Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations

Abstract

We developed a new class of physics-informed generative adversarial networks (PI-GANs) to solve in a unified manner forward, inverse and mixed stochastic problems based on a limited number of scattered measurements. Unlike standard GANs relying only on data for training, here we encoded into the architecture of GANs the governing physical laws in the form of stochastic differential equations (SDEs) using automatic differentiation. In particular, we applied Wasserstein GANs with gradient penalty (WGAN-GP) for its enhanced stability compared to vanilla GANs. We first tested WGAN-GP in approximating Gaussian processes of different correlation lengths based on data realizations collected from simultaneous reads at sparsely placed sensors. We obtained good approximation of the generated stochastic processes to the target ones even for a mismatch between the input noise dimensionality and the effective dimensionality of the target stochastic processes. We also studied the overfitting issue for both the discriminator and generator, and we found that overfitting occurs also in the generator in addition to the discriminator as previously reported. Subsequently, we considered the solution of elliptic SDEs requiring approximations of three stochastic processes, namely the solution, the forcing, and the diffusion coefficient. We used three generators for the PI-GANs, two of them were feed forward deep neural networks (DNNs) while the other one was the neural network induced by the SDE. Depending on the data, we employed one or multiple feed forward DNNs as the discriminators in PI-GANs. Here, we have demonstrated the accuracy and effectiveness of PI-GANs in solving SDEs for up to 30 dimensions, but in principle, PI-GANs could tackle very high dimensional problems given more sensor data with low-polynomial growth in computational cost.

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• PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• Adversarial Uncertainty Quantification in Physics-Informed Neural Networks
• Deep Autoencoder based Energy Method for the Bending, Vibration, and Buckling Analysis of Kirchhoff Plates
• SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics
• Enforcing Statistical Constraints in Generative Adversarial Networks for Modeling Chaotic Dynamical Systems
• Solving forward and inverse problems of the logarithmic nonlinear Schrodinger equation with PT-symmetric harmonic potential via deep learning
• On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton--Jacobi partial differential equations
• PI-VAE: Physics-Informed Variational Auto-Encoder for stochastic differential equations
• Multi-Output Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Uncertainties
• Physics Informed Deep Learning for Transport in Porous Media. Buckley Leverett Problem
• Learning the temporal evolution of multivariate densities via normalizing flows
• The Seven-League Scheme: Deep learning for large time step Monte Carlo simulations of stochastic differential equations
• Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks