Physics-Informed Generator-Encoder Adversarial Networks with Latent
  Space Matching for Stochastic Differential Equations
Ruisong Gao Min Yang Jin Zhang
We propose a new class of physics-informed neural networks, called
Physics-Informed Generator-Encoder Adversarial Networks, to effectively address
the challenges posed by forward, inverse, and mixed problems in stochastic
differential equations. In these scenarios, while the governing equations are
known, the available data consist of only a limited set of snapshots for system
parameters. Our model consists of two key components: the generator and the
encoder, both updated alternately by gradient descent. In contrast to previous
approaches of directly matching the approximated solutions with real snapshots,
we employ an indirect matching that operates within the lower-dimensional
latent feature space. This method circumvents challenges associated with
high-dimensional inputs and complex data distributions, while yielding more
accurate solutions compared to existing neural network solvers. In addition,
the approach also mitigates the training instability issues encountered in
previous adversarial frameworks in an efficient manner. Numerical results
provide compelling evidence of the effectiveness of the proposed method in
solving different types of stochastic differential equations.