  Physics-informed neural networks (PINNs) have attracted a lot of attention in
scientific computing as their functional representation of partial differential
equation (PDE) solutions offers flexibility and accuracy features. However,
their training cost has limited their practical use as a real alternative to
classic numerical methods. Thus, we propose to incorporate multi-resolution
hash encoding into PINNs to improve the training efficiency, as such encoding
offers a locally-aware (at multi resolution) coordinate inputs to the neural
network. Borrowed from the neural representation field community (NeRF), we
investigate the robustness of calculating the derivatives of such hash encoded
neural networks with respect to the input coordinates, which is often needed by
the PINN loss terms. We propose to replace the automatic differentiation with
finite-difference calculations of the derivatives to address the discontinuous
nature of such derivatives. We also share the appropriate ranges for the hash
encoding hyperparameters to obtain robust derivatives. We test the proposed
method on three problems, including Burgers equation, Helmholtz equation, and
Navier-Stokes equation. The proposed method admits about a 10-fold improvement
in efficiency over the vanilla PINN implementation.
