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

工程、物理、经济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。一个熟悉的例子是液体射流分解成大量的液滴，与背景随时间和空间变化的场相互作用。这个例子与持续的大流行之间可以得出直接联系，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，计算复杂度随着粒子数量的增加而增加。因此，开发一种高效、快速的计算算法来解决潜在的方程将对工程界产生相当大的影响，并将在包括物理学、医学和公共卫生在内的广泛学科中产生重大影响。通过加速解决这些高维、复杂的问题，机器学习有望彻底改变这一广泛的应用。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，但这些数据通常要么生成成本高昂，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新建模的，然后网络参数可以根据物理定律精确地预测系统的演化。在这项努力中——美国和法国之间的合作——ednn算法被开发用于精确和有效地解决高维偏微分方程。解决了与最优网络架构设计、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，并针对基准高保真数据进行了评估。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.