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

2026-01-15

AI Summary

This work addresses the high computational cost and slow training of conventional coordinate-based neural networks for solving differential equations, as well as the limited accuracy of existing grid-based methods in computing high-order derivatives due to their reliance on linear interpolation. The authors propose a novel representation that integrates differentiable feature grids with infinitely differentiable radial basis function (RBF) interpolation—introducing RBFs into feature grids for the first time to enable accurate high-order derivative computation. A multi-resolution collocated grid structure is designed to efficiently capture high-frequency details while preserving global gradient stability. The model is trained implicitly using the governing differential equation as the loss function. On Poisson, Helmholtz, and Kirchhoff–Love thin plate problems, it achieves 5–20× speedups over MLP-based approaches (reducing solve times from minutes to seconds) while maintaining comparable accuracy and a compact model size.

Technology Category

• Machine Learning: Representation Learning
• Computer Vision: Representation Learning for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• differential equations
• neural solvers
• feature grids
• high-order derivatives
• interpolation

Innovation

• differentiable feature grids
• radial basis function interpolation
• neural differential equation solver
• multi-resolution decomposition
• implicit representation