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

TL;DR

DInf-Grid introduces a differentiable, grid-based neural representation using radial basis functions for efficient and accurate differential equation solving, significantly outperforming traditional coordinate-based neural methods in speed while maintaining accuracy.

Contribution

The paper proposes DInf-Grid, a novel neural differential equation solver combining feature grids with RBF interpolation, enabling faster training and higher-order derivative computation.

Findings

Achieves 5-20x speed-up over MLP-based methods.

Successfully solves Poisson, Helmholtz, and Kirchhoff-Love equations.

Maintains accuracy and compactness in solutions.

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.