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

• URL: http://arxiv.org/abs/2601.10715v1
• Date: Thu, 15 Jan 2026 18:59:57 GMT
• Title: DInf-Grid: A Neural Differential Equation Solver with Differentiable Feature Grids
• Authors: Navami Kairanda, Shanthika Naik, Marc Habermann, Avinash Sharma, Christian Theobalt, Vladislav Golyanik,
• Abstract summary: We present a differentiable grid-based representation for efficiently solving differential equations (DEs)<n>Our results demonstrate a 5-20x speed-up over coordinate-based methods, solving differential equations in seconds or minutes while maintaining comparable accuracy and compactness.
• Score: 73.28614344779076
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

Related papers

• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• Enabling Automatic Differentiation with Mollified Graph Neural Operators [73.52999622724101]
  We propose the mollified graph neural operator ($m$GNO), the first method to leverage automatic differentiation and compute exact gradients on arbitrary geometries.<n>For a PDE example on regular grids, $m$GNO paired with autograd reduced the L2 relative data error by 20x compared to finite differences.<n>It can also solve PDEs on unstructured point clouds seamlessly, using physics losses only, at resolutions vastly lower than those needed for finite differences to be accurate enough.
  arXiv  Detail & Related papers  (2025-04-11T06:16:30Z)
• Learning Subgrid-scale Models with Neural Ordinary Differential Equations [0.39160947065896795]
  We propose a new approach to learning the subgrid-scale model when simulating partial differential equations (PDEs) In this approach neural networks are used to learn the coarse- to fine-grid map, which can be viewed as subgrid-scale parameterization. Our method inherits the advantages of NODEs and can be used to parameterize subgrid scales, approximate coupling operators, and improve the efficiency of low-order solvers.
  arXiv  Detail & Related papers  (2022-12-20T02:45:09Z)
• LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data [47.49194807524502]
  We propose LordNet, a tunable and efficient neural network for modeling entanglements. The experiments on solving Poisson's equation and (2D and 3D) Navier-Stokes equation demonstrate that the long-range entanglements can be well modeled by the LordNet.
  arXiv  Detail & Related papers  (2022-06-19T14:41:08Z)
• Physics-constrained Unsupervised Learning of Partial Differential Equations using Meshes [1.066048003460524]
  Graph neural networks show promise in accurately representing irregularly meshed objects and learning their dynamics. In this work, we represent meshes naturally as graphs, process these using Graph Networks, and formulate our physics-based loss to provide an unsupervised learning framework for partial differential equations (PDE) Our framework will enable the application of PDE solvers in interactive settings, such as model-based control of soft-body deformations.
  arXiv  Detail & Related papers  (2022-03-30T19:22:56Z)
• Message Passing Neural PDE Solvers [60.77761603258397]
  We build a neural message passing solver, replacing allally designed components in the graph with backprop-optimized neural function approximators. We show that neural message passing solvers representationally contain some classical methods, such as finite differences, finite volumes, and WENO schemes. We validate our method on various fluid-like flow problems, demonstrating fast, stable, and accurate performance across different domain topologies, equation parameters, discretizations, etc., in 1D and 2D.
  arXiv  Detail & Related papers  (2022-02-07T17:47:46Z)
• Unsupervised Reservoir Computing for Solving Ordinary Differential Equations [1.6371837018687636]
  unsupervised reservoir computing (RC), an echo-state recurrent neural network capable of discovering approximate solutions that satisfy ordinary differential equations (ODEs) We use Bayesian optimization to efficiently discover optimal sets in a high-dimensional hyper- parameter space and numerically show that one set is robust and can be used to solve an ODE for different initial conditions and time ranges.
  arXiv  Detail & Related papers  (2021-08-25T18:16:42Z)
• Learning optimal multigrid smoothers via neural networks [1.9336815376402723]
  We propose an efficient framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs) CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories. Numerical results on anisotropic rotated Laplacian problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.
  arXiv  Detail & Related papers  (2021-02-24T05:02:54Z)
• DiffPD: Differentiable Projective Dynamics with Contact [65.88720481593118]
  We present DiffPD, an efficient differentiable soft-body simulator with implicit time integration. We evaluate the performance of DiffPD and observe a speedup of 4-19 times compared to the standard Newton's method in various applications.
  arXiv  Detail & Related papers  (2021-01-15T00:13:33Z)
• Multipole Graph Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  One of the main challenges in using deep learning-based methods for simulating physical systems is formulating physics-based data. We propose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time.
  arXiv  Detail & Related papers  (2020-06-16T21:56:22Z)
• FiniteNet: A Fully Convolutional LSTM Network Architecture for Time-Dependent Partial Differential Equations [0.0]
  We use a fully convolutional LSTM network to exploit the dynamics of PDEs. We show that our network can reduce error by a factor of 2 to 3 compared to the baseline algorithms.
  arXiv  Detail & Related papers  (2020-02-07T21:18:46Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.