We present a new category of physics-informed neural networks called physics informed variational embedding generative adversarial network (PI-VEGAN), that effectively tackles the forward, inverse, and mixed problems of stochastic differential equations. In these scenarios, the governing equations are known, but only a limited number of sensor measurements of the system parameters are available. We integrate the governing physical laws into PI-VEGAN with automatic differentiation, while introducing a variational encoder for approximating the latent variables of the actual distribution of the measurements. These latent variables are integrated into the generator to facilitate accurate learning of the characteristics of the stochastic partial equations. Our model consists of three components, namely the encoder, generator, and discriminator, each of which is updated alternatively employing the stochastic gradient descent algorithm. We evaluate the effectiveness of PI-VEGAN in addressing forward, inverse, and mixed problems that require the concurrent calculation of system parameters and solutions. Numerical results demonstrate that the proposed method achieves satisfactory stability and accuracy in comparison with the previous physics-informed generative adversarial network (PI-WGAN).

翻译：我们提出了一类新的物理信息神经网络，称为物理信息变分嵌入生成对抗网络（PI-VEGAN），有效解决了随机微分方程的正问题、反问题及混合问题。在这些场景中，控制方程已知，但仅有少量系统参数的传感器测量值可用。我们将控制物理定律通过自动微分融入PI-VEGAN，同时引入变分编码器来近似测量值实际分布的潜在变量。这些潜在变量被集成到生成器中，以促进对随机偏方程特征的精确学习。我们的模型由三个组件组成，即编码器、生成器和判别器，每个组件通过随机梯度下降算法交替更新。我们评估了PI-VEGAN在解决需要同时计算系统参数和解决方案的正问题、反问题和混合问题中的有效性。数值结果表明，与先前的物理信息生成对抗网络（PI-WGAN）相比，所提出方法在稳定性和精度方面均取得了令人满意的性能。