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

工程学、物理学、经济学和运筹学中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。一个熟悉的例子是液体射流分解成非常大量的液滴，与背景时间相关和空间变化的场相互作用。这一例子与持续的大流行之间可以得出直接联系，在这种大流行中，气雾剂传播空气传播病原体是污染过程的一个重要方面。在这样的高维问题中，计算复杂度随着粒子数的增加而增加。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，并将在包括物理、医学和公共卫生在内的广泛学科中产生重大影响。机器学习通过加速这些高维、复杂问题的解决，有望给这种广泛的应用带来革命性的变化。传统的机器学习方法依赖于训练数据来近似控制方程的解，但这种数据通常要么生成成本很高，要么可能不可用。一个独特的例外是最近发明的进化深度神经网络(EDNN)，它不依赖于训练。相反，这些网络通过求解控制方程来预测或预测相关物理学的演变。这一独特的特征是可能的，因为控制方程是根据网络参数重塑的，然后网络参数可以根据物理定律演变，以准确地预测系统的演变。在这项工作中-美国和法国的合作-EDNN算法被开发出来，用于准确和有效地求解高维偏微分方程组。与最优网络架构的设计、解决方案的动态适应性和大规模并行的可扩展性相关的基本挑战被解决，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审查标准进行评估，被认为值得支持。

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.