CISE-ANR: Small: Evolutional deep neural network for resolution of high-dimensional partial differential equations

CISE-ANR：小型：用于求解高维偏微分方程的进化深度神经网络

• 批准号: 2214925
• 负责人: Tamer Zaki
• 金额: $ 59.89万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-01-01 至 2025-12-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2214925&HistoricalAwards=false
• 关键词: CISE; ANR; Small; Evolutional; deep

A vast number of phenomena across engineering, physics, economics and operational research are difficult to computationally predict because they depend on a large number of dimensions. A familiar example is the breakup of a liquid jet into a very large number of droplets, interacting with the background time-dependent and spatially varying field. A direct connection can be drawn between this example and the continuing pandemic where aerosol transmission of airborne pathogens is an important aspect of the contamination process. In such high-dimensional problems, the computational complexity increases with the number of particles. Therefore, developing an efficient and fast computational algorithms for solving the underlying equations will have a considerable impact on the engineering community, and will also come with significant ramifications across a wide range of disciplines including physics, medicine and public health. Machine-learning holds significant promise to revolutionize this vast range of applications by accelerating the solution of these high-dimensional, complex problems. Conventional machine-learning approaches rely on training data to approximate solutions of the governing equations, but such data are often either costly to generate or may not be available. One unique exception is the recently invented evolutional deep neural networks (EDNN) which do not rely on training. Instead, these networks forecast, or predict, the evolution of the pertinent physics by solving the governing equations. This unique feature is possible because the governing equations are recast in terms of the network parameters which can then evolve according to the physical laws to accurately predict the evolution of the system. In this effort—a collaboration between the United States and France—EDNN algorithms are developed for accurate and efficient solution of high-dimensional partial differential equations. Fundamental challenges related to the design of the optimal network architecture, dynamic adaptivity of the solution and scalability for massive parallelism are addressed, and evaluated against benchmark high-fidelity data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

工程、物理学、经济学和运筹学领域的大量现象很难通过计算来预测，因为它们依赖于大量的维度。一个常见的例子是液体射流破碎成大量液滴，并与液滴相互作用。背景时间依赖和空间变化的领域可以在这个例子和持续的流行病之间建立直接联系，其中空气传播的病原体的气溶胶传播是污染过程的一个重要方面，在这种高维问题中，计算复杂性随着时间的推移而增加。因此，开发一种高效、快速的计算算法来求解基本方程将对工程界产生相当大的影响，并且还将对包括物理、医学和公共卫生在内的广泛学科产生重大影响。通过加速这些高维复杂问题的解决，传统的机器学习方法依赖于训练数据来近似求解控制方程，从而彻底改变了广泛的应用，但此类数据通常要么生成成本高昂，要么可能无法获得。一个独特的例外是最近发明的。进化深度神经网络（EDNN）不依赖于训练，而是通过求解控制方程来预测相关物理的演化，因为控制方程是根据网络参数可以根据物理定律演化，从而准确预测系统的演化。在这项由美国和法国合作的项目中，我们开发了 EDNN 算法，用于准确、高效地求解高维偏高效微分方程。与设计有关的基本挑战。解决方案的最佳网络架构、解决方案的动态适应性和大规模并行的可扩展性，并根据基准高保真数据进行评估。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优势和更广泛的影响进行评估，被认为值得支持审查标准。