Efficient Neural Network Based Numerical Schemes for Hyperbolic Conservation Laws

基于神经网络的方法在许多科学计算问题上取得了成功，但对于其他许多问题，与经典的数值方法相比，它们仍然缺乏令人满意的实用效率。PI将探索提高基于神经网络的方法求解双曲型守恒律的效率的各种方法，双曲型守恒律是一类用于包括气体动力学和基本描述输运在内的许多重要应用的模型方程。此外，还将探索先进的优化算法。作为一种求解偏微分方程组的通用方法，基于神经网络的方法在许多应用中仍然远远低于经典的数值方法，特别是对于双曲型守恒律。PI将探索提高基于神经网络的方法求解含时双曲型守恒律的效率的方法，通过使用神经网络作为空间离散化以及通过非光滑凸优化来加强凸不变域的适当限制。神经网络的结构化确定性初始化和用于更新单元平均的有限体积方法可用于加速寻找神经网络解的优化的收敛。该项目的另一个重点是探索最近在数值偏微分方程组方面取得的突破对设计更有效的优化算法的启发。此外，还将探索用于梯度流的无条件稳定格式的优化技术。研究了一种构造基于神经网络的守恒律数值格式的新方法。将使用有限体积公式，以便经典的时间推进工具可以很容易地与神经网络空间离散化相结合，以简化加速收敛的优化问题。本课程将探讨加强凸不变域的限制器的非光滑优化算法的严格分析，以及有效的限制器的实现。最近在相场方程无条件稳定格式方面的突破将被应用于大规模优化算法，以寻找可能更有效的稳态解算器，用于数据科学中的梯度下降型算法。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。

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.