  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).
