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Scientific machine learning has been successfully applied to inverse problemsand PDE discovery in computational physics. One caveat concerning currentmethods is the need for large amounts of ("clean") data, in order tocharacterize the full system response and discover underlying physical models.Bayesian methods may be particularly promising for overcoming these challenges,as they are naturally less sensitive to the negative effects of sparse andnoisy data. In this paper, we propose to use Bayesian neural networks (BNN) inorder to: 1) Recover the full system states from measurement data (e.g.temperature, velocity field, etc.). We use Hamiltonian Monte-Carlo to samplethe posterior distribution of a deep and dense BNN, and show that it ispossible to accurately capture physics of varying complexity, withoutoverfitting. 2) Recover the parameters instantiating the underlying partialdifferential equation (PDE) governing the physical system. Using the trainedBNN, as a surrogate of the system response, we generate datasets of derivativesthat are potentially comprising the latent PDE governing the observed systemand then perform a sequential threshold Bayesian linear regression (STBLR),between the successive derivatives in space and time, to recover the originalPDE parameters. We take advantage of the confidence intervals within the BNNoutputs, and introduce the spatial derivatives cumulative variance into theSTBLR likelihood, to mitigate the influence of highly uncertain derivative datapoints; thus allowing for more accurate parameter discovery. We demonstrate ourapproach on a handful of example, in applied physics and non-linear dynamics.

Bayesian Deep Learning for Partial Differential Equation Parameter Discovery with Sparse and Noisy Data