Smooth-NGP: $C^{\infty }$-smooth Differentiable Hash Grids

Grid-based implicit methods like Instant-NGP greatly speed up neural field training, but their default multilinear interpolation yields only piecewise-smooth functions, which can destabilize higher-order derivatives and create grid-aligned artifacts. These issues limit performance in derivative-sensitive tasks, such as precise normal computation and PDE solving. We present *Smooth-NGP*, a $C^{\infty }$-smooth variant of Instant-NGP that swaps multilinear interpolation for an infinitely differentiable kernel and extends the hash grid with a sliding-window design. The smooth kernel provides stable closed-form derivatives of any order, supporting derivative-driven losses while retaining the efficiency of grid encodings. However, because the kernel is flat at cell boundaries, its derivatives go to zero there, reducing gradient flow exactly where grid artifacts tend to appear. The sliding-window mechanism resolves this by ensuring each query point falls within the interior of several shifted grid windows, preserving useful gradients across the entire domain. On image fitting, signed distance field learning, and PINN-based PDE solving, Smooth-NGP reliably improves reconstruction quality, derivative fidelity, training stability, and surpasses leading grid-based baselines. We will release the source code and experimental scripts upon publication.