PILIR: Physics-Informed Local Implicit Representation

TL;DR

PILIR introduces a local implicit neural representation that enhances physics-informed neural networks by capturing high-frequency details more efficiently, leading to faster convergence and improved accuracy in solving PDEs.

Contribution

The paper proposes a novel local implicit representation that separates global and local features, overcoming spectral bias in PINNs for better high-frequency detail learning.

Findings

PILIR effectively mitigates spectral bias in PINNs.

It achieves faster convergence for high-frequency components.

PILIR outperforms state-of-the-art methods in accuracy.

Abstract

Physics-Informed Neural Networks have become a powerful mesh-free method for solving partial differential equations, but their performance is often limited by spectral bias. Specifically, in standard MLPs used in PINNs, the global parameter coupling causes the model to prioritize learning low-frequency components, resulting in slow convergence for high-frequency details. To overcome this limitation, we introduce the Physics-Informed Local Implicit Representation (PILIR). Our approach separates the global physical domain into a discrete latent feature space and a continuous generative decoder. By using a learnable grid to encode explicit spatial locality, PILIR can capture high-frequency details locally, preventing dilution by global patterns. A generative neural operator then synthesizes these local latent features into continuous physical fields, allowing accurate reconstruction of fine-scale structures. Experiments on a range of challenging PDEs show that PILIR effectively mitigates spectral bias, thereby boosting the convergence of high-frequency details and achieving superior accuracy compared to state-of-the-art methods.