Data driven approximation of parametrized PDEs by reduced basis and neural networks
Data driven approximation of parametrized PDEs by reduced basis and neural networks
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DOI：
10.1016/j.jcp.2020.109550
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发表时间：
2020-09-01
影响因子：
4.1
通讯作者：
Pegolotti, Luca
中科院分区：
文献类型：
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作者：
Dal Santo, Niccolo;Deparis, Simone;Pegolotti, Luca
我们感兴趣的是偏微分方程的近似与数据驱动的方法的基础上减少的基础方法和机器学习。我们假设感兴趣的现象可以用一个参数化的偏微分方程来模拟，但物理参数的值是未知的或难以直接测量。我们的方法允许估计感兴趣的领域，例如材料样品的温度或流体的速度，给定域中少数点的数据。我们建议在最后一层嵌入一个简化基求解器作为奇异激活函数的神经网络来完成这项任务。减少的基础求解器帐户的基本物理现象，它是从快照从随机选择的值的物理参数在一个昂贵的离线阶段。然后，相同的全阶解用于神经网络的训练。事实上，所选择的架构类似于非对称自动编码器，其中解码器是缩减基求解器，因此它不包含可训练参数。由此产生的自动编码器的潜在空间包括馈送缩减基求解器的参数依赖量，这些参数依赖量取决于所考虑的偏微分方程，是物理参数本身的值或微分算子的仿射分解系数。
We are interested in the approximation of partial differential equations with a data-driven approach based on the reduced basis method and machine learning. We suppose that the phenomenon of interest can be modeled by a parametrized partial differential equation, but that the value of the physical parameters is unknown or difficult to be directly measured. Our method allows to estimate fields of interest, for instance temperature of a sample of material or velocity of a fluid, given data at a handful of points in the domain. We propose to accomplish this task with a neural network embedding a reduced basis solver as exotic activation function in the last layer. The reduced basis solver accounts for the underlying physical phenomenon and it is constructed from snapshots obtained from randomly selected values of the physical parameters during an expensive offline phase. The same full order solutions are then employed for the training of the neural network. As a matter of fact, the chosen architecture resembles an asymmetric autoencoder in which the decoder is the reduced basis solver and as such it does not contain trainable parameters. The resulting latent space of our autoencoder includes parameter-dependent quantities feeding the reduced basis solver, which – depending on the considered partial differential equation – are the values of the physical parameters themselves or the affine decomposition coefficients of the differential operators.