Reliable predictions of critical phenomena, such as weather, wildfires and epidemics are often founded on models described by Partial Differential Equations (PDEs). However, simulations that capture the full range of spatio-temporal scales in such PDEs are often prohibitively expensive. Consequently, coarse-grained simulations that employ heuristics and empirical closure terms are frequently utilized as an alternative. We propose a novel and systematic approach for identifying closures in under-resolved PDEs using Multi-Agent Reinforcement Learning (MARL). The MARL formulation incorporates inductive bias and exploits locality by deploying a central policy represented efficiently by Convolutional Neural Networks (CNN). We demonstrate the capabilities and limitations of MARL 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：关键现象（如天气、野火和流行病）的可靠预测通常基于由偏微分方程（PDEs）描述的模型。然而，能够捕捉此类PDEs全时空尺度的模拟往往计算代价过高。因此，采用启发式方法和经验闭合项的粗粒度模拟常被用作替代方案。我们提出了一种新颖且系统的方法，利用多智能体强化学习（MARL）来识别欠分辨PDEs中的闭合项。该MARL框架通过引入归纳偏置并利用局部性特性，采用由卷积神经网络（CNN）高效表征的中央策略。我们通过对流方程和Burgers方程数值解展示了MARL的能力与局限性。结果表明，该方法在分布内和分布外测试案例中均能实现精确预测，并且相较于全尺度解析方法具有显著的加速效果。