PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh
  Representations
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/
675a80635f94292180d2f7b9
Namgyu Kang
claimed_verified
2024-12-12T08:19:37.357Z
675a7d0ecb8625d2f4770e66
/avatars/a5a5c2cb90d9eca3b04b8dc949135c0f.svg
Namgyu Kang
user
kangnamgyu27
675a80635f94292180d2f7ba
Jaemin Oh
675a80635f94292180d2f7bb
Youngjoon Hong
675a80635f94292180d2f7bc
Eunbyung Park
PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh
  Representations
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/
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.
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