Adversarial uncertainty quantification in physics-informed neural networks

Abstract

We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physicsinformed neural networks. Specifically, we employ latent variable models to construct probabilistic representations for the system states, and put forth an adversarial inference procedure for training them on data, while constraining their predictions to satisfy given physical laws expressed by partial differential equations. Such physics-informed constraints provide a regularization mechanism for effectively training deep generative models as surrogates of physical systems in which the cost of data acquisition is high, and training data-sets are typically small. This provides a flexible framework for characterizing uncertainty in the outputs of physical systems due to randomness in their inputs or noise in their observations that entirely bypasses the need for repeatedly sampling expensive experiments or numerical simulators. We demonstrate the effectiveness of our approach through a series of examples involving uncertainty propagation in non-linear conservation laws, and the discovery of constitutive laws for flow through porous media directly from noisy data.

Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data

“…A similar approach is also applied to learn the constitutive relationship in a Darcy flow [40]. The PINN approach has been recently extended to assimilate multi-fidelity training data [41], and its UQ analyses have been explored based on arbitrary polynomial chaos [42] and adversarial inference [43]. Similar ideas of using physical constraints to regularize the DNN training have also been investigated in [44][45][46][47].…”

Weak adversarial networks for high-dimensional partial differential equations

“…More recently, an adaptive collocation strategy is presented for a method in [27]. In [28], an adversarial inference procedure is used for quantifying and propagating uncertainty in systems governed by non-linear differential equations, where the discriminator distinguishes the real observation and the approximation provided by the generative network through the given physical laws expressed by PDEs and the generator tries to fool the discriminator. To the best of our knowledge, none of the existing methods models the solution and test function in the weak solution form of the PDE as primal and adversarial networks as proposed in the present work.…”

Modeling the dynamics of PDE systems with physics-constrained deep auto-regressive networks

“…The mean squared error defined in Eq. (38) as well as the energy squared error defined in Eq. (44), both generalized to two dimensions, are evaluated for each timestep for the 200 test scenarios.…”