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是许多科学与工程问题的关键。此类模拟中的一个开放挑战在于其多分辨率动力学特性：系统中极小部分区域具有极高动态性，需要极精细的分辨率，而大部分区域变化缓慢，可通过较粗的空间尺度建模。典型的基于学习的替代模型采用均匀空间尺度，这需要按最精细尺度进行解析，导致为达到所需精度而浪费大量计算资源。本研究提出学习可控自适应多分辨率物理模拟（LAMP），作为首个完全基于深度学习的替代模型，该模型联合学习演化模型并优化恰当的空间分辨率，将更多计算资源分配给高动态区域。LAMP包含用于学习前向演化的图神经网络（GNN），以及基于GNN的演员-评论家网络用于学习空间细化和粗化的策略。我们引入学习技术，以误差与计算成本的加权和作为目标函数优化LAMP，使其能在推理时自适应于误差与计算权衡的不同相对重要性。我们在非线性偏微分方程的一维基准测试和具有挑战性的二维网格模拟中评估该方法，结果表明LAMP优于现有最先进的深度学习替代模型，能自适应权衡计算资源以改善长期预测误差：在一维非线性偏微分方程中实现平均33.7%的误差降低，并在二维网格模拟中优于MeshGraphNets结合经典自适应网格细化（AMR）方法。项目网站及数据代码参见：http://snap.stanford.edu/lamp。