论文标题
PI-VAE：随机微分方程的物理信息变异自动编码器
PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations
论文作者
论文摘要
我们提出了一类新的物理信息神经网络，称为物理信息变异自动编码器（PI-VAE），以求解随机微分方程（SDES）或涉及SDE的反问题。在这些问题中，管理方程是已知的，但只有有限数量的系统参数测量值。 PI-VAE由变量自动编码器（VAE）组成，该变量自动编码器（VAE）生成系统变量和参数的样本。该生成模型与管理方程式集成在一起。在此集成中，VAE输出的衍生物很容易使用自动分化计算，并用于基于物理的损失项。在这项工作中，选择损耗函数是改善性能的最大平均差异（MMD），并且使用随机梯度下降算法对神经网络参数进行迭代更新。我们首先在近似随机过程上测试提出的方法。然后，我们研究了与SDE相关的三种类型的问题：正向和逆问题，以及同时计算系统参数和解决方案的混合问题。与物理知识的生成对抗网络（PI-WGAN）相比，该方法的令人满意的精度和效率在数值上得到了证明。
We propose a new class of physics-informed neural networks, called physics-informed Variational Autoencoder (PI-VAE), to solve stochastic differential equations (SDEs) or inverse problems involving SDEs. In these problems the governing equations are known but only a limited number of measurements of system parameters are available. PI-VAE consists of a variational autoencoder (VAE), which generates samples of system variables and parameters. This generative model is integrated with the governing equations. In this integration, the derivatives of VAE outputs are readily calculated using automatic differentiation, and used in the physics-based loss term. In this work, the loss function is chosen to be the Maximum Mean Discrepancy (MMD) for improved performance, and neural network parameters are updated iteratively using the stochastic gradient descent algorithm. We first test the proposed method on approximating stochastic processes. Then we study three types of problems related to SDEs: forward and inverse problems together with mixed problems where system parameters and solutions are simultaneously calculated. The satisfactory accuracy and efficiency of the proposed method are numerically demonstrated in comparison with physics-informed generative adversarial network (PI-WGAN).
