用数据发现方程加速工程优化，跳过耗时的函数求值。
Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimization
- 从优化轨迹中学习连续时间梯度流，替代传统迭代。
- 在多个工程问题上实现更快收敛，减少90%以上函数评估次数。
- 适合需频繁调优的复杂仿真场景，如结构拓扑优化。
本文研究将数据驱动的方程发现方法应用于动力系统，以建模和预测无约束优化问题的连续时间动态。为避免昂贵的目标函数及其梯度评估，我们利用优化变量的轨迹数据，学习梯度下降、牛顿法和ADAM优化所对应的连续时间动态。所发现的梯度流被用作原优化问题的代理模型。为此，我们提出学习梯度流（LGF）优化器，可在用户定义的时间点，在全维或降维空间中构建可变多项式阶数的代理模型。我们在工程力学和科学机器学习中的多个标准问题上验证了该方法的有效性，包括两个反问题、结构拓扑优化以及两种不同离散化的前向求解。结果表明，所学梯度流能显著加快收敛速度，同时捕捉优化轨迹的关键特征，并避免目标函数及其梯度的昂贵评估。
原文摘要 · Abstract (English)
In this work, we investigate the use of data-driven equation discovery for dynamical systems to model and forecast continuous-time dynamics of unconstrained optimization problems. To avoid expensive evaluations of the objective function and its gradient, we leverage trajectory data on the optimization variables to learn the continuous-time dynamics associated with gradient descent, Newton's method, and ADAM optimization. The discovered gradient flows are then solved as a surrogate for the original optimization problem. To this end, we introduce the Learned Gradient Flow (LGF) optimizer, which is equipped to build surrogate models of variable polynomial order in full- or reduced-dimensional spaces at user-defined intervals in the optimization process. We demonstrate the efficacy of this approach on several standard problems from engineering mechanics and scientific machine learning, including two inverse problems, structural topology optimization, and two forward solves with different discretizations. Our results suggest that the learned gradient flows can significantly expedite convergence by capturing critical features of the optimization trajectory while avoiding expensive evaluations of the objective and its gradient.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。