Posterior Concentration of Bayesian Physics-Informed Neural Networks for Elliptic PDEs

Unlike a standard PINN--which produces an approximate Deep neural networks (DNNs) or multi-layer perceptronssolution by minimizing a PDE-residual loss and thus yields (MLPs) offer various inherent advantages over traditionalonly a point estimate, failing to quantify uncertainty inapproaches of scientific computing and data analysis, suchduced by noisy or limited data, a Bayesian PINN returns a as finite element methods, wavelets and kernel methods, full posterior distribution over solutions by combining the which are often hampered by the irregular and nonlinearuncertain information from the likelihood (data) and the data structures and the high input dimensions. In contrast, DNNs are capable of approximating a rich class of functions prior. Bayesian neural networks, originating in the seminal works of MacKay (MacKay, 1995) and Neal (Neal, 1995), with aforementioned complexities and can also easily en-have been extensively studied over the past three decades codes additional complex physical structures, such as sym- (Lampinen & Vehtari, 2001; Titterington, 2004; Graves, metry and other invariant structures.