  Physics-informed Neural Networks (PINNs) have recently achieved remarkable
progress in solving Partial Differential Equations (PDEs) in various fields by
minimizing a weighted sum of PDE loss and boundary loss. However, there are
several critical challenges in the training of PINNs, including the lack of
theoretical frameworks and the imbalance between PDE loss and boundary loss. In
this paper, we present an analysis of second-order non-homogeneous PDEs, which
are classified into three categories and applicable to various common problems.
We also characterize the connections between the training loss and actual
error, guaranteeing convergence under mild conditions. The theoretical analysis
inspires us to further propose MultiAdam, a scale-invariant optimizer that
leverages gradient momentum to parameter-wisely balance the loss terms.
Extensive experiment results on multiple problems from different physical
domains demonstrate that our MultiAdam solver can improve the predictive
accuracy by 1-2 orders of magnitude compared with strong baselines.
