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

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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.