  Solving partial differential equations (PDEs) is the canonical approach for
understanding the behavior of physical systems. However, large scale solutions
of PDEs using state of the art discretization techniques remains an expensive
proposition. In this work, a new physics-constrained neural network (NN)
approach is proposed to solve PDEs without labels, with a view to enabling
high-throughput solutions in support of design and decision-making. Distinct
from existing physics-informed NN approaches, where the strong form or weak
form of PDEs are used to construct the loss function, we write the loss
function of NNs based on the discretized residual of PDEs through an efficient,
convolutional operator-based, and vectorized implementation. We explore an
encoder-decoder NN structure for both deterministic and probabilistic models,
with Bayesian NNs (BNNs) for the latter, which allow us to quantify both
epistemic uncertainty from model parameters and aleatoric uncertainty from
noise in the data. For BNNs, the discretized residual is used to construct the
likelihood function. In our approach, both deterministic and probabilistic
convolutional layers are used to learn the applied boundary conditions (BCs)
and to detect the problem domain. As both Dirichlet and Neumann BCs are
specified as inputs to NNs, a single NN can solve for similar physics, but with
different BCs and on a number of problem domains. The trained surrogate PDE
solvers can also make interpolating and extrapolating (to a certain extent)
predictions for BCs that they were not exposed to during training. Such
surrogate models are of particular importance for problems, where similar types
of PDEs need to be repeatedly solved for many times with slight variations. We
demonstrate the capability and performance of the proposed framework by
applying it to steady-state diffusion, linear elasticity, and nonlinear
elasticity.
