  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).
