Many contemporary studies utilize grid-based models for neural field
representation, but a systematic analysis of grid-based models is still
missing, hindering the improvement of those models. Therefore, this paper
introduces a theoretical framework for grid-based models. This framework points
out that these models' approximation and generalization behaviors are
determined by grid tangent kernels (GTK), which are intrinsic properties of
grid-based models. The proposed framework facilitates a consistent and
systematic analysis of diverse grid-based models. Furthermore, the introduced
framework motivates the development of a novel grid-based model named the
Multiplicative Fourier Adaptive Grid (MulFAGrid). The numerical analysis
demonstrates that MulFAGrid exhibits a lower generalization bound than its
predecessors, indicating its robust generalization performance. Empirical
studies reveal that MulFAGrid achieves state-of-the-art performance in various
tasks, including 2D image fitting, 3D signed distance field (SDF)
reconstruction, and novel view synthesis, demonstrating superior representation
ability. The project website is available at
this https URL

该论文介绍了一种基于网格的模型的理论框架，并提出了一种名为 MulFAGrid 的新型基于网格的模型，该模型在各种任务中展示了卓越的表征能力和较低的泛化性能。

基于网格的神经场模型的归纳增强

Grounding and Enhancing Grid-based Models for Neural Fields

Predicting the evolution of spatiotemporal physical systems from sparse and
scattered observational data poses a significant challenge in various
scientific domains. Traditional methods rely on dense grid-structured data,
limiting their applicability in scenarios with sparse observations. To address
this challenge, we introduce GrINd (Grid Interpolation Network for Scattered
Observations), a novel network architecture that leverages the high-performance
of grid-based models by mapping scattered observations onto a high-resolution
grid using a Fourier Interpolation Layer. In the high-resolution space, a
NeuralPDE-class model predicts the system's state at future timepoints using
differentiable ODE solvers and fully convolutional neural networks
parametrizing the system's dynamics. We empirically evaluate GrINd on the
DynaBench benchmark dataset, comprising six different physical systems observed
at scattered locations, demonstrating its state-of-the-art performance compared
to existing models. GrINd offers a promising approach for forecasting physical
systems from sparse, scattered observational data, extending the applicability
of deep learning methods to real-world scenarios with limited data
availability.

通过将稀疏的观测数据映射到高分辨率的网格空间，并利用可微分的 ODE 求解器和完全卷积神经网络预测未来的时间点上系统的状态，GrINd 为从稀疏、分散的观测数据中预测物理系统的演变提供了一种有前景的方法。

GrINd: 网格内插网络用于离散观测

GrINd: Grid Interpolation Network for Scattered Observations

Neural Radiance Fields (NeRFs) can be dramatically accelerated by spatial
grid representations. However, they do not explicitly reason about scale and so
introduce aliasing artifacts when reconstructing scenes captured at different
camera distances. Mip-NeRF and its extensions propose scale-aware renderers
that project volumetric frustums rather than point samples but such approaches
rely on positional encodings that are not readily compatible with grid methods.
We propose a simple modification to grid-based models by training model heads
at different spatial grid resolutions. At render time, we simply use coarser
grids to render samples that cover larger volumes. Our method can be easily
applied to existing accelerated NeRF methods and significantly improves
rendering quality (reducing error rates by 20-90% across synthetic and
unbounded real-world scenes) while incurring minimal performance overhead (as
each model head is quick to evaluate). Compared to Mip-NeRF, we reduce error
rates by 20% while training over 60x faster.

通过在不同空间网格分辨率上训练模型头部，我们对基于网格的模型进行简单修改，以实现容易的扩展，提高渲染质量和性能效率。