AI 中文总结
该研究针对高维PDE约束优化的内存瓶颈，提出基于时间可逆动态低秩近似的伴随状态法，大幅降低动力学方程梯度优化的内存需求，并在等离子体物理示例中验证了方法有效性。
AI 中文摘要
针对高维问题（如动力学方程）的偏微分方程约束优化，核心挑战在于内存成本往往过高。采用伴随状态法计算梯度需要存储正向解的全部时间历程，而对于这类问题，仅存储单个正向解实例的内存成本就可能已成为限制因素，因此该方法显然不可行。本文提出一种内存高效的伴随状态法，该方法采用动态低秩近似（一种模型降阶技术）压缩正向解与伴随解，并通过时间可逆低秩积分器绕过存储全部正向解的需求。动态低秩方法会带来诸多挑战：在秩亏缺情况下可逆性会失效，且低秩轨迹可能表现出混沌行为，后者尤其会对优化问题产生一系列重要影响。我们解决了这些挑战，并证明所提方法可大幅降低动力学方程梯度优化的内存需求，具体研究了两个来自动力学等离子体物理的示例：优化束分布以抑制尾 bump 不稳定性，以及利用外部电场塑造粒子束。
英文摘要
The primary challenge of conducting PDE-constrained optimization for high-dimensional problems, such as kinetic equations, is the often prohibitive memory cost. Computing gradients using the adjoint state method would require the storage of the entire time history of the forward solution. For such problems, where the memory cost for storing a single instance of the forward solution can already be a limiting factor, this is clearly not feasible. In this paper, we propose a memory-efficient adjoint state method that compresses the forward and adjoint solution with a dynamical low-rank approximation (a model order reduction technique) and bypasses the need to store the entire forward solution by employing a time-reversible low-rank integrator. The dynamical low-rank approach introduces a number of challenges: reversibility can fail in the rank-deficient case and the low-rank trajectories can show chaotic behavior. In particular, the latter has a number of important consequences for the optimization problem. We address those challenges and show that our method can drastically reduce the memory requirement for gradient-based optimization of kinetic equations. In particular, we consider two examples from kinetic plasma physics: optimizing beam profiles to suppress a bump-on-tail instability and shaping a particle beam using external electric fields.
