Neural Cellular Automata (NCA) is a class of Cellular Automata where the update rule is parameterized by a neural network that can be trained using gradient descent. In this paper, we focus on NCA models used for texture synthesis, where the update rule is inspired by partial differential equations (PDEs) describing reaction-diffusion systems. To train the NCA model, the spatio-termporal domain is discretized, and Euler integration is used to numerically simulate the PDE. However, whether a trained NCA truly learns the continuous dynamic described by the corresponding PDE or merely overfits the discretization used in training remains an open question. We study NCA models at the limit where space-time discretization approaches continuity. We find that existing NCA models tend to overfit the training discretization, especially in the proximity of the initial condition, also called "seed". To address this, we propose a solution that utilizes uniform noise as the initial condition. We demonstrate the effectiveness of our approach in preserving the consistency of NCA dynamics across a wide range of spatio-temporal granularities. Our improved NCA model enables two new test-time interactions by allowing continuous control over the speed of pattern formation and the scale of the synthesized patterns. We demonstrate this new NCA feature in our interactive online demo. Our work reveals that NCA models can learn continuous dynamics and opens new venues for NCA research from a dynamical systems' perspective.

中文翻译：

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NoiseNCA：噪声种子改善神经细胞自动机的时空连续性

神经元胞自动机 (NCA) 是一类元胞自动机，其中更新规则由可以使用梯度下降进行训练的神经网络参数化。在本文中，我们重点关注用于纹理合成的 NCA 模型，其中更新规则受到描述反应扩散系统的偏微分方程 (PDE) 的启发。为了训练NCA模型，时空域被离散化，并使用欧拉积分对偏微分方程进行数值模拟。然而，经过训练的 NCA 是否真正学习了相应 PDE 描述的连续动态，还是仅仅过度拟合训练中使用的离散化仍然是一个悬而未决的问题。我们在时空离散化接近连续性的极限下研究 NCA 模型。我们发现现有的 NCA 模型往往会过度拟合训练离散化，特别是在初始条件（也称为“种子”）附近。为了解决这个问题，我们提出了一种利用均匀噪声作为初始条件的解决方案。我们证明了我们的方法在广泛的时空粒度范围内保持 NCA 动态一致性的有效性。我们改进的 NCA 模型通过允许连续控制模式形成的速度和合成模式的规模，实现了两种新的测试时间交互。我们在交互式在线演示中演示了这一新的 NCA 功能。我们的工作表明，NCA 模型可以学习连续动力学，并从动力系统的角度为 NCA 研究开辟了新的场所。