Block 1

Background

• Phononic crystal(PhC): Material with periodic Young's modulus E and Density ρ
• Eigenvalue problem of PhC is a highly non-linear input-output mapping problem
• The problem is to get the eigenvalues given a certain material(E,ρ,v) and certain wave numbers(Q1,Q2) by solving the following physical equations
• 公式1
• 公式2

What is CNN(或者其他题目？)

• CNN: Convolutional Neuron Network
• Convolutional layers involved: Kernels capable of capturing highly non-linear features in different hierachies
• Maxpool layers involved: Dimension Reduced without losing important information, processing speeded up, models more robust
• CNN 样例图

Why CNN is used

• Traditional methods: Invloves PDE solving and complicated process
• Trained CNN: An approach based on pure Data-driven architectures, sidesteps cumbersome algorithms to cpature the input-output map F w,b
• 公式3
• CNN can naturally represent tensors in physical problems
• Suitable for pattern recognization, widely applied in image processing
• Input of PhC problem can be interpreted as 5-channel "images"

Block 2

Forward Problem

• Goal: For the PhC problem, construct and train a CNN architecture which gives e, given a certain material M(E,ρ,v) and wave number Q(Q1,Q2)

Input and output data

• Input [128,128,5]: One 128*128 grid domain, one 5-vector on each point in domain representing E,ρ,v,Q1,Q2
• Output [50,1]: One 50-vector for 50 eigenvalues
• One input designates one specific combination of material and wave number.
• One output corresponds to the eigenvalues given one specific M and Q combination.
• All data normed by:
• 标准化公式 (x-x拔)/(xmax-xmin)

CNN architecture

• Input[128,128,5]→Conv2d(Relu)→Maxpool→Conv2d(Relu)→Maxpool→Conv2d(Relu)→Maxpool→Dense(Relu)→Dense(Linear)→Ouput[50,1]
• 放图

Result

• CNN is trained upon datasets with increasing numbers of training and validation data. A series of models acquired and tested by a size=22 test dataset.
• As the number of training data goes on, the MAE performance of prediction in test set is below
• 训练数量 图

Block 3

inverse problem

在inverse problem中，我们致力于通过神经网络来解决Phononics Crystal的eigenvalue problem的逆运算。 原公式为【】，通过正向网络可以得到，y=f(x)，现在我们希望通过神经网络训练出一个模型，将原正向的输出，作为现在的输入，原输入作为现在的输出，找到一个对应的映射关系————x=g(y)，其中，输入y是一个维度为50的特征值向量，输出x是一个二维平面（128*128）上的每个点的5个物理参数（杨氏模量，泊松系数，以及材料密度）。