Introduction
Neural networks are increasingly used to solve complex differential equations, but standard architectures are notoriously slow to train. In physics-informed machine learning, neural networks are trained to solve Partial Differential Equations (PDEs)—like those governing heat diffusion, wave propagation, or cloth deformation. The standard approach uses coordinate-based Multi-Layer Perceptrons (MLPs) like Siren. While accurate and infinitely differentiable, MLPs are computationally heavy because every weight in the network must be updated for every spatial coordinate queried.
Recently, the AI graphics community solved a similar speed issue using feature grids (like Instant-NGP or K-Planes), which localize learning and train in seconds. However, these grid methods rely on linear interpolation. Linear interpolation is mathematically blocky: its first derivative is discontinuous, and its second derivative is zero everywhere. Because many physical laws (like the Poisson or Helmholtz equations) rely on second-order derivatives (Laplacians), standard fast grids completely fail when trying to solve these PDEs.
To bridge this gap, this paper introduces $\boldsymbol{\partial^\infty}$-Grid, a novel architecture that combines the blazing speed of localized feature grids with the infinite differentiability of Radial Basis Functions (RBFs).
The $\boldsymbol{\partial^\infty}$-Grid Architecture
The authors set out to build a representation that is both as fast as a feature grid and as smooth and differentiable as an MLP. They achieve this through three core innovations:
- RBF Interpolation: Instead of using standard linear interpolation to guess values between grid points, the authors use Gaussian Radial Basis Functions (RBFs). Because Gaussians are smooth and infinitely differentiable ($C^\infty$), the network can easily compute the higher-order derivatives (Jacobians, Laplacians) required by PDE loss functions using standard automatic differentiation.
- Efficient Local Neighborhoods: Standard RBFs calculate weights across the entire domain, which is computationally ruinous. The authors optimize this by precomputing overlapping local "rings" (neighborhoods) based on the RBF's shape parameter, ensuring smooth interpolation without the global math overhead.
- Multi-Resolution Grids: The authors stack multiple grids at different scales. In standard graphics tasks, this captures fine details. Here, it serves a second, crucial purpose: it acts as a highway for global gradient propagation, allowing the PDE solver to converge stably across the entire physical domain rather than getting stuck in local minima.
Unlike competing RBF-based grid methods (like NeuRBF) that use adaptive, off-grid centers creating microscopic discontinuities, $\boldsymbol{\partial^\infty}$-Grid's fixed, overlapping grid structure is explicitly engineered to survive continuous PDE supervision.
Experimental Validation and Results
Because this paper focuses on solving PDEs, it evaluates the model against several physical boundary value problems rather than traditional static datasets. These include the Poisson equation for image reconstruction, the Helmholtz equation for wavefields, and Kirchhoff-Love thin shell theory for cloth simulation.
The results demonstrate massive efficiency gains. $\boldsymbol{\partial^\infty}$-Grid achieves a 5–20× speedup over Siren (the standard MLP baseline) across various tasks:
- Image Reconstruction (Poisson): Recovering an image from its gradients took Siren 10 minutes; $\boldsymbol{\partial^\infty}$-Grid did it in just 25 seconds with a higher PSNR (32.24 vs 32.12).
- Wave Simulation (Helmholtz): The model solved complex steady-state wavefields from a point source in ~3.5 minutes, compared to Siren's ~21.5 minutes, while perfectly matching the closed-form analytical solution.
- Cloth Simulation: $\boldsymbol{\partial^\infty}$-Grid successfully modeled highly non-linear membrane and bending strains to simulate realistic cloth wrinkles. Standard grid methods (Instant-NGP, K-Planes) completely failed to converge on this task due to their inability to compute higher-order derivatives.
Limitations and Future Work
While highly effective for 2D and lower-dimensional PDEs, the method faces the "Curse of Dimensionality." RBF interpolation requires querying $6^d$ neighbors per scale. While fast in 2D, it becomes computationally expensive in 3D (like Eikonal SDF tasks) and beyond. The authors suggest exploring planar projections (like HexPlane) to mitigate this in the future.
Additionally, the model's accuracy is sensitive to the RBF shape parameter ($\epsilon$); if tuned poorly, it can over-smooth the physics. The method is also susceptible to Runge's phenomenon (oscillations near domain boundaries). Finally, the current implementation relies on standard PyTorch; developing a custom CUDA kernel could push the speedups even further.