Implicit Neural Representations with Periodic Activation Functions

TL;DR

This paper introduces Sinusoidal Representation Networks (Sirens), which use periodic activation functions to effectively model complex signals and their derivatives, enabling solutions to PDEs and learning priors.

Contribution

The paper proposes Sirens, a novel neural network architecture with periodic activations, for better implicit signal representation and PDE solving capabilities.

Findings

Successfully models images, wavefields, video, and sound with derivatives.

Enables solving boundary value problems like Eikonal, Poisson, Helmholtz, and wave equations.

Demonstrates learning priors over Siren functions using hypernetworks.

Abstract

Implicitly defined, continuous, differentiable signal representations parameterized by neural networks have emerged as a powerful paradigm, offering many possible benefits over conventional representations. However, current network architectures for such implicit neural representations are incapable of modeling signals with fine detail, and fail to represent a signal's spatial and temporal derivatives, despite the fact that these are essential to many physical signals defined implicitly as the solution to partial differential equations. We propose to leverage periodic activation functions for implicit neural representations and demonstrate that these networks, dubbed sinusoidal representation networks or Sirens, are ideally suited for representing complex natural signals and their derivatives. We analyze Siren activation statistics to propose a principled initialization scheme and demonstrate the representation of images, wavefields, video, sound, and their derivatives. Further, we show how Sirens can be leveraged to solve challenging boundary value problems, such as particular Eikonal equations (yielding signed distance functions), the Poisson equation, and the Helmholtz and wave equations. Lastly, we combine Sirens with hypernetworks to learn priors over the space of Siren functions.