarchy in some specific case studies, which exemplify the multiscale path from the vehiclewise implementation of a driver-assist control to its optimal hydrodynamic design.

GSOM↗

Data-Driven Learning for the Mori--Zwanzig Formalism: A Generalization of the Koopman Learning Framework

A theoretical framework which unifies the conventional Mori--Zwanzig formalism and the approximate Koopman learning of deterministic dynamical systems from noiseless observation is presented. In this framework, the Mori--Zwanzig formalism, developed in statistical mechanics to tackle the hard problem of construction of reduced-order dynamics for high-dimensional dynamical systems, can be considered as a natural generalization of the Koopman description of the dynamical system. We next show that, similar to the approximate Koopman learning methods, data-driven methods can be developed for the Mori--Zwanzig formalism with Mori's linear projection operator. We have developed two algorithms to extract the key operators, the Markov and the memory kernel, using time series of a reduced set of observables in a dynamical system. We have adopted the Lorenz `96 system as a test problem and solved for the above operators. These operators exhibit complex behaviors, which are unlikely to be captured by traditional modeling approaches in Mori--Zwanzig analysis. The nontrivial generalized fluctuation-dissipation relationship, which relates the memory kernel with the two-time correlation statistics of the orthogonal dynamics, was numerically verified as a validation of the solved operators. Here we present numerical evidence that the generalized Langevin equation, a key construct in the Mori--Zwanzig formalism, is more advantageous in predicting the evolution of the reduced set of observables than the conventional approximate Koopman operators.

97 MATHEMATICS AND COMPUTING↗

Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems

Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multi-time case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the Linear Quadratic Regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Lastly, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, post-training calibration, transfer learning, and sparse dynamics identification.

97 MATHEMATICS AND COMPUTING↗

Constructing Neural Network Based Models for Simulating Dynamical Systems

Dynamical systems see widespread use in natural sciences like physics, biology, and chemistry, as well as engineering disciplines such as circuit analysis, computational fluid dynamics, and control. For simple systems, the differential equations governing the dynamics can be derived by applying fundamental physical laws. However, for more complex systems, this approach becomes exceedingly difficult. Data-driven modeling is an alternative paradigm that seeks to learn an approximation of the dynamics of a system using observations of the true system. In recent years, there has been an increased interest in applying data-driven modeling techniques to solve a wide range of problems in physics and engineering. Here this article provides a survey of the different ways to construct models of dynamical systems using neural networks. In addition to the basic overview, we review the related literature and outline the most significant challenges from numerical simulations that this modeling paradigm must overcome. Based on the reviewed literature and identified challenges, we provide a discussion on promising research areas.

97 MATHEMATICS AND COMPUTING↗

Structural Models and Sequence Alignment Results of the Rhodospirillum rubrum Proteome

This dataset contains the structural models for the primary transcripts of the Rhodospirillum rubrum proteome as well as sequence alignment results for a subset of the encoded proteins. For each protein, the five models inferred from AlphaFold 2 are provided. The largest pTM-scoring model for each protein was energy minimized; this minimized structure as well as its AlphaFold pickle output file are also provided. This set of structures represent an alternate source of models for the R. rubrum proteome to those available in the AlphaFold Protein Structure Database. For proteins that have been annotated as hypothetical, sequence alignment results from the HHblits and SAdLSA alignment methods are provided. These methods are often more capable to resolve sequence homology than other methods. Therefore, the results from both HHblits and SAdLSA are provided to identify possible homologs for these challenging proteins. Numerous sequence databases are utilized for these alignments. References AlphaFold v2 Multimer: https://doi.org/10.1101/2021.10.04.463034. References HHBlits: https://doi.org/10.1186/s12859-019-3019-7. References SAdLSA: https://doi.org/10.3389/fbinf.2021.689960.

59 BASIC BIOLOGICAL SCIENCES↗

Structural Models and Sequence Alignment Results of the Desulfovibrio vulgaris Proteome

This dataset contains the structural models for the primary transcripts of the Desulfovibrio vulgaris proteome as well as sequence alignment results for a subset of the encoded proteins. For each protein, the five models inferred from AlphaFold 2 are provided. The largest pTM-scoring model for each protein was energy minimized; this minimized structure as well as its AlphaFold pickle output file are also provided. This set of structures represent an alternate source of models for the D. vulgaris proteome to those available in the AlphaFold Protein Structure Database (AFDB). This is a bit more complicated since the proteins reporting in the AFDB originate from an outdated form of the D. vulgaris sequence. The different versions of the D. vulgaris gene annotation are collected in the Chronology subdirectory; further consideration of these changes on the structural space of the proteome are currently underway. For proteins that have been annotated as hypothetical, sequence alignment results from the HHblits and SAdLSA alignment methods are provided. These methods are often more capable to resolve sequence homology than other methods. Therefore, the results from both HHblits and SAdLSA are provided to identify possible homologs for these challenging proteins. Numerous sequence databases are utilized for these alignments. References AlphaFold v2 Multimer: https://doi.org/10.1101/2021.10.04.463034. References HHblits: hhtps://doi.org/10.1186/s12859-019-3019-7. References SAdLSA: hhtps://doi.org/10.3389/fbinf.2021.689960.

59 BASIC BIOLOGICAL SCIENCES↗

Event-scale predictions of water and nitrogen exports in coastal watersheds

Focal Area(s): The purpose of this white paper is to call for the improvement of the predictability and explainability of the magnitude and timing of water and nutrient exports in coastal watersheds. In this regard, hybrid machine learning approaches and process-based numerical models provide new opportunities to maximize predictability, emulate process-based model components, and explicitly address model uncertainty associated with event-scale predictions in coastal watersheds.

54 ENVIRONMENTAL SCIENCES↗

Mesoscale Organization in Cumulus-Coupled Stratocumulus

Marine cloud systems cover a substantial portion of the world’s oceans. Most of these clouds form relatively close to the ocean surface, typically within one to two kilometers, a region referred to by meteorologists as the marine boundary layer. They are composed predominantly of liquid water, although ice particles can occur in mid- and high-latitude marine clouds during winter. In satellite imagery, these clouds appear bright against the darker ocean surface below, reflecting a large fraction of incoming sunlight back into space that would otherwise warm the ocean. Because marine boundary layer clouds cover such an extensive area of the ocean, they exert a significant influence on Earth’s overall transfer of solar energy absorbed by the surface and thermal energy emitted to space, a balance known as the planetary radiation budget. Marine boundary layer clouds are typically thin, and their formation and dissipation depend on a delicate balance between processes acting at the ocean surface below and the warm, dry air above. They are notoriously difficult to simulate accurately in weather forecast models, which often produce too few marine low clouds in midlatitudes and clouds in tropical regions that are excessively bright, meaning they reflect too much solar radiation. The marine boundary layer is frequently characterized by widespread overcast cloud cover that often transitions from a continuous, single-layer deck to more broken cloud fields toward the tropics. These transitions typically proceed through an intermediate stage in which shallow, broken clouds form beneath the overlying stratiform cloud deck. Once broken clouds develop below the overcast, they frequently self-organize into cloud clusters known as marine boundary layer convective complexes (MBLCCs), although the mechanisms governing the formation and organization of MBLCCs remain poorly understood. Accurately representing these transitions in long-range weather forecast models is essential because they influence the properties of air masses advected over the continental United States and Europe, and they become increasingly important for forecasts on seasonal and longer timescales. We employed two complementary approaches to investigate the processes controlling MBLCCs and their impact on marine cloud cover. Long-term observations from the U.S. Department of Energy’s Eastern North Atlantic (ENA) Observatory provided a unique dataset that allowed us to characterize fundamental properties of MBLCCs, including their typical size and frequency of occurrence. These observations were combined with high-resolution numerical simulations performed on supercomputers to examine the evolution of MBLCCs during cold-air outbreaks over the ENA region.

54 ENVIRONMENTAL SCIENCES↗

Nonlinear Anti-(Parity-Time) Symmetric Dimer

In the present work we propose a nonlinear anti- P T -symmetric dimer, that at the linear level has been experimentally created in the realm of electric circuit resonators. We find four families of solutions, the so-called upper and lower branches, both in a symmetric and in an asymmetric (symmetry-broken) form. We unveil analytically and confirm numerically the critical thresholds for the existence of such branches and explore the bifurcations (such as saddle-node ones) that delimit their existence, as well as transcritical ones that lead to their potential exchange of stability. We find that out of the four relevant branches, only one, the upper symmetric branch, corresponds to a spectrally and dynamically robust solution. We subsequently leverage detailed direct numerical computations in order to explore the dynamics of the different states, corroborating our spectral analysis results.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Galerkin formulation of path integrals in lattice field theory

We present a mathematical framework for Galerkin formulations of path integrals in lattice field theory. The framework is based on using the degrees of freedom (DOFs) associated to a Galerkin discretization as the fundamental lattice variables. We formulate standard concepts in lattice field theory, such as the partition function and correlation functions, in terms of the DOFs. For example, using continuous finite element spaces, we show that the two-point spatial correlation function can be defined between any two points on the domain (as opposed to at just lattice sites) and furthermore, this two-point function satisfies a weak propagator (or Green’s function) identity, in analogy to the continuum case, as well as a convergence estimate obtained from the standard finite element techniques. Furthermore, this framework leads naturally to higher-order formulations of lattice field theories by considering higher-order finite element spaces for the Galerkin discretization. We consider analytical and numerical examples of scalar field theory to investigate how increasing the order of piecewise polynomial finite element spaces affect the approximation of lattice observables. Finally, we sketch an outline of this Galerkin framework in the context of gauge field theories.

97 MATHEMATICS AND COMPUTING↗