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，但在许多其他问题上，与经典数值方法相比，它们仍然缺乏令人满意的实用效率。PI将探索各种方法来提高基于神经网络的方法求解双曲守恒定律的效率，双曲守恒定律是一类模型方程，用于许多重要的应用，包括气体动力学和基本描述运输。此外，还将探索先进的优化算法。作为求解偏微分方程的一种通用方法，基于神经网络的方法在许多应用中仍然比经典数值方法效率低，特别是在求解双曲守恒律时。PI将探索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强制凸不变域的适当限制来提高基于神经网络的求解时变双曲守恒律的方法的效率的方法。神经网络的结构化确定性初始化和更新单元平均的有限体积方法可用于加速神经网络解的优化收敛。该项目的另一个重点是探索数值偏微分方程的最新突破对设计更有效的优化算法的启发。此外，还将探讨梯度流无条件稳定格式的优化技术。本文将研究一种基于神经网络的守恒律数值格式的高效构造方法。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相结合，以简化加速收敛的优化问题。严格分析非光滑优化算法的限制器强制凸不变域以及有效的限制器实现将被探索。相场方程无条件稳定格式的最新突破将应用于大规模优化算法，以寻求数据科学中梯度下降型算法可能更有效的稳态求解器。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。

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.