Physics-informed variational inference for uncertainty quantification of stochastic differential equations

Abstract

We propose a physics-informed learning based on variational autoencoder (VAE) to solve data-driven stochastic differential equations when the governing equation is known and a limited number of measurements are available. Our model integrates VAE with the given physical laws expressed by stochastic partial differential equations, allowing the encoder to infer the randomness of the solution. The decoder employs a separate structure of two neural networks, where one network learns the spatial behavior and the other network learns the random behavior of the solution, making both training and inference computationally efficient. We use an evidence lower bound (ELBO) as the loss function, which incorporates the given physical laws by using automatic differentiation to compute the differential operators. The proposed model can be used to solve data-driven forward and inverse stochastic differential equations in a unified framework. We demonstrate the efficiency of the proposed model for learning stochastic processes and solving various types of stochastic partial differential equations.