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Physics-informed Neural Networks (PINNs) have recently achieved remarkableprogress in solving Partial Differential Equations (PDEs) in various fields byminimizing a weighted sum of PDE loss and boundary loss. However, there areseveral critical challenges in the training of PINNs, including the lack oftheoretical frameworks and the imbalance between PDE loss and boundary loss. Inthis paper, we present an analysis of second-order non-homogeneous PDEs, whichare classified into three categories and applicable to various common problems.We also characterize the connections between the training loss and actualerror, guaranteeing convergence under mild conditions. The theoretical analysisinspires us to further propose MultiAdam, a scale-invariant optimizer thatleverages gradient momentum to parameter-wisely balance the loss terms.Extensive experiment results on multiple problems from different physicaldomains demonstrate that our MultiAdam solver can improve the predictiveaccuracy by 1-2 orders of magnitude compared with strong baselines.

MultiAdam: Parameter-wise Scale-invariant Optimizer for Multiscale Training of Physics-informed Neural Networks