Deep Particle Algorithms and Advection-Reaction-Diffusion Transport Problems

该项目研究物理和生物学数学模型中随机相互作用粒子表示的学习和生成分布的计算方法。将开发一种集成粒子模拟和机器学习的新型计算工具，用于科学和工程领域的广泛应用，例如细菌和昆虫大规模模式形成的建模和预测，癌细胞入侵，野火蔓延和能源生产中的湍流燃烧。从粒子模拟生成的数据中训练的轻量级深度神经网络显著加快了预测速度，并有可能扩展到具有更广泛影响的各个领域的现场数据。计划中的教育和拓展活动可以帮助学生，特别是那些在多个校园中被忽视的学生，追求更高的学位和职业。所研究的数学模型是三维时空相关的平流-反应-扩散偏微分方程，特别是当它们的解在未知位置出现大梯度或浓度时，传统的基于网格的方法很难计算。该项目旨在通过深度学习、最佳传输和场耦合随机相互作用粒子动力学（即所谓的深度粒子）的集成框架来解决这一挑战。该方法是无网格的，自适应的，不需要粒子分布之间有可逆的映射。该项目还研究了所得算法的收敛和加速方面。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。

The project studies computational methods for learning and generating distributions of stochastic interacting particle representations of mathematical models in physics and biology. A new class of computational tools integrating particle simulation and machine learning will be developed for a wide range of applications in science and engineering such as modeling and prediction of large scale pattern formation of bacteria and insects, cancer cell invasion, wildfire spreading, and turbulent combustion in energy production. The light weight deep neural networks trained from data generated by particle simulations speed up prediction significantly, and potentially extend to field data in various domains with broader impacts. The planned education and out-reach activities help students, especially under-represented students, on multiple campuses to pursue advanced degrees and careers.The mathematical models under study are three space dimensional time dependent advection-reaction-diffusion partial differential equations challenging to compute by traditional mesh based methods especially when their solutions develop large gradients or concentrations at unknown locations. The project aims to address this challenge through an integrated framework of deep learning, optimal transport, and field-coupled stochastic interacting particle dynamics, the so called Deep Particle. The approach is mesh free, self-adaptive and does not require particle distributions to have invertible mappings between them. The project also investigates convergence and acceleration aspects of the resulting algorithms.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.