Simulating the time evolution of physical systems is pivotal in many scientific and engineering problems. An open challenge in simulating such systems is their multi-resolution dynamics: a small fraction of the system is extremely dynamic, and requires very fine-grained resolution, while a majority of the system is changing slowly and can be modeled by coarser spatial scales. Typical learning-based surrogate models use a uniform spatial scale, which needs to resolve to the finest required scale and can waste a huge compute to achieve required accuracy. In this work, we introduce Learning controllable Adaptive simulation for Multi-resolution Physics (LAMP) as the first full deep learning-based surrogate model that jointly learns the evolution model and optimizes appropriate spatial resolutions that devote more compute to the highly dynamic regions. LAMP consists of a Graph Neural Network (GNN) for learning the forward evolution, and a GNN-based actor-critic for learning the policy of spatial refinement and coarsening. We introduce learning techniques that optimizes LAMP with weighted sum of error and computational cost as objective, allowing LAMP to adapt to varying relative importance of error vs. computation tradeoff at inference time. We evaluate our method in a 1D benchmark of nonlinear PDEs and a challenging 2D mesh-based simulation. We demonstrate that our LAMP outperforms state-of-the-art deep learning surrogate models, and can adaptively trade-off computation to improve long-term prediction error: it achieves an average of 33.7% error reduction for 1D nonlinear PDEs, and outperforms MeshGraphNets + classical Adaptive Mesh Refinement (AMR) in 2D mesh-based simulations. Project website with data and code can be found at: http://snap.stanford.edu/lamp.
翻译:模拟物理系统的时间演化在许多科学和工程问题中至关重要。模拟这样的系统的一个公开挑战是它们的多分辨率动态:系统的一小部分非常动态,需要非常精细的分辨率,而大部分系统变化缓慢,可以用更粗的空间尺度模拟。典型的基于学习的代理模型使用统一的空间尺度,它需要解析至所需最好的分辨率,并且可能会浪费巨大的计算来实现所需的准确性。在这项工作中,我们引入了学习可控的自适应多分辨率物理仿真(LAMP)作为第一个完整的基于深度学习的代理模型,它同时学习演化模型并优化适当的空间分辨率,将更多计算资源用于高动态区域。LAMP由图结构神经网络(GNN)组成,用于学习前向演化,以及基于GNN的Actor-Critic,用于学习空间细化和粗化的策略。我们引入了Learning Techniques,用加权误差和计算成本的总和作为目标来优化LAMP,让LAMP能够适应推理时的误差与计算权衡的变化。我们在非线性偏微分方程(1D)基准测试和具有挑战性的2D网格仿真中评估了我们的方法。我们证明了我们的LAMP优于最先进的深度学习代理模型,并可以自适应地权衡计算量以提高长期预测误差:它在1D非线性偏微分方程中的平均误差降低了33.7%,并优于MeshGraphNets+经典自适应网格细化(AMR)在2D网格仿真中。项目网站以及数据和代码可以在以下网址找到:http://snap.stanford.edu/lamp。