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.

翻译：我们提出了一种新颖的基于可微分网格的表示方法，用于高效求解微分方程。目前广泛用于神经求解器的架构，如正弦神经网络，是基于坐标的多层感知机，其计算密集且训练缓慢。尽管基于网格的隐式表示替代方案（例如Instant-NGP和K-Planes）通过利用信号结构实现了更快的训练，但它们对线性插值的依赖限制了其计算高阶导数的能力，使其不适合求解微分方程。我们的方法通过将特征网格的效率与无限可微的径向基函数插值相结合，克服了这些限制。为了有效捕捉高频解并实现全局梯度稳定且更快的计算，我们引入了具有共置网格的多分辨率分解。我们提出的表示方法DInf-Grid，使用微分方程作为损失函数进行隐式训练，从而能够精确建模物理场。我们在多种任务上验证了DInf-Grid的有效性，包括用于图像重建的泊松方程、用于波场的亥姆霍兹方程以及用于布料模拟的基尔霍夫-洛夫边值问题。我们的结果表明，与基于坐标的MLP方法相比，DInf-Grid实现了5到20倍的加速，能够在数秒或数分钟内求解微分方程，同时保持相当的精度和紧凑性。