DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids

TL;DR

The paper addresses the challenge of solving differential equations with neural solvers by proposing a differentiable grid-based representation, the $\boldsymbol{\partial^\infty}$-Grid, which combines feature grids with infinitely differentiable Gaussian radial basis function interpolation and a multi-resolution, co-located grid design. The approach enables higher-order derivatives and stable global gradient flow, allowing PDE residuals to supervise training directly. It demonstrates competitive accuracy while delivering 5-20x speedups over coordinate-based MLP solvers on Poisson image reconstruction, Helmholtz wavefields, and Kirchhoff-Love cloth simulations. This method offers practical impact for fast, differentiable PDE solving and high-frequency field reconstruction in graphics and physics-informed learning.

Abstract

We present a novel differentiable grid-based representation for efficiently solving differential equations (DEs). Widely used architectures for neural solvers, such as sinusoidal neural networks, are coordinate-based MLPs that are both computationally intensive and slow to train. Although grid-based alternatives for implicit representations (e.g., Instant-NGP and K-Planes) train faster by exploiting signal structure, their reliance on linear interpolation restricts their ability to compute higher-order derivatives, rendering them unsuitable for solving DEs. Our approach overcomes these limitations by combining the efficiency of feature grids with radial basis function interpolation, which is infinitely differentiable. To effectively capture high-frequency solutions and enable stable and faster computation of global gradients, we introduce a multi-resolution decomposition with co-located grids. Our proposed representation, DInf-Grid, is trained implicitly using the differential equations as loss functions, enabling accurate modelling of physical fields. We validate DInf-Grid on a variety of tasks, including the Poisson equation for image reconstruction, the Helmholtz equation for wave fields, and the Kirchhoff-Love boundary value problem for cloth simulation. Our results demonstrate a 5-20x speed-up over coordinate-based MLP-based methods, solving differential equations in seconds or minutes while maintaining comparable accuracy and compactness.

Figures (16)

• Figure 1: Our proposed method, $\boldsymbol{\partial^\infty}$-Grid, is a feature grid-based representation capable of accurately modelling signals and their higher-order derivatives. We demonstrate its effectiveness in solving complex differential equations, including the Helmholtz equation (top) and neural cloth simulation (bottom) kairanda2023neuralclothsim (reference adapted from clyde2017modeling), achieving both high speed and accuracy.
• Figure 2: Illustration of $\boldsymbol{\partial^\infty}$-Grid. Our grid-based representation accurately represents fields $\mathbf{u}:\Omega \to \mathbb{R}^m$ and efficiently solves differential equations $\mathcal{F}$. We first sample query coordinates $\mathbf{x}$ from the input domain $\Omega$ (here, 2D). We apply infinitely differentiable RBF interpolation $\varphi ( \mathbf{x})$ to extract smooth, query-dependent features from the multi-scale learnable feature grids $\{\mathbf{F}_s\}_{s\in[0,\dots,S]}$. These interpolated features $\{\mathbf{f}_s(\mathbf{x})\}_s$ at different scales are concatenated as $\mathbf{f}(\mathbf{x} )$ and then passed through a decoder $\mathbf{d}(\mathbf{f}(\mathbf{x} );\Theta)$ to produce the final signal $\mathbf{u} (\mathbf{x})$ (e.g., image or cloth deformation). The converged neural field (solution) is obtained by optimising the governing equation $\mathcal{F}$ as the loss.
• Figure 3: Illustration of effective neighbourhoods $\mathcal{N}_2(\mathbf{x})$ in $d=1, 2$, determined by the shape parameter $\varepsilon$. An additional ring accounts for stratified sampling.
• Figure 4: Image Reconstruction. Comparison of $\boldsymbol{\partial^\infty}$-Grid with Siren sitzmann2020implicit for the reconstruction from gradient and Laplacian fields (Poisson equation). Results are visualised at training time marked by "•" and "•", and the supervision signal is highlighted with a green border. Our method achieves higher PSNR with faster convergence for high-resolution images, while Siren struggles, especially for Laplacian.
• Figure 5: Helmholtz Equation. Comparison for solving the Helmholtz equation (second-order PDE) for a single point source at the centre. Prior hybrid architectures, such as K-Planes keil2023kplanes and Instant-NGP muller2022instant, break down as their linear components equate to zero for second-order derivatives. Our method matches reference with significantly faster training time compared to Siren, as shown in the $\ell_1$ error plot on the right and $\ell_2$-error visualisations in \ref{['fig:helmholtz_error']}.