Good Lattice Training: Physics-Informed Neural Networks Accelerated by
  Number Theory
Takashi Matsubara Takaharu Yaguchi
Physics-informed neural networks (PINNs) offer a novel and efficient approach
to solving partial differential equations (PDEs). Their success lies in the
physics-informed loss, which trains a neural network to satisfy a given PDE at
specific points and to approximate the solution. However, the solutions to PDEs
are inherently infinite-dimensional, and the distance between the output and
the solution is defined by an integral over the domain. Therefore, the
physics-informed loss only provides a finite approximation, and selecting
appropriate collocation points becomes crucial to suppress the discretization
errors, although this aspect has often been overlooked. In this paper, we
propose a new technique called good lattice training (GLT) for PINNs, inspired
by number theoretic methods for numerical analysis. GLT offers a set of
collocation points that are effective even with a small number of points and
for multi-dimensional spaces. Our experiments demonstrate that GLT requires
2--20 times fewer collocation points (resulting in lower computational cost)
than uniformly random sampling or Latin hypercube sampling, while achieving
competitive performance.