Reliable predictions of critical phenomena, such as weather, wildfires and epidemics often rely on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales described by such PDEs are often prohibitively expensive. Consequently, coarse-grained simulations are usually deployed that adopt various heuristics and empirical closure terms to account for the missing information. We propose a novel and systematic approach for identifying closures in under-resolved PDEs using grid-based Reinforcement Learning. This formulation incorporates inductive bias and exploits locality by deploying a central policy represented efficiently by a Fully Convolutional Network (FCN). We demonstrate the capabilities and limitations of our framework through numerical solutions of the advection equation and the Burgers' equation. Our results show accurate predictions for in- and out-of-distribution test cases as well as a significant speedup compared to resolving all scales.

翻译：对天气、野火和流行病等关键现象的可靠预测通常依赖于偏微分方程（PDEs）描述的模型。然而，捕获此类PDE所描述的全部时空尺度的模拟往往计算成本过高。因此，通常采用粗粒度模拟，其采用各种启发式方法和经验闭合项来补偿缺失的信息。我们提出了一种新颖且系统的方法，利用基于网格的强化学习来识别欠分辨PDE中的闭合项。该框架通过部署一个由全卷积网络（FCN）高效表示的中心策略，融入了归纳偏置并利用了局部性。我们通过对平流方程和Burgers方程进行数值求解，展示了我们框架的能力与局限性。结果表明，我们的方法在分布内和分布外测试案例中均能实现准确预测，并且与解析所有尺度的模拟相比，能获得显著的加速。