Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In particular, in addition to the deep neural network (DNN) for the solution, a second DNN is considered that represents the residual of the PDE. The residual is then combined with the mismatch in the given data of the solution in order to formulate the loss function. This framework is effective but is lacking uncertainty quantification of the solution due to the inherent randomness in the data or due to the approximation limitations of the DNN architecture. Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty. We first account for the parametric uncertainty when the parameter in the differential equation is represented as a stochastic process. Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors. We can then make predictions from new sensor measurements very efficiently with the trained DNNs. Moreover, we employ dropout to correct the overfitting and also to quantify the uncertainty of DNNs in approximating the modal functions. We then design an active learning strategy based on the dropout uncertainty to place new sensors in the domain in order to improve * Corresponding Author the predictions of DNNs. Several numerical tests are conducted for both the forward and the inverse problems to quantify the effectiveness of PINNs combined with uncertainty quantification. This NN-aPC new paradigm of physics-informed deep learning with uncertainty quantification can be readily applied to other types of stochastic PDEs in multi-dimensions.

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. Here again we assumed data realizations collected from simultaneous reads at a limited number of sensors for the multiple stochastic processes. 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. For the case where we have one group of data, we employed one feed forward DNN as the discriminator while for the case of multiple groups of data we employed multiple discriminators in PI-GANs. We solved forward, inverse, and mixed problems without changing the framework of PI-GANs, obtaining both the means and standard deviations of the stochastic solution and the diffusion coefficient in good agreement with benchmarks. Here, we have demonstrated the 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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