Variational Autoencoding of PDE Inverse Problems
This work addresses the challenge of incorporating prior physical knowledge into machine learning for PDE inverse problems, which is important for small data regimes and interpretability, though it is incremental as it builds on existing variational and surrogate methods.
The authors tackled the problem of combining physical models with data-driven surrogates for PDE inverse problems, resulting in a framework that provides accelerated Bayesian inference and acts as a physics-informed regularizer, demonstrated with accuracy and computational efficiency in real-world settings.
Specifying a governing physical model in the presence of missing physics and recovering its parameters are two intertwined and fundamental problems in science. Modern machine learning allows one to circumvent these, via emulators and surrogates, but in doing so disregards prior knowledge and physical laws that are especially important for small data regimes, interpretability, and decision making. In this work we fold the mechanistic model into a flexible data-driven surrogate to arrive at a physically structured decoder network. This provides accelerated inference for the Bayesian inverse problem, and can act as a drop-in regulariser that encodes a-priori physical information. We employ the variational form of the PDE problem and introduce stochastic local approximations as a form of model based data augmentation. We demonstrate both the accuracy and increased computational efficiency of the framework on real world settings and structured spatial processes.