title: PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh
    Representations
summary: The approximation of Partial Differential Equations (PDEs) using neural
  networks has seen significant advancements through Physics-Informed Neural
  Networks (PINNs). Despite their straightforward optimization framework and
  flexibility in implementing various PDEs, PINNs often suffer from limited
  accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which
  struggle to effectively learn high-frequency and non-linear components.
  Recently, parametric mesh representations in combination with neural networks
  have been investigated as a promising approach to eliminate the inductive
  biases of neural networks. However, they usually require very high-resolution
  grids and a large number of collocation points to achieve high accuracy while
  avoiding overfitting issues. In addition, the fixed positions of the mesh
  parameters restrict their flexibility, making it challenging to accurately
  approximate complex PDEs. To overcome these limitations, we propose
  Physics-Informed Gaussians (PIGs), which combine feature embeddings using
  Gaussian functions with a lightweight neural network. Our approach uses
  trainable parameters for the mean and variance of each Gaussian, allowing for
  dynamic adjustment of their positions and shapes during training. This
  adaptability enables our model to optimally approximate PDE solutions, unlike
  models with fixed parameter positions. Furthermore, the proposed approach
  maintains the same optimization framework used in PINNs, allowing us to benefit
  from their excellent properties. Experimental results show the competitive
  performance of our model across various PDEs, demonstrating its potential as a
  robust tool for solving complex PDEs. Our project page is available at
  https://namgyukang.github.io/Physics-Informed-Gaussians/
paper_authors._id: 675a80635f94292180d2f7b9
paper_authors.name: Namgyu Kang
paper_authors.status: claimed_verified
paper_authors.statusLastChangedAt: 2024-12-12T08:19:37.357Z
paper_authors._id: 675a7d0ecb8625d2f4770e66
paper_authors.avatarUrl: /avatars/a5a5c2cb90d9eca3b04b8dc949135c0f.svg
paper_authors.fullname: Namgyu Kang
paper_authors.type: user
paper_authors.user: kangnamgyu27
paper_authors._id: 675a80635f94292180d2f7ba
paper_authors.name: Jaemin Oh
paper_authors._id: 675a80635f94292180d2f7bb
paper_authors.name: Youngjoon Hong
paper_authors._id: 675a80635f94292180d2f7bc
paper_authors.name: Eunbyung Park
paper_title: PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh
    Representations
paper_summary: The approximation of Partial Differential Equations (PDEs) using neural
  networks has seen significant advancements through Physics-Informed Neural
  Networks (PINNs). Despite their straightforward optimization framework and
  flexibility in implementing various PDEs, PINNs often suffer from limited
  accuracy due to the spectral bias of Multi-Layer Perceptrons (MLPs), which
  struggle to effectively learn high-frequency and non-linear components.
  Recently, parametric mesh representations in combination with neural networks
  have been investigated as a promising approach to eliminate the inductive
  biases of neural networks. However, they usually require very high-resolution
  grids and a large number of collocation points to achieve high accuracy while
  avoiding overfitting issues. In addition, the fixed positions of the mesh
  parameters restrict their flexibility, making it challenging to accurately
  approximate complex PDEs. To overcome these limitations, we propose
  Physics-Informed Gaussians (PIGs), which combine feature embeddings using
  Gaussian functions with a lightweight neural network. Our approach uses
  trainable parameters for the mean and variance of each Gaussian, allowing for
  dynamic adjustment of their positions and shapes during training. This
  adaptability enables our model to optimally approximate PDE solutions, unlike
  models with fixed parameter positions. Furthermore, the proposed approach
  maintains the same optimization framework used in PINNs, allowing us to benefit
  from their excellent properties. Experimental results show the competitive
  performance of our model across various PDEs, demonstrating its potential as a
  robust tool for solving complex PDEs. Our project page is available at
  https://namgyukang.github.io/Physics-Informed-Gaussians/
paper_ai_summary: Physics-Informed Gaussians combine Gaussian feature embeddings with lightweight neural networks to mitigate spectral bias and offer flexible, high-accuracy solutions for Partial Differential Equations.
paper_ai_keywords: Physics-Informed Neural Networks PINNs Multi-Layer Perceptrons MLPs parametric mesh representations neural networks inductive biases Physics-Informed Gaussians PIGs Gaussian functions trainable parameters mean variance high-frequency non-linear components collocation points overfitting dynamic adjustment trainable parameters optimization framework Partial Differential Equations PDEs