Collaborative Research: CISE-ANR: CCF/AF: Small: Evolutional Deep Neural Network for Resolution of High-Dimensional Partial Differential Equations – EDNNForest

Machine-learning (ML) holds significant promise in revolutionizing a wide range of applications, in particular in the domain of multi-scale and multi-physics problems. Success in realizing the promise of ML is predicated on the availability of training data, which are often obtained from scientific computations. Conventional approaches to solving the equations of physics require difficult and specialized software development, grid generation and adaptation, and the use of specialized data and software pipelines that differ from those adopted in ML. A disruptive new approach that was recently proposed by the US team is Evolutional Deep Neural Networks (EDNN, pronounced ``Eden") which leverages the software and hardware infrastructure used in ML to replace conventional computational methods, and to tackle their shortcomings. EDNN is unique because it does not rely on training to express known solutions, but rather the network parameters evolve using the governing physical laws such that the network can predict the evolution of the physical system. In the proposed effort, the EDNN framework will be extended to solve high-dimensional partial differential equations, used to model a vast range of phenomena in economics, finance, operational research, and multi-phase fluid dynamics, where population balance equations govern phenomena as diverse as aerosol transmission of airborne pathogens or mixing enhancement in energy conversion devices. The simulation of such flows is an open issue of particular interest to the US and French teams, a strong motivation for the proposed collaboration. We will demonstrate the ease of software development using automatic differentiation tools and the capacity of EDNN to eliminate the curse of dimensionality and the tyranny of moment closure. Success stands to disrupt and transform the decades-old computational approach to solving nonlinear differential equations and to remove the barriers to generation of training data required for ML.