LEARNING CONTROLLABLE ADAPTIVE SIMULATION FOR MULTI-RESOLUTION PHYSICS

Abstract

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 Multiresolution 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 GNNbased 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.

1. INTRODUCTION

Simulating the time evolution of a physical system is of vital importance in science and engineering (Lynch, 2008; Carpanese, 2021; Sircombe et al., 2006; Courant et al., 1967; Lelievre & Stoltz, 2016) . Usually, the physical system has a multi-resolution nature: a small fraction of the system is highly dynamic, and requires very fine-grained resolution to simulate accurately, while a majority of the system is changing slowly. Examples include hazard prediction in weather forecasting (Majumdar et al., 2021) , disruptive instabilities in the plasma fluid in nuclear fusion (Kates-Harbeck et al., 2019) , air dynamics near the boundary for jet engine design (Athanasopoulos et al., 2009) , and more familiar examples such as wrinkles in a cloth (Pfaff et al., 2021) and fluid near the boundary for flow through the cylinder (Vlachas et al., 2022) . Due to the typical huge size of such systems, it is pivotal that those systems are simulated not only accurately, but also with as small of a computational cost as possible. A uniform spatial resolution that pays similar attention to regions with vastly different dynamics, will waste significant compute on slow-changing regions while may be insufficient for highly dynamic regions. 



To accelerate physical simulations, deep learning (DL)-based surrogate models have recently emerged as a promising alternative to complement (Um et al., 2020) or replace (Li et al., 2021) classical solvers. They reduce computation and accelerate the simulation with larger spatial (Um * Equal contribution. 1

