Efficient Neural Network Based Numerical Schemes for Hyperbolic Conservation Laws

Purdue University, West Lafayette IN

Investigators

Abstract

Neural network based methods have achieved success for many scientific computing problems, but for many other problems, they still lack satisfying and practical efficiency when compared to classical numerical methods. The PI will explore various approaches for enhancing efficiency of neural network based methods for solving hyperbolic conservation laws, which is a class of model equations used in many important applications including gas dynamics and basically describe transport. In addition, advanced optimization algorithms will be explored. As a generic approach for solving PDEs, neural network based methods are still way less efficient than classical numerical methods in many applications, especially for hyperbolic conservation laws. The PI will explore methods for enhancing efficiency of neural network based methods for solving time-dependent hyperbolic conservation laws by using neural network as a spatial discretization along with suitable limiters for enforcing convex invariant domain by non-smooth convex optimization. A structured deterministic initialization of a neural network and a finite volume method for updating cell averages can be used to accelerate convergence of optimization for finding neural network solutions. Another focus of the project is to explore inspirations of recent breakthroughs in numerical PDEs toward designing more efficient optimization algorithms. In addition, optimization techniques from unconditionally stable schemes for gradient flow will be explored. A novel approach for constructing efficient neural network based numerical schemes for conservation laws will be investigated. A finite volume formulation will be used so that classical time marching tools can be easily combined with a neural network spatial discretization to simplify the optimization problem for acceleration of convergence. Rigorous analysis of non-smooth optimization algorithms for a limiter enforcing convex invariant domain along with efficient limiter implementation will be explored. Recent breakthroughs in unconditionally stable schemes for phase field equations will be applied to large scale optimization algorithms to seek possibly more efficient steady state solvers for gradient descent type algorithms in data science. 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.