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At least 1,117 records · Page 62

Portage: A Modular Data Remap Library for Multiphysics Applications on Advanced Architectures

Portage is a scalable and extensible remap library for numerical simulations. It supports state-of-the-art remap schemes for meshes and particles in 2D and 3D up to a second-order accuracy. Portage ensures critical properties such as local/global conservation and bounds preservation for mesh remap. It enables multi-material field remap through a dedicated plugin, and leverages the hybrid parallelism exposed by advanced architectures using multi-processing and multi-threading.

97 MATHEMATICS AND COMPUTING↗

A First Look at the Upcoming SISO Space Reference FOM

Spaceflight is difficult, dangerous and expensive; human spaceflight even more so. In order to mitigate some of the danger and expense, professionals in the space domain have relied, and continue to rely, on computer simulation. Simulation is used at every level including concept, design, analysis, construction, testing, training and ultimately flight. As space systems have grown more complex, new simulation technologies have been developed, adopted and applied. Distributed simulation is one those technologies. Distributed simulation provides a base technology for segmenting these complex space systems into smaller, and usually simpler, component systems or subsystems. This segmentation also supports the separation of responsibilities between participating organizations. This segmentation is particularly useful for complex space systems like the International Space Station (ISS), which is composed of many elements from many nations along with visiting vehicles from many nations. This is likely to be the case for future human space exploration activities. Over the years, a number of distributed simulations have been built within the space domain. While many use the High Level Architecture (HLA) to provide the infrastructure for interoperability, HLA without a Federation Object Model (FOM) is insufficient by itself to insure interoperability. As a result, the Simulation Interoperability Standards Organization (SISO) is developing a Space Reference FOM. The Space Reference FOM Product Development Group is composed of members from several countries. They contribute experiences from projects within NASA, ESA and other organizations and represent government, academia and industry. The initial version of the Space Reference FOM is focusing on time and space and will provide the following: (i) a flexible positioning system using reference frames for arbitrary bodies in space, (ii) a naming conventions for well-known reference frames, (iii) definitions of common time scales, (iv) federation agreements for common types of time management with focus on time stepped simulation, and (v) support for physical entities, such as space vehicles and astronauts. The Space Reference FOM is expected to make collaboration politically, contractually and technically easier. It is also expected to make collaboration easier to manage and extend.

Mueller, Bjorn↗

Data-driven emulation of modal aerosol microphysics via neural operator-based modeling

The complexity and the small characteristic scales of aerosol microphysical processes pose a big challenge for accurate and efficient Earth system simulations at regional and global scales. In this work, we construct and evaluate a surrogate model: the aerosol deep operator network (ADON), a physics-inspired dual-net architecture for emulating the aerosol microphysics parameterization suite in the version 2 of the Energy Earth System Model (E3SMv2). The current version of the surrogate model is trained on a dataset comprising 9.8 million samples obtained from a global E3SMv2 simulation with the horizontal resolution of about one degree under cloud-free conditions. Incorporating domain spatial and temporal coordinates, as well as principle components extracted from training data, the dual-net surrogate model effectively captures the intricate representations of aerosol and the relationship with atmospheric state variables, achieving an R-squared score over $$95.7\%$$ for all the lognormal aerosol modes in the extrapolated regime. The validated model provides feature importance of input variables and their impact on the predictive capacity of the surrogate model in relation to the E3SM. The computational cost of online inference time deployed on CPUs and GPUs with lower precisions highlights ADON’s efficiency and potential in robust predictive modeling for large-scale Earth system computations.

Bai, Zhe↗

A Model for Backscattering from Quasi Periodic Corn Canopies at L-Band

In this study, a model for backscattering at L-band from a corn canopy is proposed. The canopy consists of a quasi-periodic distribution of stalks and a random distribution of leaves. The Distorted Born Approximation (DBA) is employed to calculate the single scattered return from the corn field. The new feature of the method is that the coherence of the stalks in the row direction is incorporated in the model in a systematic fashion. Since the wavelength is on the order of the distance between corn stalks in a row, grating lobe behavior is observed at certain azimuth angles of incidence. The results are compared with experimental values measured in Huntsville, Alabama in 1998. The mean field and the effective dielectric constant of the canopy are obtained by using the Foldy approximation. The stalks are placed in the effective medium in a two dimensional lattice to simulate the row structure of a corn field. In order to mimic a real corn field, a quasi-periodic stalk distribution is assumed where the stalks are given small random perturbations about their lattice locations. Corn leaves are also embedded in the effective medium and the backscattered field from the stalks and the leaves is computed. The backscattering coefficient is calculated and averaged over successive stalk position perturbations. It is assumed that soil erosion has smoothed the soil sufficiently so that it can be assumed flat. Corn field backscatter data was collected from cornfields during the Huntsville 98 experimental campaign held at Alabama A&M University Research Station, Huntsville, Alabama in 1998 using the NASA/GW truck mounted radar. Extensive ground truth data was collected. This included soil moisture measurements and corn plant architectural data to be used in the model. In particular, the distances between the stalks in a single row have been measured. The L-band radar backscatter data was collected for both H and V polarizations and for look angles of 15o and 45o over a two week period under varying soil moisture conditions. These measured backscattering values will be compared with the model backscattering values and a discussion of the results will be presented.

Lang, R.↗

Explicitly antisymmetrized neural network layers for variational Monte Carlo simulation

The combination of neural networks and quantum Monte Carlo methods has arisen as a promising path forward for highly accurate electronic structure calculations. Previous proposals have combined equivariant neural network layers with a final antisymmetric layer in order to satisfy the antisymmetry requirements of the electronic wavefunction. However, to date it is unclear if one can represent antisymmetric functions of physical interest, and it is difficult to precisely measure the expressiveness of the antisymmetric layer. Here, this work attempts to address this problem by introducing explicitly antisymmetrized universal neural network layers. This approach has a computational cost which increases factorially with respect to the system size, but we are nonetheless able to apply it to small systems to better understand how the structure of the antisymmetric layer affects its performance. We first introduce a generic antisymmetric (GA) neural network layer, which we use to replace the entire antisymmetric layer of the highly accurate ansatz known as the FermiNet. We demonstrate that the resulting FermiNet-GA architecture can yield effectively the exact ground state energy for small atoms and molecules. We then consider a factorized antisymmetric (FA) layer which more directly generalizes the FermiNet by replacing the products of determinants with products of antisymmetrized neural networks. We find, interestingly, that the resulting FermiNet-FA architecture does not significantly outperform the FermiNet. This strongly suggests that the sum of products of antisymmetries is a key limiting aspect of the FermiNet architecture. To explore this further, we investigate a slight modification of the FermiNet, called the full determinant mode, which replaces each product of determinants with a single combined determinant. We find that the full single-determinant FermiNet closes a large part of the gap between the standard single-determinant FermiNet and FermiNet-GA on small atomic and molecular problems. Surprisingly, on the nitrogen molecule at a dissociating bond length of 4.0 Bohr, the full single-determinant FermiNet can outperform the largest standard FermiNet calculation with 64 determinants.

97 MATHEMATICS AND COMPUTING↗

The Origins of Transmembrane Ion Channels

Even though membrane proteins that mediate transport of ions and small molecules across cell walls are among the largest and least understood biopolymers in contemporary cells, it is still possible to shed light on their origins and early evolution. The central observation is that transmembrane portions of most ion channels are simply bundles of -helices. By combining results of experimental and computer simulation studies on synthetic models and natural channels, mostly of non-genomic origin, we show that the emergence of -helical channels was protobiologically plausible, and did not require highly specific amino acid sequences. Despite their simple structure, such channels could possess properties that, at the first sight, appear to require markedly larger complexity. Specifically, we explain how the antiamoebin channels, which are made of identical helices, 16 amino acids in length, achieve efficiency comparable to that of highly evolved channels. We further show that antiamoebin channels are extremely flexible, compared to modern, genetically coded channels. On the basis of our results, we propose that channels evolved further towards high structural complexity because they needed to acquire stable rigid structures and mechanisms for precise regulation rather than improve efficiency. In general, even though architectures of membrane proteins are not nearly as diverse as those of water-soluble proteins, they are sufficiently flexible to adapt readily to the functional demands arising during evolution.

Pohorille, Andrew↗

Machine-learning modeling of magnetization dynamics in quasi-equilibrium and driven metallic spin systems

Here, we present a perspective on recent progress in machine-learning (ML) force-field approaches for large-scale Landau–Lifshitz–Gilbert (LLG) simulations of metallic spin systems. Building on a generalization of the Behler–Parrinello (BP) architecture originally developed for quantum molecular dynamics, we develop scalable and transferable ML models that faithfully capture the complex, environment-dependent electron-mediated exchange fields characteristic of itinerant magnets. A central ingredient of this framework is the implementation of symmetry-aware magnetic descriptors based on group-theoretical bispectrum formalisms. Leveraging these ML force fields, LLG simulations faithfully reproduce hallmark non-collinear magnetic orders—such as the 120° and tetrahedral states—on the triangular lattice, and successfully capture the complex spin textures emerging in the mixed-phase states of a square-lattice double-exchange model under thermal quench. We further discuss a generalized potential theory that extends the BP formalism to incorporate both conservative and nonconservative electronic torques, thereby enabling ML models to learn nonequilibrium exchange fields from computationally demanding microscopic approaches such as nonequilibrium Green’s-function techniques. This extension yields quantitatively accurate predictions of voltage-driven domain-wall motion and establishes a foundation for quantum-accurate, multiscale modeling of nonequilibrium spin dynamics and spintronic functionalities.

Descriptors↗

Planning for the semiconductor manufacturer of the future

Texas Instruments (TI) is currently contracted by the Air Force Wright Laboratory and the Defense Advanced Research Projects Agency (DARPA) to develop the next generation flexible semiconductor wafer fabrication system called Microelectronics Manufacturing Science & Technology (MMST). Several revolutionary concepts are being pioneered on MMST, including the following: new single-wafer rapid thermal processes, in-situ sensors, cluster equipment, and advanced Computer Integrated Manufacturing (CIM) software. The objective of the project is to develop a manufacturing system capable of achieving an order of magnitude improvement in almost all aspects of wafer fabrication. TI was awarded the contract in Oct., 1988, and will complete development with a fabrication facility demonstration in April, 1993. An important part of MMST is development of the CIM environment responsible for coordinating all parts of the system. The CIM architecture being developed is based on a distributed object oriented framework made of several cooperating subsystems. The software subsystems include the following: process control for dynamic control of factory processes; modular processing system for controlling the processing equipment; generic equipment model which provides an interface between processing equipment and the rest of the factory; specification system which maintains factory documents and product specifications; simulator for modelling the factory for analysis purposes; scheduler for scheduling work on the factory floor; and the planner for planning and monitoring of orders within the factory. This paper first outlines the division of responsibility between the planner, scheduler, and simulator subsystems. It then describes the approach to incremental planning and the way in which uncertainty is modelled within the plan representation. Finally, current status and initial results are described.

Fargher, Hugh E.↗

A 3D Simulation Platform for Decentralized Decision-Making in Advanced Air Mobility

This paper presents a general purpose, plug-and-play simulation platform for the use of future aviation stakeholders, such as urban airspace planners, air vehicle operators, ground operation managers, air traffic controllers and aviation researchers. The presented simulator platform is envisioned to serve as a toolkit to visualize, evaluate, and configure future advanced air mobility (AAM) operations. Highlighting features of this toolkit include a modular architecture that allows multiple smart unmanned aerial systems (UASs) to remotely connect to the simulation server and participate in decentralized decision-making scenario simulations. As an example of the decentralized decision-making scenario, an inter-agent negotiation-based conflict resolution use case is considered in this paper, where the UASs leverage the on-board/on-the-edge artificial intelligence (AI) capability to continually build situational awareness, and use this information to predict future conflicts and resolve them through machine-to-machine negotiation. As such operations are non-existent at scale currently, the presented simulation platform offers a viable and cost-effective alternative for assessing the efficacy of AAM research outcomes and challenges in future shared airspace usage. The simulation platform allows plug-n-play connectivity with AI and non-AI compute modules representing individual UAS’s flight control. Each module can interact with the simulation platform independently to communicate current and desired future states, situational awareness, and conflict resolution utilization costs for inter-agent negotiation. The simulation environment orchestrates realistic operational scenarios with spatiotemporal details, dynamic events, tactical conflict-resolution methods, interfaces for customizing air traffic control parameters, and information exchange uncertainties. In the future, this can serve as a community focused cloud simulation platform, incorporating multi-stakeholder airspace constraints from regulatory, government, city, and local agencies.

Aditya N Das↗

A 3D Simulation Platform for Decentralized Decision-Making in Advanced Air Mobility

This paper presents a general purpose, plug-and-play simulation platform for the use of future aviation stakeholders, such as urban airspace planners, air vehicle operators, ground operation managers, air traffic controllers and aviation researchers. The presented simulator platform is envisioned to serve as a toolkit to visualize, evaluate, and configure future advanced air mobility (AAM) operations. Highlighting features of this toolkit include a modular architecture that allows multiple smart unmanned aerial systems (UASs) to remotely connect to the simulation server and participate in decentralized decision-making scenario simulations. As an example of the decentralized decision-making scenario, an inter-agent negotiation-based conflict resolution use case is considered in this paper, where the UASs leverage the on-board/on-the-edge artificial intelligence (AI) capability to continually build situational awareness, and use this information to predict future conflicts and resolve them through machine-to-machine negotiation. As such operations are non-existent at scale currently, the presented simulation platform offers a viable and cost-effective alternative for assessing the efficacy of AAM research outcomes and challenges in future shared airspace usage. The simulation platform allows plug-n-play connectivity with AI and non-AI compute modules representing individual UAS’s flight control. Each module can interact with the simulation platform independently to communicate current and desired future states, situational awareness, and conflict resolution utilization costs for inter-agent negotiation. The simulation environment orchestrates realistic operational scenarios with spatiotemporal details, dynamic events, tactical conflict-resolution methods, interfaces for customizing air traffic control parameters, and information exchange uncertainties. In the future, this can serve as a community focused cloud simulation platform, incorporating multi-stakeholder airspace constraints from regulatory, government, city, and local agencies.

Aditya Das↗

Multidisciplinary concurrent optimization framework for multi-phase building design process

Modern day building design projects require multidisciplinary expertise from architects and engineers across various phases of the design (conceptual, preliminary, and detailed) and construction processes. The Architecture Engineering and Construction (AEC) community has recently shifted gears toward leveraging design optimization techniques to make well-informed decisions in the design of buildings. However, most of the building design optimization efforts are either multidisciplinary optimization confined to just a specific design phase (conceptual/preliminary/detailed) or single disciplinary optimization (structural/thermal/daylighting/energy) spanning across multiple phases. Complexity in changing the optimization setup as the design progresses through subsequent phases, interoperability issues between modeling and physics-based analysis tools used at later stages, and the lack of an appropriate level of design detail to get meaningful results from these sophisticated analysis tools are few challenges that limit multi-phase multidisciplinary design optimization (MDO) in the AEC field. Here this paper proposes a computational building design platform leveraging concurrent engineering techniques such as interactive problem structuring, simulation-based optimization using meta models for energy and daylighting (machine learning based) and tradespace visualization. The proposed multi-phase concurrent MDO framework is demonstrated by using it to design and optimize a sample office building for energy and daylighting objectives across multiple phases. Furthermore, limitations of the proposed framework and future avenues of research are listed.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

Toward higher-radix switches with co-packaged optics for improved network locality in data center and HPC networks [Invited]

In this work, we study the network locality improvements that can be achieved by using co-packaged optics in data center and high-performance computing (HPC) networks. The increased escape bandwidth offered by co-packaged optics can enable switches with speeds of 51.2 Tb/s and beyond. From a network architecture perspective, the key advantages of introducing co-packaged optics at the switch points include the implementation of large-scale topologies of >12,000 end points with 4× higher bisection bandwidth and the reduction of the required number of switches by >40% compared with state-of-the-art approaches. From a network operation perspective, improved network locality and faster operation can be achieved since the higher-radix switches can mitigate the impact of network contention. Placing applications under fewer leaf switches reduces the number of packets that cross the spine switches in a leaf-spine topology. The proposed scheme is evaluated via discrete-event simulations: we initially evaluate the network locality properties of the system by using virtual-machine traces from a production data center, and we subsequently quantify the performance improvements by simulating an all-to-all pattern for a variety of message sizes over a number of nodes. The results suggest that co-packaged optics form a promising solution for keeping up with bandwidth scaling in future networks. The virtual-machine analysis shows that large-scale applications can be placed under up to 50% fewer first-level switches, while the network analysis shows speedups of up to 7.1, which translates to execution time reductions of up to 26% and 42.7% for applications with communication ratios of 0.3 and 0.5, respectively.

99 GENERAL AND MISCELLANEOUS↗

A design rationale for NASA TileWorld

Automated systems that can operate in unrestricted real-world domains are still well beyond current computational capabilities. This paper argues that isolating essential problem characteristics found in real-world domains allows for a careful study of how particular control systems operate. By isolating essential problem characteristics and studying their impact on autonomous system performance, we should be able to more quickly deliver systems for practical real-world problems. For our research on planning, scheduling, and control, we have selected three particular domain attributes to study: exogenous events, uncertain action outcome, and metric time. We are not suggesting that studies of these attributes in isolation are sufficient to guarantee the obvious goals of good methodology, brilliant architectures, or first-class results; however, we are suggesting that such isolation facilitates the achievement of these goals. To study these attributes, we have developed the NASA TileWorld. We describe the NASA TileWorld simulator in general terms, present an example NASA TileWorld problem, and discuss some of our motivations and concerns for NASA TileWorld.

Philips, Andrew B.↗

Software for Simulating Air Traffic

Future Air Traffic Management Concepts Evaluation Tool (FACET) is a system of software for performing computational simulations for evaluating advanced concepts of advanced air-traffic management. FACET includes a program that generates a graphical user interface plus programs and databases that implement computational models of weather, airspace, airports, navigation aids, aircraft performance, and aircraft trajectories. Examples of concepts studied by use of FACET include aircraft self-separation for free flight; prediction of air-traffic-controller workload; decision support for direct routing; integration of spacecraft-launch operations into the U.S. national airspace system; and traffic- flow-management using rerouting, metering, and ground delays. Aircraft can be modeled as flying along either flight-plan routes or great-circle routes as they climb, cruise, and descend according to their individual performance models. The FACET software is modular and is written in the Java and C programming languages. The architecture of FACET strikes a balance between flexibility and fidelity; as a consequence, FACET can be used to model systemwide airspace operations over the contiguous U.S., involving as many as 10,000 aircraft, all on a single desktop or laptop computer running any of a variety of operating systems. Two notable applications of FACET include: (1) reroute conformance monitoring algorithms that have been implemented in one of the Federal Aviation Administration s nationally deployed, real-time, operational systems; and (2) the licensing and integration of FACET with the commercially available Flight Explorer, which is an Internet- based, real-time flight-tracking system.

Sridhar, Banavar↗

A massively parallel and scalable multi-CPU material point method

Harnessing the power of modern multi-GPU architectures, we present a massively parallel simulation system based on the Material Point Method (MPM) for simulating physical behaviors of materials undergoing complex topological changes, self-collision, and large deformations. Our system makes three critical contributions. First, we introduce a new particle data structure that promotes coalesced memory access patterns on the GPU and eliminates the need for complex atomic operations on the memory hierarchy when writing particle data to the grid. Second, we propose a kernel fusion approach using a new Grid-to-Particles-to-Grid (G2P2G) scheme, which efficiently reduces GPU kernel launches, improves latency, and significantly reduces the amount of global memory needed to store particle data. Finally, we introduce optimized algorithmic designs that allow for efficient sparse grids in a shared memory context, enabling us to best utilize modern multi-GPU computational platforms for hybrid Lagrangian-Eulerian computational patterns. We demonstrate the effectiveness of our method with extensive benchmarks, evaluations, and dynamic simulations with elastoplasticity, granular media, and fluid dynamics. In comparisons against an open-source and heavily optimized CPU-based MPM codebase [Fang et al. 2019] on an elastic sphere colliding scene with particle counts ranging from 5 to 40 million, our GPU MPM achieves over 100x per-time-step speedup on a workstation with an Intel 8086K CPU and a single Quadro P6000 GPU, exposing exciting possibilities for future MPM simulations in computer graphics and computational science. Moreover, compared to the state-of-the-art GPU MPM method [Hu et al. 2019a], we not only achieve 2x acceleration on a single GPU but our kernel fusion strategy and Array-of-Structs-of-Array (AoSoA) data structure design also generalizes to multi-GPU systems. Our multi-GPU MPM exhibits near-perfect weak and strong scaling with 4 GPUs, enabling performant and large-scale simulations on a 10243 grid with close to 100 million particles with less than 4 minutes per frame on a single 4-GPU workstation and 134 million particles with less than 1 minute per frame on an 8-GPU workstation.

Wang, Xinlei↗

ClimateNet: an expert-labeled open dataset and deep learning architecture for enabling high-precision analyses of extreme weather

Abstract. Identifying, detecting, and localizing extreme weather events is a crucial first step in understanding how they may vary under different climate change scenarios. Pattern recognition tasks such as classification, object detection, and segmentation (i.e., pixel-level classification) have remained challenging problems in the weather and climate sciences. While there exist many empirical heuristics for detecting extreme events, the disparities between the output of these different methods even for a single event are large and often difficult to reconcile. Given the success of deep learning (DL) in tackling similar problems in computer vision, we advocate a DL-based approach. DL, however, works best in the context of supervised learning – when labeled datasets are readily available. Reliable labeled training data for extreme weather and climate events is scarce. We create “ClimateNet” – an open, community-sourced human-expert-labeled curated dataset that captures tropical cyclones (TCs) and atmospheric rivers (ARs) in high-resolution climate model output from a simulation of a recent historical period. We use the curated ClimateNet dataset to train a state-of-the-art DL model for pixel-level identification – i.e., segmentation – of TCs and ARs. We then apply the trained DL model to historical and climate change scenarios simulated by the Community Atmospheric Model (CAM5.1) and show that the DL model accurately segments the data into TCs, ARs, or “the background” at a pixel level. Further, we show how the segmentation results can be used to conduct spatially and temporally precise analytics by quantifying distributions of extreme precipitation conditioned on event types (TC or AR) at regional scales. The key contribution of this work is that it paves the way for DL-based automated, high-fidelity, and highly precise analytics of climate data using a curated expert-labeled dataset – ClimateNet. ClimateNet and the DL-based segmentation method provide several unique capabilities: (i) they can be used to calculate a variety of TC and AR statistics at a fine-grained level; (ii) they can be applied to different climate scenarios and different datasets without tuning as they do not rely on threshold conditions; and (iii) the proposed DL method is suitable for rapidly analyzing large amounts of climate model output. While our study has been conducted for two important extreme weather patterns (TCs and ARs) in simulation datasets, we believe that this methodology can be applied to a much broader class of patterns and applied to observational and reanalysis data products via transfer learning.

54 ENVIRONMENTAL SCIENCES↗

Data communication requirements for the advanced NAS network

The goal of the Numerical Aerodynamic Simulation (NAS) Program is to provide a powerful computational environment for advanced research and development in aeronautics and related disciplines. The present NAS system consists of a Cray 2 supercomputer connected by a data network to a large mass storage system, to sophisticated local graphics workstations, and by remote communications to researchers throughout the United States. The program plan is to continue acquiring the most powerful supercomputers as they become available. In the 1987/1988 time period it is anticipated that a computer with 4 times the processing speed of a Cray 2 will be obtained and by 1990 an additional supercomputer with 16 times the speed of the Cray 2. The implications of this 20-fold increase in processing power on the data communications requirements are described. The analysis was based on models of the projected workload and system architecture. The results are presented together with the estimates of their sensitivity to assumptions inherent in the models.

Levin, Eugene↗

Physics-informed machine learning

Despite great progress in simulating multiphysics problems using the numerical discretization of partial differential equations (PDEs), one still cannot seamlessly incorporate noisy data into existing algorithms, mesh generation remains complex, and high-dimensional problems governed by parameterized PDEs cannot be tackled. Moreover, solving inverse problems with hidden physics is often prohibitively expensive and requires different formulations and elaborate computer codes. Machine learning has emerged as a promising alternative, but training deep neural networks requires big data, not always available for scientific problems. Instead, such networks can be trained from additional information obtained by enforcing the physical laws (for example, at random points in the continuous space-time domain). Such physics-informed learning integrates (noisy) data and mathematical models, and implements them through neural networks or other kernel-based regression networks. Moreover, it may be possible to design specialized network architectures that automatically satisfy some of the physical invariants for better accuracy, faster training and improved generalization. Furthermore, we review some of the prevailing trends in embedding physics into machine learning, present some of the current capabilities and limitations and discuss diverse applications of physics-informed learning both for forward and inverse problems, including discovering hidden physics and tackling high-dimensional problems.

97 MATHEMATICS AND COMPUTING↗