图1 不同科学领域的大致空间尺度和时间尺度

Fig.1 The spatial and temporal scales for different scientific disciplines

图2 系统动力学的描述设置

Fig.2 Notation for the system's dynamics

图3 不同仿真方法在光谱上的分布(第一性原理的传统数值方法(最左端)、数据驱动(最右端)以及其优势(蓝色)和局限性(橙色)。)

Fig.3 Different simulation methods lying on a spectrum (On the leftmost are classical numerical methods based on first principles. On the rightmost are the pure data-driven methods. Also denoted are the pros and cons for different methods.)

图4 传统数值方法将偏微分方程在时间和空间上离散化以便于数值求解

Fig.4 Classical numerical method discretizes the PDE in time and space to facilitate numerically solving them

图5 AI仿真模型学习代理模型fθ，最小化预测误差

Fig.5 AI-based simulation method learns a surrogate model fθ and minimizes prediction error

图6 GNS将物理系统建模成粒子系统，模拟水流(左)、沙子(中)、胶泥(右)的含时演化[14] (a) GNS预测结果；(b) 经过渲染后的结果

Fig.6 GNS models a physical system as a particle-based system, simulating the temporal dynamics of water (left), sand (middle), and gel (right)[14] (a) prediction results of GNS; (b) after rending

图7 GNS[14]网络架构: 系统在t时刻的状态(各粒子的位置和速度)作为输入，经过(c)编码器, (d)消息传递, (e)解码器，预测t+1时刻的加速度，经过欧拉法预测t+1时刻的系统状态。

Fig.7 GNS[14] network architecture: The input to the GNS is the state of the system at time t, including every particle's position and velocity. Then through (c) encoder, (d) message passing, and (e) decoder, it predicts the acceleration at time step t+1. Euler's method is used to predict the state at time step t+1.

图8 (a) HGNS架构[15]; (b) HGNS对水流体积的预测结果(截面)与真实值比较(可见其在20个月内预测效果与真实值精确符合。)

Fig.8 (a) HGNS architecture[15]; (b) its prediction of water volume (cross-section) vs. ground-truth (We see that its prediction matches well with ground-truth.)

图9 GraphCast架构[16](其构建的多尺度图神经网络，可以同时汇聚局域和多尺度的信息。) (a)encoder; (b) processor; (c) decoder; (d) simultaneous multi-mesh message-passing

Fig.9 Architecture of GraphCast[16] (Its multiscale graph network can aggregate local and multiscale information.) (a)encoder; (b) processor; (c) decoder; (d) simultaneous multi-mesh message-passing

图10 神经算子可以将无限维的函数映射到另一个无限维函数

Fig.10 Neural operator can map an infinite-dimensional function to another infinite dimensional function

图11 傅立叶神经算子[26]的网络架构

Fig.11 Architecture for the Fourier neural operators[26]

图12 科学设计和控制的任务设置(根据目标J，优化初始条件u0、控制序列m[0, T]、参数a或者边界$\partial \mathbb{X} $。)

Fig.12 Task setting for scientific design and control (Based on the objective J, it optimizes the initial condition u0, control sequence m[0, T], parameter a or boundary $\partial \mathbb{X} $.)

图13 在文[28]对飞机翼型的设计，优化目标为减小阻力(提出的基于代理模型+反向传播的方法达到传统求解器(DAFoam)优化的性能，并且速度更快。)(a)initial design; (b) optimized design; (c) drag; (d)reward

Fig.13 The airfoil design task in Ref.[28], where the design objective is minimizing drag (The surrogate model+backpropagation proposed by the authors attains similar performance as the classical solver (DAFoam), while significantly faster.) (a)initial design; (b) optimized design; (c) drag; (d) reward

图14 文[30]的实验结果(对于逆问题，如果在原空间进行反向传播优化系统初始条件(第一行，without prior)，则会优化出具有高频噪声的对抗样本。而根据文章所提出的在隐空间进行设计，由于有了先验知识，则能优化出与真实值非常一致的初始条件。)

Fig.14 Experimental results Ref.[30] (For the inverse problem, if backpropagation is performed in the original space to optimize the initial conditions of the system (first row, without prior), it will obtain adversarial samples with high-frequency noise. On the other hand, if performing optimization in the latent space as introduced in the paper, due to the prior knowledge, it will attain the initial conditions that match well with the ground-truth.)

图15 强化学习设置

Fig.15 Task setup for reinforcement learning

图16 瑞士的TCV磁约束可控核聚变托卡马克装置

Fig.16 Tokamakà configuration variable (TCV) fusion device in Switzerland

图17 文[31]的深度强化学习与托卡马克装置交互示意图(a)deployment; (b)TCV; (c)vessel cross section

Fig.17 Illustration of the interaction between the deep reinforcement learning method[31] and the tokamak device in TCV (a)deployment; (b)TCV; (c)vessel cross section

图18 文[35]中训练和推理的方法架构(a)训练时；(b)推理时

Fig.18 Illustration of training and inference pipeline[35] (a)train; (b) compositional design

图19 同时生成的系统仿真轨迹和机翼形状[35] (a)气流水平速度场；(b)气流垂直速度场; (c)压强场(中间红色和蓝色表示机翼形状，并发现在训练中未见的“编队飞行”模式。)

Fig.19 The system's state trajectory and airfoil shape by simultaneously generated[35] (a) horizontal velocity field; (b) vertical velocity field; (c) pressure field. (It also discovers formation flying, which does not appear in training.)

图20 (a) AI费曼架构及(b)其重新发现的部分方程

Fig.20 Architecture of (a) AI Feynman and (b) some of its re-discovered equations

图21 KAN架构[54]

Fig.21 Architecture of KAN[54]

图22 AI笛卡尔架构[64]

Fig.22 Pipeline of AI-descartes[64]