Adversarial Uncertainty Quantification in Physics-Informed Neural Networks (1811.04026v1)

Abstract: We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physics-informed 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.

• The paper introduces a framework that integrates PINNs with adversarial inference to quantify uncertainty in physical systems governed by nonlinear differential equations.
• It leverages latent variable models and deep generative approaches to approximate probabilistic system states effectively in small data regimes.
• Numerical examples, such as Burgers' equation and porous media flow, validate the framework's robust predictive performance and precise uncertainty quantification.

Adversarial Uncertainty Quantification in Physics-Informed Neural Networks

The paper by Yibo Yang and Paris Perdikaris introduces an innovative deep learning framework tailored for capturing and propagating uncertainty in physical systems governed by non-linear differential equations. This framework capitalizes on the integration of physics-informed neural networks (PINNs) and adversarial inference mechanisms structured to comply with established physical laws, typically represented through partial differential equations (PDEs). The central innovation lies in utilizing latent variable models to create probabilistic representations of system states, therefore, enabling these models to serve as surrogates, especially useful when dataset sizes are minute or data acquisition costs are prohibitive.

The authors leverage deep generative models constrained by known physical laws, thereby regularizing the training of these models by embedding physical knowledge directly into the learning algorithm. This strategy significantly minimizes the reliance on traditional, data-intensive training processes often required in machine learning, empowering the models to effectively operate even with small data regimes.

From an implementation perspective, the framework extends the capabilities of classical PINNs by incorporating probabilistic elements that enable uncertainty quantification. This is achieved via adversarial learning, where the aim is to enable the model to infer accurate probabilistic outputs considering randomness in inputs or measurement noise. The adversarial training involves constructing generative models that approximate the desired solution while ensuring that the generated samples adhere to an underlying physics-constrained probability distribution.

The paper showcases the versatility of this approach through diverse examples, such as uncertainty propagation in non-linear conservation laws and the fundamental characterization of constitutive laws for flows through porous media based on incomplete, noisy data. Notably, the authors demonstrate that their proposed method offers not only a robust mean prediction for solutions but also a precise quantification of uncertainties involved, thus providing a comprehensive uncertainty characterization that aids subsequent decision-making processes.

Strong numerical results elucidate the efficacy of this framework. For example, in the application involving Burgers' equation, the model effectively captures and explores the uncertainties surrounding shock formation from noisy initial conditions. Additionally, in the derivation of constitutive relationships for porous media flow, the method accurately identifies state-dependent diffusion coefficients despite considerable noise, underscoring its practical applicability to real-world, data-constrained problems.

The paper's contributions extend beyond the immediate practical implications. Theoretically, it bridges a crucial gap in machine learning by reconciling the traditionally deterministic nature of PINNs with probabilistic modeling, thereby enhancing the robustness and interpretability of predictions. Furthermore, by efficiently combining generative adversarial networks (GANs) with variational inference techniques, the authors pioneer a path forward for uncertainty quantification in complex systems—a significant step towards adoption in scientific computing and engineering domains.

Future work could extend this methodology to even more complex systems, potentially scaling up to high-dimensional and high-fidelity models. The integration of more sophisticated generative models could further refine uncertainty quantification approaches, possibly amplifying performance and accuracy. Additionally, exploring alternative adversarial training strategies or regularization techniques might offer new avenues for reducing computational costs and enhancing model convergence.

In conclusion, the proposed framework epitomizes a productive merger between modern machine learning techniques and traditional scientific computing, leading to a methodologically sound approach for handling uncertainty in data-scarce environments. This work is poised to influence numerous fields where uncertainty quantification is critical, driving innovation in the way researchers approach data-driven, physics-informed modeling challenges.