  Physics-informed neural networks (PINNs) are a new tool for solving boundary
value problems by defining loss functions of neural networks based on governing
equations, boundary conditions, and initial conditions. Recent investigations
have shown that when designing loss functions for many engineering problems,
using first-order derivatives and combining equations from both strong and weak
forms can lead to much better accuracy, especially when there are heterogeneity
and variable jumps in the domain. This new approach is called the mixed
formulation for PINNs, which takes ideas from the mixed finite element method.
In this method, the PDE is reformulated as a system of equations where the
primary unknowns are the fluxes or gradients of the solution, and the secondary
unknowns are the solution itself. In this work, we propose applying the mixed
formulation to solve multi-physical problems, specifically a stationary
thermo-mechanically coupled system of equations. Additionally, we discuss both
sequential and fully coupled unsupervised training and compare their accuracy
and computational cost. To improve the accuracy of the network, we incorporate
hard boundary constraints to ensure valid predictions. We then investigate how
different optimizers and architectures affect accuracy and efficiency. Finally,
we introduce a simple approach for parametric learning that is similar to
transfer learning. This approach combines data and physics to address the
limitations of PINNs regarding computational cost and improves the network's
ability to predict the response of the system for unseen cases. The outcomes of
this work will be useful for many other engineering applications where deep
learning is employed on multiple coupled systems of equations for fast and
reliable computations.
