When: 10AM on December 2, 2022
Where: Joyce Cummings Center, Room 260
We propose a physics-constrained neural network (NN) approach to solve partial differential equations (PDEs) without labels. We express the loss function of these NNs in terms of the residual of PDEs obtained through an efficient, discrete, convolution operator-based, and vectorized implementation. We explore an encoder-decoder N 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. Both Dirichlet and Neumann BC are specified as inputs to NNs, and we explore whether a single NN can solve for similar physics; i.e., the same PDE, but with different BCs and on a number of problem domains. The trained BNN PDE solvers demonstrate a degree of success at extrapolated predictions for BC that they were not exposed to during training. We demonstrate the capacity and performance of the proposed framework by applying it to different steady-state and equilibrium boundary value problems with physics that spans diffusion, linear and nonlinear elasticity. Such NN solution frameworks assume particular importance in problems where high-throughput solutions of PDEs with different boundary conditions and on varying domains are desired in support of design and decision-making.